the brou e-malle-rouquier conjecture for the exceptional ... · the brou e-malle-rouquier...
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![Page 1: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/1.jpg)
The Broue-Malle-Rouquier conjecture for the exceptionalgroups of rank 2
Ph.D thesis defense
supervised by Ivan Marin
Presented by Eirini Chavli
12 May 2016
![Page 2: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/2.jpg)
Introduction
Real reflection group
Generalized braid group Iwahori Hecke algebra
![Page 3: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/3.jpg)
Introduction
![Page 4: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/4.jpg)
Introduction
Complex reflection group
Complex braid group Generic Hecke algebra
![Page 5: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/5.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections
i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 6: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/6.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 7: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/7.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 8: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/8.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 9: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/9.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.
G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 10: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/10.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 11: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/11.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,
where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 12: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/12.jpg)
Complex reflection groups
Definition
A complex reflection group of rank n is a finite subgroup of GLn(C), whichis generated by pseudo-reflections i.e. elements of finite order whose vectorspace of fixed points is a hyperplane (dimCker(s − 1) = n − 1).
Examples:
Any finite Coxeter groupC = 〈s1, ..., sn | s2i = 1, si sjsi . . .︸ ︷︷ ︸
mij−factors
= sjsi sj . . .︸ ︷︷ ︸mij−factors
〉
G4 = 〈s1, s2 | s31 = s32 = 1, s1s2s1 = s2s1s2〉.G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉.
Coxeter-like presentation: 〈s1, . . . , sn | so(si )i = 1, {vi = wi}i∈I 〉,where I is a finite set of relations such that, for each i ∈ I , vi and wi arepositive words with the same length in elements s1, . . . , sn.
![Page 13: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/13.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.
By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 14: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/14.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.
Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 15: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/15.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 16: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/16.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 17: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/17.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 18: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/18.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 19: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/19.jpg)
Complex braid groups
Let W be a complex reflection group and let X be the set of all the pointsof Cn that are not fixed by a pseudo-reflection.By Steinberg’s theorem we have that the action of W on X is free.Therefore, it defines a Galois covering X → X/W , which gives rise to thefollowing exact sequence, for every x ∈ X .
1→ π1(X , x)→ π1(X/W , x)→W → 1
Definition (Broue-Malle-Rouquier 1998)
The complex braid group B associated to W is the fundamental groupπ1(X/W , x).
Examples
For the symmetric group Sn, B is the usual braid group on n strands.
For any finite Coxeter group B is a generalized braid group.
![Page 20: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/20.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).
We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
x
H
•
•0
![Page 21: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/21.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
x
H
•
•0
![Page 22: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/22.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
x
H
•
•0
![Page 23: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/23.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
x
H
•
•x0
•0
![Page 24: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/24.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
x
H
•H⊥
•x0
•0
![Page 25: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/25.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
s · x
x
H
•
•H⊥
••
x0
s · x0•0
![Page 26: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/26.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
s · x
x
H
•
•H⊥
••
x0
s · x0•0
![Page 27: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/27.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
s · x
x
H
•
•H⊥
••
x0
s · x0•0
The complex braid group B is generated by distinguished braidedreflections, whose image inside W are the distinguished pseudo-reflectionsthat generate W .
![Page 28: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/28.jpg)
Complex braid groups
Let s be a distinguished pseudo-reflection of order m and H =ker(s − 1).We get elements inside B = π1(X/W , x), called distinguished braidedreflections, as follows:
s · x
x
H
•
•H⊥
••
x0
s · x0•0
The complex braid group B is generated by distinguished braidedreflections, whose image inside W are the distinguished pseudo-reflectionsthat generate W .
![Page 29: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/29.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 30: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/30.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).
o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 31: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/31.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 32: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/32.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 33: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/33.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 34: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/34.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 35: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/35.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 36: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/36.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 37: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/37.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 38: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/38.jpg)
Generic Hecke algebras
Let (us,j)1≤j≤o(s) be a set of indeterminates, where
s runs along the distinguished pseudo-reflections of W (if s ∼ s ′ thenus,j = us′,j).o(s) denotes the order of s.
Let R := Z[u±1s,1 , ..., u±1s,o(s)].
Definition
The generic-Hecke algebra HW associated to W is the algebraRB/(σ − us,1) . . . (σ − us,o(s)), where σ runs among the distinguishedbraided pseudo-reflections associated to s.
Examples
G4 = 〈s1, s2 : s31 = s32 = 1, s1s2s1 = s2s1s2〉
HG4 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2,3∏
j=1
(σi − uj) = 0〉
G10 = 〈s1, s2 : s31 = s42 = 1, s1s2s1s2 = s2s1s2s1〉
HG10 = 〈σ1, σ2 : σ1σ2σ1σ2 = σ2σ1σ2σ1,3∏
j=1
(σ1 − uj) =4∏
j=1
(σ2 − vj) = 0〉
![Page 39: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/39.jpg)
The BMR freeness conjecture
Let W be a complex reflection group of rank n.
Conjecture (Broue-Malle-Rouquier 1998)
HW is a free R-module of rank |W |.
Proposition (Broue-Malle-Rouquier 1998)
If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.
Every complex reflection group is a direct product of so-called irreducibleones.
Conjecture (equivalent form)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
![Page 40: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/40.jpg)
The BMR freeness conjecture
Let W be a complex reflection group of rank n.
Conjecture (Broue-Malle-Rouquier 1998)
HW is a free R-module of rank |W |.
Proposition (Broue-Malle-Rouquier 1998)
If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.
Every complex reflection group is a direct product of so-called irreducibleones.
Conjecture (equivalent form)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
![Page 41: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/41.jpg)
The BMR freeness conjecture
Let W be a complex reflection group of rank n.
Conjecture (Broue-Malle-Rouquier 1998)
HW is a free R-module of rank |W |.
Proposition (Broue-Malle-Rouquier 1998)
If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.
Every complex reflection group is a direct product of so-called irreducibleones.
Conjecture (equivalent form)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
![Page 42: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/42.jpg)
The BMR freeness conjecture
Let W be a complex reflection group of rank n.
Conjecture (Broue-Malle-Rouquier 1998)
HW is a free R-module of rank |W |.
Proposition (Broue-Malle-Rouquier 1998)
If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.
Every complex reflection group is a direct product of so-called irreducibleones.
Conjecture (equivalent form)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
![Page 43: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/43.jpg)
The BMR freeness conjecture
Let W be a complex reflection group of rank n.
Conjecture (Broue-Malle-Rouquier 1998)
HW is a free R-module of rank |W |.
Proposition (Broue-Malle-Rouquier 1998)
If HW is spanned over R by |W | elements, then it is a free R-module ofrank |W |.
Every complex reflection group is a direct product of so-called irreducibleones.
Conjecture (equivalent form)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
![Page 44: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/44.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to
The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).
The finite set of 34 exceptions.
![Page 45: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/45.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to
The infinite family G (de, e, n) of groups of n × n monomialmatrices
such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).
The finite set of 34 exceptions.
![Page 46: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/46.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to
The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C)
andtheir product to µd(C).
The finite set of 34 exceptions.
![Page 47: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/47.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to
The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).
The finite set of 34 exceptions.
![Page 48: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/48.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belong either to
The infinite family G (de, e, n) of groups of n × n monomialmatrices such that their non-zero coefficients belong to µde(C) andtheir product to µd(C).
The finite set of 34 exceptions.
![Page 49: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/49.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
![Page 50: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/50.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
![Page 51: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/51.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
![Page 52: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/52.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
![Page 53: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/53.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
We have some recent results from Marin and Pfeiffer (2015).
![Page 54: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/54.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
The finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
We have some recent results from Marin and Pfeiffer (2015).
![Page 55: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/55.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
A finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
We have some recent results from Marin and Pfeiffer (2015).
![Page 56: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/56.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
A finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
We have some recent results from Marin and Pfeiffer (2015).
Marin has proved the case of G26 (2012).
![Page 57: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/57.jpg)
Classification of irreducible complex reflection groups
The finite irreducible complex reflection groups were classified by Shephardand Todd (1954). They belond either to
The infinite family G (de, e, n) of groups of n × n monomials suchthat their non-zero coefficients belond to µde(C) and their product toµd(C). X (Ariki, Ariki-Koike 1993)
A finite set of 34 exceptions.
We only have to deal with the 34 exceptional groups, nicknamed as
G4, ...G11,G12,G13, ...,G22, G23,G24,G25,G26,G27,G28,G29,G30,G31,G32,G33,G34,G35,G36,G37.
Among them there are 6 finite Coxeter groups.
We have some recent results from Marin and Pfeiffer (2015).
Marin has proved the case of G26 (2012).
![Page 58: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/58.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.
It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 59: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/59.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 60: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/60.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 61: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/61.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.
X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 62: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/62.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 63: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/63.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.
X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 64: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/64.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 65: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/65.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.
X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 66: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/66.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 67: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/67.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.
X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 68: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/68.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 69: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/69.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue - Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G4, G8, G16.
![Page 70: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/70.jpg)
A first approach: The finite quotients of Bn
Let Bn be the braid group on n strands.It admits a presentation with generators s1, ..., sn−1 and relations
si si+1si = si+1si si+1, for i = 1, ..., n − 1
si sj = sjsi , for |i − j | > 1
Theorem (Coxeter 1957)
The quotient W of Bn by the relations ski = 1 is a finite group if and onlyif 1
k + 1n >
12 .
n = 2. Then, W is a cyclic group.X
k = 2. Then, W = Sn.X
n = 5 and k = 3. Then W = G32.X (Marin 2011)
n = 4 and k = 3. Then W = G25.X (Broue- Malle 1994, Marin 2011)
n = 3 and k = 3, 4, 5. Then W = G41, G8, G16. X (C. 2013)
1The case of G4 has been proven by Broue - Malle 1994, Berceanu - Funar 1994,Marin 2011.
![Page 71: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/71.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.
Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 72: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/72.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 73: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/73.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 74: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/74.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 75: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/75.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).
Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 76: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/76.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 77: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/77.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn, (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosets W1\Gn/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw = sa1i1 sa2i2. . . sarir , where ai ∈ Z.
We define the element Tfw inside HGn to be the product σa1i1 σa2i2. . . σarir
inside Hk (Tf1 = 1).Let u1 be the subalgebra of HGn (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1
![Page 78: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/78.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn
(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1︸ ︷︷ ︸U
Since 1 ∈ U, it is enough to prove that σiU ⊂ U i.e. it is enough to provethat σ2U ⊂ U.
![Page 79: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/79.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn
(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1︸ ︷︷ ︸U
Since 1 ∈ U, it is enough to prove that σiU ⊂ U
i.e. it is enough to provethat σ2U ⊂ U.
![Page 80: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/80.jpg)
The cases of G4, G8, G16
Let W1 be the subgroup of Gn (n = 4, 8, 16) which is generated by s1.Let J denote a system of representatives of the double cosetsW1\Wk/W1.
Gn =⊔w∈J
W1 · w ·W1.
For every w ∈ J we fix a factorization fw of w into a product of thegenerators s1 and s2 of Gn and we define an element Tfw inside HGn
(Tf1 = 1).Let u1 be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σ1.
HGn
?=
∑w∈J
u1 · Tfw · u1︸ ︷︷ ︸U
Since 1 ∈ U, it is enough to prove that σiU ⊂ U i.e. it is enough to provethat σ2U ⊂ U.
![Page 81: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/81.jpg)
An application: A classification of small dimensionalirreducible representations of B3
There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:
If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .
Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .
Example: The irreducible representations of dimension 3
A =
λ3 0 0λ1λ3 + λ22 λ2 0
λ2 1 λ1
, B =
λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3
,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.
![Page 82: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/82.jpg)
An application: A classification of small dimensionalirreducible representations of B3
There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:
If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .
Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .
Example: The irreducible representations of dimension 3
A =
λ3 0 0λ1λ3 + λ22 λ2 0
λ2 1 λ1
, B =
λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3
,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.
![Page 83: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/83.jpg)
An application: A classification of small dimensionalirreducible representations of B3
There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:
If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .
Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .
Example: The irreducible representations of dimension 3
A =
λ3 0 0λ1λ3 + λ22 λ2 0
λ2 1 λ1
, B =
λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3
,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.
![Page 84: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/84.jpg)
An application: A classification of small dimensionalirreducible representations of B3
There is a classification from I. Tuba and H. Wenzl in 1999 of theirreducible representations of B3 = 〈σ1, σ2 : σ1σ2σ1 = σ2σ1σ2〉 ofdimension k at most 5:
If σ1 7→ A, σ2 7→ B is such a representation, we have that A and B can bechosen of order triangular form and that their coefficients are completelydetermined by the eigenvalues and by a choice of a kth root of detA thatwe call r .
Moreover, such irreducible representations exist if the eigenvalues do notannihilate some polynomials Pk in eigenvalues and r .
Example: The irreducible representations of dimension 3
A =
λ3 0 0λ1λ3 + λ22 λ2 0
λ2 1 λ1
, B =
λ1 −1 λ20 λ2 −λ1λ3 − λ220 0 λ3
,where (λ21 + λ2λ3)(λ22 + λ1λ3)(λ23 + λ1λ2) 6= 0.
![Page 85: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/85.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Questions
What is the reason we do not expect to have similar results for anydimension beyond 5?
Why are the matrices in this neat form?
What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?
![Page 86: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/86.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Questions
What is the reason we do not expect to have similar results for anydimension beyond 5?
Why are the matrices in this neat form?
What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?
![Page 87: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/87.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Questions
What is the reason we do not expect to have similar results for anydimension beyond 5?
Why are the matrices in this neat form?
What is the nature of the polynomials Pk and why do they provide anecessary condition for a representation of this form to be irreducible?
![Page 88: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/88.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .
By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 89: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/89.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.
ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 90: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/90.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2),
(ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 91: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/91.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 92: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/92.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 93: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/93.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 94: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/94.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let ρ : B3 → GLk(C) be a representation of B3 of dimension k ≤ 5 overC. Let A := ρ(σ1) and B := ρ(σ2), with eigenvalues λ1, . . . , λk .By the theorem of Cayley-Hamilton there is a monic polynomialQ(X ) = (X − λ1) . . . (X − λk) ∈ C[X ] of degree k such thatQ(A) = Q(B) = 0.ρ(σ1)ρ(σ2)ρ(σ1) = ρ(σ2)ρ(σ1)ρ(σ2), (ρ(σi )− λ1Ik) . . . (ρ(σi )− λk Ik) = 0.
Let Hk = 〈σ1, σ2 | σ1σ2σ1 = σ2σ1σ2, (σi − u1) . . . (σi − uk) = 0〉 be theHecke algebra of S3,G4,G8 and G16 respectively over the ringRk := Z[u±1 , ..., u
±k ].
Let θ : Rk → C be a specialization of Hk defined by ui 7→ λi .
We only need to describe the irreducible CHk := Hk ⊗θ C-modules ofdimension k .
![Page 95: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/95.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra,
i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 96: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/96.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 97: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/97.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e.
t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 98: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/98.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 99: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/99.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 100: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/100.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 101: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/101.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 102: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/102.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Assumption
Hk is a symmetric algebra, i.e. there is a linear map t : Hk → Rk suchthat:
t is a trace function, i.e. t(ab) = t(ba), for all a, b ∈ Hk .
The bilinear form Hk × Hk → Rk , (aa′ 7→ t(aa′)) is non-degenerate.
Theorem (Malle 1999)
There is a field K ⊃ Rk such that KHk := Hk ⊗RkK is split semisimple.
Theorem (Geck 1990)
t =∑
χ∈Irr(KHk )
1
sχχ,
where sχ is the Schur element of χ with respect to t.
![Page 103: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/103.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 104: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/104.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 105: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/105.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 106: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/106.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 107: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/107.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 108: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/108.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 109: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/109.jpg)
An application: A classification of small dimensionalirreducible representations of B3
Let R0(KHk) be the Grothendieck group of finite-dimensionalKHk -modules.
Let R+0 (KHk) be the subset of R0(KHk) consisting of elements [V ], where
V is a finite-dimensional KHk -module.
We can define the decomposition map dθ associated to θ.
R+0 (KHk)
dθ−→ R+0 (CHk)
Let S be an irreducible k-dimensional CHk module.
Theorem (C. 2014)
There is an irreducible KHk -module M of dimension k such thatdθ([M]) = [S ].
θ(sχM) 6= 0.
![Page 110: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/110.jpg)
The cases of G4, G8, G16
Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .
We know that Z (B3) = 〈z〉.
HG4 = u1 + u1z−1 + u1u2u1
HG8 = u1 + u1z + u1z−1 + u1z
−2 + u1u2u1 + u1σ−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1
HG16 = u1 + u1z + u1z−1 + u1z
2 + u1z−2 + u1z
3 + u1z−3+
+u1z4 + u1z
−4 + u1z−5 + u1u2u1 + u1σ
−12 σ2
1σ−12 u1+
+u1σ2σ−21 σ2u1 + u1σ
22σ
21σ
22u1 + u1σ
−22 σ−21 σ−22 u1+
+u1σ2σ−21 σ2
2u1 + u1σ−12 σ2
1σ−22 u1 + u1σ
−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1 + u1σ
−22 σ−21 σ2
2u1 + u1σ22σ
21σ−22 u1+
+u1σ22σ−21 σ2
2u1 + u1σ−22 σ2
1σ−22 u1 + u1σ
−22 σ1σ
−12 u1+
+u1σ−12 σ1σ
−22 u1 + u1σ
−22 σ2
1σ−12 σ1σ
−12 u1+
+u1σ22σ−21 σ2σ
−11 σ2u1 + u1σ2σ
−21 σ2
2σ−21 σ2
2u1++u1σ
−12 σ2
1σ−22 σ2
1σ−22 u1
![Page 111: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/111.jpg)
The cases of G4, G8, G16
Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.
HG4 = u1 + u1z−1 + u1u2u1
HG8 = u1 + u1z + u1z−1 + u1z
−2 + u1u2u1 + u1σ−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1
HG16 = u1 + u1z + u1z−1 + u1z
2 + u1z−2 + u1z
3 + u1z−3+
+u1z4 + u1z
−4 + u1z−5 + u1u2u1 + u1σ
−12 σ2
1σ−12 u1+
+u1σ2σ−21 σ2u1 + u1σ
22σ
21σ
22u1 + u1σ
−22 σ−21 σ−22 u1+
+u1σ2σ−21 σ2
2u1 + u1σ−12 σ2
1σ−22 u1 + u1σ
−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1 + u1σ
−22 σ−21 σ2
2u1 + u1σ22σ
21σ−22 u1+
+u1σ22σ−21 σ2
2u1 + u1σ−22 σ2
1σ−22 u1 + u1σ
−22 σ1σ
−12 u1+
+u1σ−12 σ1σ
−22 u1 + u1σ
−22 σ2
1σ−12 σ1σ
−12 u1+
+u1σ22σ−21 σ2σ
−11 σ2u1 + u1σ2σ
−21 σ2
2σ−21 σ2
2u1++u1σ
−12 σ2
1σ−22 σ2
1σ−22 u1
![Page 112: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/112.jpg)
The cases of G4, G8, G16
Let ui be the subalgebra of HGn (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.
HG4 = u1 + u1z−1 + u1u2u1
HG8 = u1 + u1z + u1z−1 + u1z
−2 + u1u2u1 + u1σ−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1
HG16 = u1 + u1z + u1z−1 + u1z
2 + u1z−2 + u1z
3 + u1z−3+
+u1z4 + u1z
−4 + u1z−5 + u1u2u1 + u1σ
−12 σ2
1σ−12 u1+
+u1σ2σ−21 σ2u1 + u1σ
22σ
21σ
22u1 + u1σ
−22 σ−21 σ−22 u1+
+u1σ2σ−21 σ2
2u1 + u1σ−12 σ2
1σ−22 u1 + u1σ
−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1 + u1σ
−22 σ−21 σ2
2u1 + u1σ22σ
21σ−22 u1+
+u1σ22σ−21 σ2
2u1 + u1σ−22 σ2
1σ−22 u1 + u1σ
−22 σ1σ
−12 u1+
+u1σ−12 σ1σ
−22 u1 + u1σ
−22 σ2
1σ−12 σ1σ
−12 u1+
+u1σ22σ−21 σ2σ
−11 σ2u1 + u1σ2σ
−21 σ2
2σ−21 σ2
2u1++u1σ
−12 σ2
1σ−22 σ2
1σ−22 u1
![Page 113: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/113.jpg)
The cases of G4, G8, G16
Let ui be the subalgebra of HGn , (n = 4, 8, 16) which is generated by σi .We know that Z (B3) = 〈z〉.
HG4 = u1 + u1z−1 + u1u2u1
HG8 = u1 + u1z + u1z−1 + u1z
−2 + u1u2u1 + u1σ−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1
HG16 = u1 + u1z + u1z−1 + u1z
2 + u1z−2 + u1z
3 + u1z−3+
+u1z4 + u1z
−4 + u1z−5 + u1u2u1 + u1σ
−12 σ2
1σ−12 u1+
+u1σ2σ−21 σ2u1 + u1σ
22σ
21σ
22u1 + u1σ
−22 σ−21 σ−22 u1+
+u1σ2σ−21 σ2
2u1 + u1σ−12 σ2
1σ−22 u1 + u1σ
−12 σ1σ
−12 u1+
+u1σ2σ−11 σ2u1 + u1σ
−22 σ−21 σ2
2u1 + u1σ22σ
21σ−22 u1+
+u1σ22σ−21 σ2
2u1 + u1σ−22 σ2
1σ−22 u1 + u1σ
−22 σ1σ
−12 u1+
+u1σ−12 σ1σ
−22 u1 + u1σ
−22 σ2
1σ−12 σ1σ
−12 u1+
+u1σ22σ−21 σ2σ
−11 σ2u1 + u1σ2σ
−21 σ2
2σ−21 σ2
2u1++u1σ
−12 σ2
1σ−22 σ2
1σ−22 u1
![Page 114: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/114.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.
We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 115: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/115.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C))
' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 116: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/116.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).
Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 117: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/117.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.
The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 118: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/118.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 119: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/119.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 120: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/120.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 121: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/121.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 122: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/122.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SU2(C)/Z (SU2(C)) ' SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...,G15.
Icosahedral family G16, ...,G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 123: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/123.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...G11, ...,G15.
Icosahedral family G16, ...,G19, ...G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
![Page 124: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/124.jpg)
The exceptional groups of rank 2
Let W be an exceptional group of rank 2.We know that W := W /Z (W ) ≤ SO3(R).Up to classification of the subgroups of SO3(R), W is the tetrahedral, theoctahedral or the icosahedral group.The groups G4, ...,G22 fall into three families:
Tetrahedral family G4, ...,G7.
Octahedral family G8, ...G11, ...,G15.
Icosahedral family G16, ...,G19, ...G22.
In each family there is a maximal group W such that every W in thisfamily is a subgroup of W .
Proposition (C. 2013)
If HW is torsion free and the BMR conjecture is true for HW
, then theconjecture holds for HW , as well.
![Page 125: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/125.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 126: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/126.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 127: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/127.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1,
i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 128: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/128.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
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A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 130: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/130.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 131: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/131.jpg)
A more global approach: Etingof-Rains 2004
We know that W is the group of even elements in a finite Coxeter groupC of rank 3 of type A3,B3,H3, respectively, with Coxeter matrix (mij).
Let tij ,k be set of indeterminates, where i 6= j , k ∈ Z/mijZ andt−1ij ,k = tji ,−k . Let R := Z[tij ,k ].
A(C ) = 〈 Y1,Y2,Y3 | Y 2i = 1, i 6= j implies
mij∏k=1
(YiYj − tij ,k) = 0 〉
A+(C ) = 〈Aij := YiYj , i 6= j〉
Example: When C is the finite Coxeter group of type B3 the R-algebraA+(C ) can be presented as follows:
< (A13 − t13,1)(A13 − t13,2) = 0A13,A32,A21 (A32 − t32,1) . . . (A32 − t32,3) = 0, A13A32A21 = 1
(A21 − t21,1) . . . (A21 − t21,4) = 0 >
![Page 132: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/132.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
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A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3.
Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 134: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/134.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3.
Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 135: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/135.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 136: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/136.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 137: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/137.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 138: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/138.jpg)
A more global approach: Etingof-Rains 2004
For every reduced word wx that represents x inside W , we denote by Twx
the corresponding element in A+(C ).
Example: Let C be the finite Coxeter group of type B3. Letwx = y1y2y1y3. Then, Twx = A12A13.
Proposition (Etingof-Rains 2004)
A+(C ) is generated as R-module by the elements Twx , x ∈W .
Let R+ = R[z , z−1], where z is the action of the image of the center of Binside HW .
Proposition (Etingof-Rains 2004)
There is a ring morphism φ : R � R+, inducing Φ : A+(C )⊗θ R+ � HW ,considering HW as R+-module.
![Page 139: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/139.jpg)
The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .
wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3
From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)
Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
For every x ∈W we fix a reduced word wx in letters y1, y2 and y3 thatrepresents x in W .wx = y2y3From the reduced word wx one can obtain a word wx :
wx =
{wx for x = 1
wx(y1y1)n1(y2y2)n2(y3y3)n3 for x 6= 1.
wx = y2y3(y1y1)2(y2y2)Moving the pairs yiyi somewhere inside wx and using the braid relationsbetween the generators yi one can obtain a word wx :
`(wx) = `(wx), where `(w) denotes the length of the word w .
Let m be an odd number. Whenever in the word wx there is a letter yiat the mth-position from left to right, then in the (m + 1)th-positionthere is a letter yj , j 6= i .
wx = wx if and only if wx = wx . In particular, w1 = w1.
wx = y2y1y1y2y2y1y1y3
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The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
RzkTwx ,
where Twx is the corresponding element in A+(C ).
![Page 147: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/147.jpg)
The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
RzkTwx ,
where Twx is the corresponding element in A+(C ).
![Page 148: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/148.jpg)
The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
RzkTwx ,
where Twx is the corresponding element in A+(C ).
![Page 149: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/149.jpg)
The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
Rzk Twx ,
where Twx is the corresponding element in A+(C )⊗θ R+.
![Page 150: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/150.jpg)
The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
RzkΦ(Twx ),
where Twx is the corresponding element in A+(C )⊗θ R+.
Theorem (C. 2015)
HW = U, and, hence, the BMR freeness conjecture is true for the groupsbelonging to the first two families.
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The conjecture for the first two families
Let W be an exceptional group belonging to the tetrahedral or octahedralfamily.
For every x ∈ W we found a specific word wx and we set
U :=∑x∈W
|Z(W )|−1∑k=0
RzkΦ(Twx ),
where Twx is the corresponding element in A+(C )⊗θ R+.
Theorem (C. 2015)
HW = U, and, hence, the BMR freeness conjecture is true for the groupsbelonging to the first two families.
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Remaining cases
Conjecture (equivalent version)
HW is spanned over R by |W | elements, where W is an irreduciblecomplex reflection group.
G4, ...,G7︸ ︷︷ ︸T
, G8, . . . ,G15︸ ︷︷ ︸O
, G16,G17,G18,G19,G20,G21,G22︸ ︷︷ ︸I
, G23,G24, ...G36,G37.
![Page 153: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/153.jpg)
Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2),
where u2 is the subalgebra of HG6 generated
by σ2.
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
![Page 156: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/156.jpg)
Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉
HG9 =7∑
k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra
of HG9 generated by σ2.
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉
HG9 =7∑
k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra
of HG9 generated by σ2.
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉
HG9 =7∑
k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra
of HG9 generated by σ2.
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉
HG9 =7∑
k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra
of HG9 generated by σ2.
• G17 = 〈s1, s2 | s21 = s52 = 1, (s1s2)3 = (s2s1)3〉
HG17
?=
19∑k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1 + zk
5∑i=1
u2wi ), where u2 is
the subalgebra of HG17 generated by σ2 and wi the image of an element ofB inside HG17 .
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Open questions
• G6 = 〈s1, s2 | s21 = s32 = 1, (s1s2)3 = (s2s1)3〉
HG6 =3∑
k=0
(zku2 + zku2σ1u2), where u2 is the subalgebra of HG6 generated
by σ2.
• G9 = 〈s1, s2 | s21 = s42 = 1, (s1s2)3 = (s2s1)3〉
HG9 =7∑
k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1), where u2 is the subalgebra
of HG9 generated by σ2.
• G17 = 〈s1, s2 | s21 = s52 = 1, (s1s2)3 = (s2s1)3〉
HG17
?=
19∑k=0
(zku2 + zku2σ1u2 + zku2σ1σ−22 σ1 + zk
5∑i=1
u2wi ), where u2 is
the subalgebra of HG17 generated by σ2 and wi the image of an element ofB inside HG17 .
![Page 161: The Brou e-Malle-Rouquier conjecture for the exceptional ... · The Brou e-Malle-Rouquier conjecture for the exceptional groups of rank 2 Ph.D thesis defense supervised by Ivan Marin](https://reader034.vdocuments.site/reader034/viewer/2022042917/5f5becd112a1255cc049d81c/html5/thumbnails/161.jpg)
Open questions
• G7 = 〈s1, s2, s3 | s21 = s32 = s33 = 1, s1s2s3 = s2s3s1 = s3s1s2〉
HG7 =11∑k=0
(zku3u2 + zkσ2σ−13 u2), where u2 and u3 are the subalgebras of
HG7 generated by σ2 and σ3, respectively.
• G11 = 〈s1, s2, s3 | s21 = s32 = s43 = 1, s1s2s3 = s2s3s1 = s3s1s2〉
HG11 =23∑k=0
(zku3u2 + zku3σ2σ−13 u2), where u2 and u3 are the subalgebras
of HG7 generated by σ2 and σ3, respectively.
• G19 = 〈s1, s2, s3 | s21 = s32 = s53 = 1, s1s2s3 = s2s3s1 = s3s1s2〉
HG19
?=
59∑k=0
(zku3u2 + zku3σ2σ−13 u2 + zk
2∑i=1
u3wiu2), where u2 and u3 are
the subalgebras of HG19 generated by σ2 and σ3, respectively and wi theimage of an element of B inside HG19 .