sorting monoids on coxeter groups - bucknell...
TRANSCRIPT
1 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Sorting monoids on Coxeter groups
Florent Hivert1 Anne Schilling2 Nicolas M. Thiery2,3
1LITIS/LIFAR, Universite Rouen, France
2University of California at Davis, USA
3Laboratoire de Mathematiques d’Orsay, Universite Paris Sud, France
FPSAC’10, San Francisco, August 3rd of 2010
arXiv:0711.1561 [math.RT] (FPSAC’06)arXiv:0804.3781 [math.RT] (FPSAC’08)arXiv:0912.2212 [math.CO] (FPSAC’10)
+ research in progress
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
1234
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
1234
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
1243
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
1423
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4123
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4123
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4132
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4312
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4312
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .
Relations: s2i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
2 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Bubble (anti) sort algorithm
4321
Underlying combinatorics: right permutahedron
123
213 132
312231
321
1234
2134 1324 1243
2314 3124 2143 1342 1423
2341 3214 2413 3142 4123 1432
3241 2431 3412 4213 4132
3421 4231 4312
4321
Elementary transpositions: s1, s2, s3, . . .Relations: s2
i = 1, (s1s2)3 = 1, (s2s3)3 = 1, (s1s3)2 = 1
3 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Coxeter groups
Definition (Coxeter group W )
Generators : s1, s2, . . . (simple reflections)
Relations: s2i = 1 and si sj · · ·︸ ︷︷ ︸
mi,j
= sjsi · · ·︸ ︷︷ ︸mi,j
, for i 6= j
Reduced words
4 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
π1 π2
π1 π2
π2 π1
π2 π1
π1 π2
π1 π2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
S3 H0(S3) H0(S3)
4 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
π1 π2
π1 π2
π2 π1
π2 π1
π1 π2
π1 π2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
S3 H0(S3) H0(S3)
4 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
S3 H0(S3) H0(S3)
4 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
S3 H0(S3) H0(S3)
4 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
123
213 132
312231
321
π1 π2
π2 π1
π1 π2
S3 H0(S3) H0(S3)
5 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
Definition (0-Hecke monoid H0(W ) of a Coxeter group W )
Generators : 〈π1, π2, . . . 〉 (simple reflections)Relations: π2
i = πi and braid relations
Theorem
|H0(W )| = |W |+ lots of nice properties
Motivation: simple combinatorial model (bubble sort)appears in Iwahori-Hecke algebras, Schur symmetric functions,Schubert, Macdonald, Kazhdan-Lusztig polynomials,(affine) Stanley symmetric functions, mathematical physics,Schur-Weyl duality for quantum groups, representations of GL(Fq),. . .
5 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
Definition (0-Hecke monoid H0(W ) of a Coxeter group W )
Generators : 〈π1, π2, . . . 〉 (simple reflections)Relations: π2
i = πi and braid relations
Theorem
|H0(W )| = |W |+ lots of nice properties
Motivation: simple combinatorial model (bubble sort)appears in Iwahori-Hecke algebras, Schur symmetric functions,Schubert, Macdonald, Kazhdan-Lusztig polynomials,(affine) Stanley symmetric functions, mathematical physics,Schur-Weyl duality for quantum groups, representations of GL(Fq),. . .
5 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
0-Hecke monoid
Definition (0-Hecke monoid H0(W ) of a Coxeter group W )
Generators : 〈π1, π2, . . . 〉 (simple reflections)Relations: π2
i = πi and braid relations
Theorem
|H0(W )| = |W |+ lots of nice properties
Motivation: simple combinatorial model (bubble sort)appears in Iwahori-Hecke algebras, Schur symmetric functions,Schubert, Macdonald, Kazhdan-Lusztig polynomials,(affine) Stanley symmetric functions, mathematical physics,Schur-Weyl duality for quantum groups, representations of GL(Fq),. . .
6 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Classical orders on Coxeter groups
123
213 132
312231
321
s1 s2
s2 s1
s1s2
123
213 132
312231
321
s1×s1 s2×s2
×s2 ×s1s2× s1×
s1×s2× ×s1 ×s2
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
Left order Left-Right order
Right order
Bruhat order
Suffix Factor Prefix Subword
6 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Classical orders on Coxeter groups
123
213 132
312231
321
s1 s2
s2 s1
s1s2
123
213 132
312231
321
s1×s1 s2×s2
×s2 ×s1s2× s1×
s1×s2× ×s1 ×s2
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
Left order Left-Right order
Right order
Bruhat order
Suffix Factor
Prefix
Subword
6 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Classical orders on Coxeter groups
123
213 132
312231
321
s1 s2
s2 s1
s1s2
123
213 132
312231
321
s1×s1 s2×s2
×s2 ×s1s2× s1×
s1×s2× ×s1 ×s2
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
Left order
Left-Right order
Right order
Bruhat order
Suffix
Factor
Prefix
Subword
6 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Classical orders on Coxeter groups
123
213 132
312231
321
s1 s2
s2 s1
s1s2
123
213 132
312231
321
s1×s1 s2×s2
×s2 ×s1s2× s1×
s1×s2× ×s1 ×s2
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
Left order Left-Right order Right order
Bruhat order
Suffix Factor Prefix
Subword
6 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Classical orders on Coxeter groups
123
213 132
312231
321
s1 s2
s2 s1
s1s2
123
213 132
312231
321
s1×s1 s2×s2
×s2 ×s1s2× s1×
s1×s2× ×s1 ×s2
123
213 132
312231
321
s1 s2
s2 s1
s1 s2
123
213 132
312231
321
Left order Left-Right order Right order Bruhat order
Suffix Factor Prefix Subword
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
7 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Blocks of permutations
Definition (Block of a permutation w)
• Type A: sub-permutation matrix
• Type free: J,K such that WJw = wWK
• Example: w := 36475812
••
••
••
••
• Simple permutation: cf. [Albert, Atkinson 05] + dim 2 posets• {blocks of w}: sub-lattice of the Boolean lattice
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
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Theorem
• Intervals are lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
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Theorem
• Intervals are lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
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Theorem
• Intervals are lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
•
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Theorem
• Intervals are lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
•
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•
•
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Theorem
• Intervals are (distributive?) lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
8 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The cutting poset
Definition (HST09: Cutting poset (W ,v))
u v w if u = wJ with J block
•
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•
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•
••
•
•
••
•
••
••
•
••
•
•
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•
•
••
•
•
•••
Theorem
• Intervals are (distributive?) lattices
• Mobius function: inclusion-exclusion along minimal blocks
• Meet-semi lattice?
9 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The Big Picture
Hζ(W )
Hζ(W )
Hζ(Sn) ∧
H−1(W )
H−1(W )
TLn
Hq(W )
Hq(W )
Hq(Sn) ∧
H0(W )〈π1, π2, . . . 〉
H0(W )〈π0π1, π2, . . . 〉
NDPFn
H0(W )〈π1, π2, . . . 〉
H0(W )〈π0π1, π2, . . . 〉
NDPFn
NDPFB
HWQ[π1, π2, . . . , s1, s2, . . . ]Q[π1, π2, . . . , π1, π2, . . . ]
Q[π0π1, π2, . . . ]
NDFn
W〈s1, s2, . . . 〉
W〈s0s1, s2, . . . 〉
Sn ∧
M1
NDPF (Bruhat(W )) End(<L (W ))
〈π1, π2, . . . , π1, π2, . . . 〉 〈π1, π2, . . . , s1, s2, . . . 〉 〈π0π1, π2, . . . 〉
End(BooleanLattice)
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
10 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The biHecke monoid
Question
Size of M(W ) = 〈π1, π2, . . . , π1, π2, . . . 〉|M(Sn)| = 1, 3, 23, 477, 31103, ?
• How to attack such a problem?
• Generators and relations?
• Representation theory?
Theorem (HST08)
M(W ) admits |W | simple / indecomposable projective modules
• Why do we care?
|M(W )| =∑
w∈W
dim Sw . dim Pw
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
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. 7−→
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
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. 7−→
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. .•
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
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. 7−→
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. .•
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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•
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. 7−→
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. .•.
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. .•
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.•
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
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. 7−→
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. .•.
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. .•
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.•
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Lemma
For f ∈ M(W ) and w ∈W : (siw).f = w .f or si (w .f )
Proof.
Exchange property / associativity
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
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. 7−→
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Corollary
• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f
• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f
• M(W ) is aperiodic
• f in M(W ) is determined by its fibers and f (1)
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
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. ••.
•
.•
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. 7−→
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. .•
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Corollary
• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f
• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f
• M(W ) is aperiodic
• f in M(W ) is determined by its fibers and f (1)
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
.
. ••.
•
.•
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•.
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. 7−→
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. .•.
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. .•
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Corollary
• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f
• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f
• M(W ) is aperiodic
• f in M(W ) is determined by its fibers and f (1)
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
•.
.
. ••.
•
.•
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•.
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. 7−→
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. .•.
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.•
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. .•
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.•
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Corollary
• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f
• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f
• M(W ) is aperiodic
• f in M(W ) is determined by its fibers and f (1)
11 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Key combinatorial lemma
132
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. .•.
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. .•
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.•
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Corollary
• Preservation of left order: u ≤L v =⇒ u.f ≤L v .f
• Preservation of Bruhat order: u ≤B v =⇒ u.f ≤B v .f
• M(W ) is aperiodic
• f in M(W ) is determined by its fibers and f (1)
12 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Some elements of the monoid1
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2
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.•• .
.•
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•
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132
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2123
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Lemma
The image set of an idempotent is an interval in left order
12 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Some elements of the monoid1
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2
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.•• .
.•
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•. .
.•
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•
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132
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2123
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•. .
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Lemma
The image set of an idempotent is an interval in left order
12 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Some elements of the monoid1
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•.
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2
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.•
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132
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2123
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Lemma
The image set of an idempotent is an interval in left order
12 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Some elements of the monoid1
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•.
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2
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.•• .
.•
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.•
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132
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2123
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Lemma
The image set of an idempotent is an interval in left order
12 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Some elements of the monoid1
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2
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132
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2123
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Lemma
The image set of an idempotent is an interval in left order
13 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
×1
2×
×1
×2
2×
×2
×2
2×2×1
×2
2×21×1
2×
×2
2×
1×
2×
×2×1
×2
×1
1×
×1 ×2
×1
×1
×1
×2
×2
1×
×1
×2
×1
1×
×2
1×
×1
1×
×2
×1×2
2×
×1
×1
2×
×1
×1
×1
×2
1×
1×1
×1
×1
2×
×2
1×1
×2
2×2 2×
2×
×1
×2
1×1
×1
2×2
1×
1×
×2
×2
1×
×2
1×
×2
×2
×1
2×
2×
1×
×1
×2
1×
1×
2×
×2
×1
2×
×2
×1
.
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.7→
••
••
••
1
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2
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2
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••
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1
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•.
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12
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12
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.7→
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21
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.7→
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21
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.7→
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21
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.7→
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.•
21
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.7→
•.
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12
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.7→
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.•
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12
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.7→
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121
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.7→
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.•
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121
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.7→
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.•
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121
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.7→
•.
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•.
212
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.7→
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.•
212
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.7→
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212
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.7→
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.•
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212
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.7→
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212
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.7→
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121
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.7→
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.•
.•
121
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.7→
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14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
14 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Green relations
Theorem
• Regular J -classes are indexed by W
• J -order on regular classes: left-right order on W
• R-classes: intervals in right order on W
• R-order on regular R-classes: ≈ right order on W
• L-order on regular L-classes: ≈ left order on W
Problems
• L,R,J -order between non regular classes?
• L-classes?
15 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Representation theory of M(W )
Corollary
M(W ) admits |W | simple modules / indecomposable projectivemodules
Problem
Dimension of simple and indecomposable projective modules?
15 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Representation theory of M(W )
Corollary
M(W ) admits |W | simple modules / indecomposable projectivemodules
Problem
Dimension of simple and indecomposable projective modules?
16 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The “Borel” submonoid M1
Definition
Submonoid M1 := {f ∈ M, f (1) = 1}
Properties
• Weakly increasing and contracting on Bruhat =⇒ J -trivial
• Idempotents: (ew )w∈W
• Generated by ew for w grassmanian, e.g. atom for (W ,∨L)
• |W | simple modules of dimension 1
• Semi-simple quotient: monoid algebra of (W ,∨L)
• Conjugacy order among idempotents: <L
• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?
Problem
Inducing these results to M?
16 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The “Borel” submonoid M1
Definition
Submonoid M1 := {f ∈ M, f (1) = 1}
Properties
• Weakly increasing and contracting on Bruhat =⇒ J -trivial
• Idempotents: (ew )w∈W
• Generated by ew for w grassmanian, e.g. atom for (W ,∨L)
• |W | simple modules of dimension 1
• Semi-simple quotient: monoid algebra of (W ,∨L)
• Conjugacy order among idempotents: <L
• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?
Problem
Inducing these results to M?
16 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
The “Borel” submonoid M1
Definition
Submonoid M1 := {f ∈ M, f (1) = 1}
Properties
• Weakly increasing and contracting on Bruhat =⇒ J -trivial
• Idempotents: (ew )w∈W
• Generated by ew for w grassmanian, e.g. atom for (W ,∨L)
• |W | simple modules of dimension 1
• Semi-simple quotient: monoid algebra of (W ,∨L)
• Conjugacy order among idempotents: <L
• dim Pw = |{f ∈ M1, f (w) ≤L w}| =?
Problem
Inducing these results to M?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
17 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
R-classes and translation algebras
1234
1324 1243
3124 1342 1423
3142 1432 4123
3412 4132
4312
1234
1324
3124 1342
3142
3412
1243
1423
4123 1432
4132
4312
Definition (Translation algebra)
HW (w) := Q[π1, π2, . . . , π1, π2, . . . ] acting on Q.[1,w ]R
• Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} =⇒ Submodules PJ
• HW (w): max. algebra stabilizing all PJ =⇒ Repr. theory• HW (w) quotient of Q[M(W )]; top: simple module Sw of M• Dimension: inclusion-exclusion along the cutting poset• Generating series calculation?
18 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Work in progress
• L-classes? Projective modules? Cartan Matrix?
• Generalization to R-trivial and aperiodic monoids(collaboration with Denton and Berg, Bergeron, Saliola)
• Fast implementation in Sage(interface with Semigroupe, ...)
19 / 19
Bubble sort and Coxeter groups The cutting poset The biHecke monoid Combinatorics Representation theory
Sage-Combinat meeting tonight
Sage’s mission:
“To create a viable high-quality and open-source alternativeto MapleTM, MathematicaTM, MagmaTM, and MATLABTM”
...“and to foster a friendly community of users and developers”
Tonight, Thurston Hall, Room 236
• 7pm-8pm: Introduction to Sage and Sage-Combinat
• 8pm-10pm: Help on installation & getting startedBring your laptop!
• Design discussions