faithful irreducible characters of the wreath productpages.uoregon.edu/raies/latex/sample...
TRANSCRIPT
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Problem StatementAlgorithm
Implementation
Faithful Irreducible Charactersof the Wreath Product
Dan Raies
The University of Akron
April 26, 2012
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Introduction
For a fixed prime p we study subgroups of the iterated wreathproduct group Zp o Zp o Zp.
In order to do this, we first build a group P that is isomorphic toZp o Zp o Zp but has a structure with which we can work.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer
Let U =
0, 1, . . . , p2 − 1
and let B be an elementary abelianp-group of rank p2 and let xu | u ∈ U be a generating set for B.An arbitrary element of B has the form
x =∏u∈U
xkuu .
Let F be the set of all functions from U to Zp and note that F isa vector space of dimension p2 over Zp according to the standardoperations of functions.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer
Let U =
0, 1, . . . , p2 − 1
and let B be an elementary abelianp-group of rank p2 and let xu | u ∈ U be a generating set for B.An arbitrary element of B has the form
x =∏u∈U
xkuu .
Let F be the set of all functions from U to Zp and note that F isa vector space of dimension p2 over Zp according to the standardoperations of functions.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 5: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/5.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ = γu | u ∈ U such that
γu (t) = δtu.
2. The basis E = eu | u ∈ U such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 6: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/6.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ = γu | u ∈ U such that
γu (t) = δtu.
2. The basis E = eu | u ∈ U such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 7: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/7.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ = γu | u ∈ U such that
γu (t) = δtu.
2. The basis E = eu | u ∈ U such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 8: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/8.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Group Actions
There is a isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Note that there is a natural action of Sym (U) on B according to(∏u∈U
xkuu
)π
=∏u∈U
xkuuπ .
This induces an action of Sym (U) on F such that
f π (u) = f(uπ−1).
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Group Actions
There is a isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Note that there is a natural action of Sym (U) on B according to(∏u∈U
xkuu
)π
=∏u∈U
xkuuπ .
This induces an action of Sym (U) on F such that
f π (u) = f(uπ−1).
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Group Actions
There is a isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Note that there is a natural action of Sym (U) on B according to(∏u∈U
xkuu
)π
=∏u∈U
xkuuπ .
This induces an action of Sym (U) on F such that
f π (u) = f(uπ−1).
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer
Let U = 0, 1, . . . , p − 1. We then define a particular setxu | u ∈ U
⊂ Sym (U). In the case where p = 3,
x0 = (630)
x1 = (741)
x2 = (852) .
We let B be the group generated byxu | u ∈ U
. It follows that
B is an elementary abelian p-group of rank p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer
Let U = 0, 1, . . . , p − 1. We then define a particular setxu | u ∈ U
⊂ Sym (U). In the case where p = 3,
x0 = (630)
x1 = (741)
x2 = (852) .
We let B be the group generated byxu | u ∈ U
. It follows that
B is an elementary abelian p-group of rank p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer
We can then use definitions similar to those for B.
Define F to be the set of all functions from U to Zp. Note thatthis is a vector space of dimension p over Zp.
There is an isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer
We can then use definitions similar to those for B.
Define F to be the set of all functions from U to Zp. Note thatthis is a vector space of dimension p over Zp.
There is an isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 15: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/15.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer
We can then use definitions similar to those for B.
Define F to be the set of all functions from U to Zp. Note thatthis is a vector space of dimension p over Zp.
There is an isomorphism η : F → B given by
η (f ) =∏u∈U
xf (u)u .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 16: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/16.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ =γu | u ∈ U
such that
γu (t) = δtu.
2. The basis E =eu | u ∈ U
such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 17: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/17.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ =γu | u ∈ U
such that
γu (t) = δtu.
2. The basis E =eu | u ∈ U
such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 18: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/18.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Bases
There are two useful bases for the vector space F .
1. The standard basis Γ =γu | u ∈ U
such that
γu (t) = δtu.
2. The basis E =eu | u ∈ U
such that
eu (t) =
(t
u
)mod p.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 19: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/19.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Third Layer
We wish to define a particular element w ∈ Sym (U). In the casewhen p = 3,
w = (876) (543) (210) .
It is easily verified for each u ∈ U that w xuw−1 = xu−1 which
induces an action of 〈w〉 on B via conjugation.
This allows us to form the group P = B o 〈w〉 which is isomorphicto Zp o Zp.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Third Layer
We wish to define a particular element w ∈ Sym (U). In the casewhen p = 3,
w = (876) (543) (210) .
It is easily verified for each u ∈ U that w xuw−1 = xu−1 which
induces an action of 〈w〉 on B via conjugation.
This allows us to form the group P = B o 〈w〉 which is isomorphicto Zp o Zp.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 21: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/21.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Third Layer
We wish to define a particular element w ∈ Sym (U). In the casewhen p = 3,
w = (876) (543) (210) .
It is easily verified for each u ∈ U that w xuw−1 = xu−1 which
induces an action of 〈w〉 on B via conjugation.
This allows us to form the group P = B o 〈w〉 which is isomorphicto Zp o Zp.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 22: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/22.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Iterated Wreath Product
Since Sym (U) acts on B and P ⊂ Sym (U) there is an inheritedaction of P on B.
According to this action we let P = B o P which is isomorphic toZp o Zp o Zp.
As sets, P = B × B × 〈w〉.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Iterated Wreath Product
Since Sym (U) acts on B and P ⊂ Sym (U) there is an inheritedaction of P on B.
According to this action we let P = B o P which is isomorphic toZp o Zp o Zp.
As sets, P = B × B × 〈w〉.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 24: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/24.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Iterated Wreath Product
Since Sym (U) acts on B and P ⊂ Sym (U) there is an inheritedaction of P on B.
According to this action we let P = B o P which is isomorphic toZp o Zp o Zp.
As sets, P = B × B × 〈w〉.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 25: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/25.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Subgroups
Fix j ∈
0, 1, . . . , p2
.
Define Fj = 〈e0, e1, . . . , ej−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp2−1 ⊂ Fp2 = F .
Define Dj =η (f ) | f ∈ Fp2−j
. Note that
1 = Dp2 ⊂ Dp2−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally, |B : Dj | = pj .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 26: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/26.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Subgroups
Fix j ∈
0, 1, . . . , p2
.
Define Fj = 〈e0, e1, . . . , ej−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp2−1 ⊂ Fp2 = F .
Define Dj =η (f ) | f ∈ Fp2−j
. Note that
1 = Dp2 ⊂ Dp2−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally, |B : Dj | = pj .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 27: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/27.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
First Layer - Subgroups
Fix j ∈
0, 1, . . . , p2
.
Define Fj = 〈e0, e1, . . . , ej−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp2−1 ⊂ Fp2 = F .
Define Dj =η (f ) | f ∈ Fp2−j
. Note that
1 = Dp2 ⊂ Dp2−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally, |B : Dj | = pj .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 28: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/28.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Subgroups
Fix k ∈ 0, 1, . . . , p.
Define Fk = 〈e0, e1, . . . , ek−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp−1 ⊂ Fp = F .
Define Dk =η (f ) | f ∈ Fp−k
. Note that
1 = Dp ⊂ Dp−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally,∣∣B : Dk
∣∣ = pk .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Subgroups
Fix k ∈ 0, 1, . . . , p.
Define Fk = 〈e0, e1, . . . , ek−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp−1 ⊂ Fp = F .
Define Dk =η (f ) | f ∈ Fp−k
. Note that
1 = Dp ⊂ Dp−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally,∣∣B : Dk
∣∣ = pk .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 30: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/30.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Second Layer - Subgroups
Fix k ∈ 0, 1, . . . , p.
Define Fk = 〈e0, e1, . . . , ek−1〉. Note that
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fp−1 ⊂ Fp = F .
Define Dk =η (f ) | f ∈ Fp−k
. Note that
1 = Dp ⊂ Dp−1 ⊂ · · · ⊂ D1 ⊂ D0 = B.
Additionally,∣∣B : Dk
∣∣ = pk .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 31: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/31.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Third Layer - Subgroups
Fix v = w xp−1. It is easily verified that vp ∈ Dk but v /∈ Dk fork ∈ 0, 1, . . . , p − 1. Then define
Tk =⟨Dk , v
⟩.
Note that∣∣Tk : Dk
∣∣ = p.
Finally, for j ∈
0, 1, . . . , p2
and k ∈ 0, 1, . . . , p define
Hjk = Dj o Tk .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Third Layer - Subgroups
Fix v = w xp−1. It is easily verified that vp ∈ Dk but v /∈ Dk fork ∈ 0, 1, . . . , p − 1. Then define
Tk =⟨Dk , v
⟩.
Note that∣∣Tk : Dk
∣∣ = p.
Finally, for j ∈
0, 1, . . . , p2
and k ∈ 0, 1, . . . , p define
Hjk = Dj o Tk .
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 33: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/33.jpg)
Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Question of Interest
ProblemFind the number of faithful irreducible characters of Hjk of eachdegree.
I There are no such characters when j ≥ p2 − p or k = p.
I The only possible character degrees are pm+1 form ∈ 1, 2, . . . , p − k.
Define ΩJ =
0, 1, . . . , p2 − p − 1
, ΩK = 0, 1, . . . , p − 1, andΩkM = 1, 2, . . . , p − k. Henceforth, assume j ∈ ΩJ , k ∈ ΩK , and
m ∈ ΩkM .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Question of Interest
ProblemFind the number of faithful irreducible characters of Hjk of eachdegree.
I There are no such characters when j ≥ p2 − p or k = p.
I The only possible character degrees are pm+1 form ∈ 1, 2, . . . , p − k.
Define ΩJ =
0, 1, . . . , p2 − p − 1
, ΩK = 0, 1, . . . , p − 1, andΩkM = 1, 2, . . . , p − k. Henceforth, assume j ∈ ΩJ , k ∈ ΩK , and
m ∈ ΩkM .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Question of Interest
ProblemFind the number of faithful irreducible characters of Hjk of eachdegree.
I There are no such characters when j ≥ p2 − p or k = p.
I The only possible character degrees are pm+1 form ∈ 1, 2, . . . , p − k.
Define ΩJ =
0, 1, . . . , p2 − p − 1
, ΩK = 0, 1, . . . , p − 1, andΩkM = 1, 2, . . . , p − k. Henceforth, assume j ∈ ΩJ , k ∈ ΩK , and
m ∈ ΩkM .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Question of Interest
ProblemFind the number of faithful irreducible characters of Hjk of eachdegree.
I There are no such characters when j ≥ p2 − p or k = p.
I The only possible character degrees are pm+1 form ∈ 1, 2, . . . , p − k.
Define ΩJ =
0, 1, . . . , p2 − p − 1
, ΩK = 0, 1, . . . , p − 1, andΩkM = 1, 2, . . . , p − k. Henceforth, assume j ∈ ΩJ , k ∈ ΩK , and
m ∈ ΩkM .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Character Theory
I Qmjk =
χ ∈ Irr (Hjk)
∣∣∣∣ χ (1) = pm+1, kerχ = 1
I Dm
jk =
λ ∈ Irr (Dj)
∣∣∣∣ irred. consts. of λHjk are in Qmjk
I Bmjk =
µ ∈ Irr (B)
∣∣∣∣ µ|Dj∈ Dm
jk
Dj
Dj Dk
Hjk
B
pjpp−k
p
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Character Theory
I It turns out that ∣∣Qmjk
∣∣ =∣∣Dm
jk
∣∣pp−k−2m−1and that ∣∣Bmjk ∣∣ =
∣∣Dmjk
∣∣pj .
Dj
Dj Dk
Hjk
B
pjpp−k
p
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Primary Challenge
Claim: If λ ∈ Irr (Dj) and N = IHjk(λ) then the irreducible
constituents of λHjk have degree |Hjk : N|.
1. N splits over Dj and λ is linear so λ extends to µ ∈ Irr (N).
2. It turns out that N/Dj is abelian so by Gallagher’s Theorem,every character that lies over λ is an extension of λ, and thusmust be linear.
3. By Clifford Correspondence, induction is then a bijection fromIrr (N | λ) to Irr (Hjk | λ).
4. Thus every character of Irr (Hjk | λ) has degree |Hjk : N|.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Primary Challenge
Claim: If λ ∈ Irr (Dj) and N = IHjk(λ) then the irreducible
constituents of λHjk have degree |Hjk : N|.1. N splits over Dj and λ is linear so λ extends to µ ∈ Irr (N).
2. It turns out that N/Dj is abelian so by Gallagher’s Theorem,every character that lies over λ is an extension of λ, and thusmust be linear.
3. By Clifford Correspondence, induction is then a bijection fromIrr (N | λ) to Irr (Hjk | λ).
4. Thus every character of Irr (Hjk | λ) has degree |Hjk : N|.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Primary Challenge
Claim: If λ ∈ Irr (Dj) and N = IHjk(λ) then the irreducible
constituents of λHjk have degree |Hjk : N|.1. N splits over Dj and λ is linear so λ extends to µ ∈ Irr (N).
2. It turns out that N/Dj is abelian so by Gallagher’s Theorem,every character that lies over λ is an extension of λ, and thusmust be linear.
3. By Clifford Correspondence, induction is then a bijection fromIrr (N | λ) to Irr (Hjk | λ).
4. Thus every character of Irr (Hjk | λ) has degree |Hjk : N|.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Primary Challenge
Claim: If λ ∈ Irr (Dj) and N = IHjk(λ) then the irreducible
constituents of λHjk have degree |Hjk : N|.1. N splits over Dj and λ is linear so λ extends to µ ∈ Irr (N).
2. It turns out that N/Dj is abelian so by Gallagher’s Theorem,every character that lies over λ is an extension of λ, and thusmust be linear.
3. By Clifford Correspondence, induction is then a bijection fromIrr (N | λ) to Irr (Hjk | λ).
4. Thus every character of Irr (Hjk | λ) has degree |Hjk : N|.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Group DefinitionsSubgroupsQuestionCharacter Theory
Primary Challenge
Claim: If λ ∈ Irr (Dj) and N = IHjk(λ) then the irreducible
constituents of λHjk have degree |Hjk : N|.1. N splits over Dj and λ is linear so λ extends to µ ∈ Irr (N).
2. It turns out that N/Dj is abelian so by Gallagher’s Theorem,every character that lies over λ is an extension of λ, and thusmust be linear.
3. By Clifford Correspondence, induction is then a bijection fromIrr (N | λ) to Irr (Hjk | λ).
4. Thus every character of Irr (Hjk | λ) has degree |Hjk : N|.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Algorithm
To demonstrate the algorithm fix p = 3.
Recall the following definitions:
I x0 = (630), x1 = (741), and x2 = (852)
I An element of B has the form (630)k0 (741)k1 (852)k2
I An element of F has the form k0γ0 + k1γ1 + k2γ2.
If π ∈ B the Ψπ : F → F is a linear transformation defined byΨπ (g) = gπ − g for g ∈ F .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Algorithm
To demonstrate the algorithm fix p = 3.
Recall the following definitions:
I x0 = (630), x1 = (741), and x2 = (852)
I An element of B has the form (630)k0 (741)k1 (852)k2
I An element of F has the form k0γ0 + k1γ1 + k2γ2.
If π ∈ B the Ψπ : F → F is a linear transformation defined byΨπ (g) = gπ − g for g ∈ F .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Algorithm
To demonstrate the algorithm fix p = 3.
Recall the following definitions:
I x0 = (630), x1 = (741), and x2 = (852)
I An element of B has the form (630)k0 (741)k1 (852)k2
I An element of F has the form k0γ0 + k1γ1 + k2γ2.
If π ∈ B the Ψπ : F → F is a linear transformation defined byΨπ (g) = gπ − g for g ∈ F .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Algorithm
To demonstrate the algorithm fix p = 3.
Recall the following definitions:
I x0 = (630), x1 = (741), and x2 = (852)
I An element of B has the form (630)k0 (741)k1 (852)k2
I An element of F has the form k0γ0 + k1γ1 + k2γ2.
If π ∈ B the Ψπ : F → F is a linear transformation defined byΨπ (g) = gπ − g for g ∈ F .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Algorithm
To demonstrate the algorithm fix p = 3.
Recall the following definitions:
I x0 = (630), x1 = (741), and x2 = (852)
I An element of B has the form (630)k0 (741)k1 (852)k2
I An element of F has the form k0γ0 + k1γ1 + k2γ2.
If π ∈ B the Ψπ : F → F is a linear transformation defined byΨπ (g) = gπ − g for g ∈ F .
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating β-values
For an integer j and a subgroup N of B define
βj(N)
=∣∣∣µ ∈ B ∣∣∣ N ⊆ I
B
(µDj
)∣∣∣ .
The first step of the algorithm is to calculate βj(N)
for each
integer j and each subgroup N of B. This is also the most intricateand difficult part of the algorithm.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating β-values
For an integer j and a subgroup N of B define
βj(N)
=∣∣∣µ ∈ B ∣∣∣ N ⊆ I
B
(µDj
)∣∣∣ .The first step of the algorithm is to calculate βj
(N)
for each
integer j and each subgroup N of B. This is also the most intricateand difficult part of the algorithm.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Images and Kernels
Before calculating these β-values we must first calculate ImΨπ andKerΨπ for each π ∈ B.
Consider the permutation π1 = (741) ∈ B.
1. KerΨπ1 = 〈e0, e1, e2, e3 + 2e4, e5, e6 + 2e7, e8〉2. ImΨπ1 = 〈e1 + e2, e4 + e5〉
Consider the permutation π2 = (852) ∈ B.
1. KerΨπ2 = 〈e0, e1, e2, e3 + e4, e4 + e5, e6 + e7, e7 + e8〉2. ImΨπ2 = 〈e2, e5〉
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Images and Kernels
Before calculating these β-values we must first calculate ImΨπ andKerΨπ for each π ∈ B.
Consider the permutation π1 = (741) ∈ B.
1. KerΨπ1 = 〈e0, e1, e2, e3 + 2e4, e5, e6 + 2e7, e8〉2. ImΨπ1 = 〈e1 + e2, e4 + e5〉
Consider the permutation π2 = (852) ∈ B.
1. KerΨπ2 = 〈e0, e1, e2, e3 + e4, e4 + e5, e6 + e7, e7 + e8〉2. ImΨπ2 = 〈e2, e5〉
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Images and Kernels
Before calculating these β-values we must first calculate ImΨπ andKerΨπ for each π ∈ B.
Consider the permutation π1 = (741) ∈ B.
1. KerΨπ1 = 〈e0, e1, e2, e3 + 2e4, e5, e6 + 2e7, e8〉2. ImΨπ1 = 〈e1 + e2, e4 + e5〉
Consider the permutation π2 = (852) ∈ B.
1. KerΨπ2 = 〈e0, e1, e2, e3 + e4, e4 + e5, e6 + e7, e7 + e8〉2. ImΨπ2 = 〈e2, e5〉
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π1〉. We wish to calculate β3(N).
1. First calculate ImΨπ1 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ1 ∩ F3 = 〈e1 + e2〉.2. Next calculate the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 .
I Note that Ψπ1 (e4 + e5) = e1 + e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + 2e7, e8〉.I We call this full pre-image Lπ1 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ1 ∩ F3 under Ψπ1 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π1〉. We wish to calculate β3(N).
1. First calculate ImΨπ1 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ1 ∩ F3 = 〈e1 + e2〉.
2. Next calculate the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 .I Note that Ψπ1 (e4 + e5) = e1 + e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + 2e7, e8〉.I We call this full pre-image Lπ1 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ1 ∩ F3 under Ψπ1 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π1〉. We wish to calculate β3(N).
1. First calculate ImΨπ1 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ1 ∩ F3 = 〈e1 + e2〉.2. Next calculate the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 .
I Note that Ψπ1 (e4 + e5) = e1 + e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + 2e7, e8〉.I We call this full pre-image Lπ1 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ1 ∩ F3 under Ψπ1 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π1〉. We wish to calculate β3(N).
1. First calculate ImΨπ1 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ1 ∩ F3 = 〈e1 + e2〉.2. Next calculate the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 .
I Note that Ψπ1 (e4 + e5) = e1 + e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + 2e7, e8〉.I We call this full pre-image Lπ1 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ1 ∩ F3 under Ψπ1 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π2〉. We wish to calculate β3(N).
1. First calculate ImΨπ2 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ2 ∩ F3 = 〈e2〉.2. Next calculate the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
I Note that Ψπ2 (e5) = e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + e7, e7 + e8〉.I We call this full pre-image Lπ2 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ2 ∩ F3 under Ψπ2 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π2〉. We wish to calculate β3(N).
1. First calculate ImΨπ2 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ2 ∩ F3 = 〈e2〉.
2. Next calculate the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .I Note that Ψπ2 (e5) = e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + e7, e7 + e8〉.I We call this full pre-image Lπ2 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ2 ∩ F3 under Ψπ2 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π2〉. We wish to calculate β3(N).
1. First calculate ImΨπ2 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ2 ∩ F3 = 〈e2〉.2. Next calculate the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
I Note that Ψπ2 (e5) = e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + e7, e7 + e8〉.I We call this full pre-image Lπ2 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ2 ∩ F3 under Ψπ2 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
One-Dimensional Example
Consider the subgroup N = 〈π2〉. We wish to calculate β3(N).
1. First calculate ImΨπ2 ∩ F3.I Recall that F3 = 〈e0, e1, e2〉.
Then ImΨπ2 ∩ F3 = 〈e2〉.2. Next calculate the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
I Note that Ψπ2 (e5) = e2.
The full pre-image is 〈e0, e1, e2, e3, e4, e5, e6 + e7, e7 + e8〉.I We call this full pre-image Lπ2 .
3. Finally, β3(N)
is the number of good functions in the full
pre-image of ImΨπ2 ∩ F3 under Ψπ2 . Hence β3(N)
= 2 · 37.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 62: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/62.jpg)
Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Two-Dimensional Example
Consider the subgroups N = 〈π1, π2〉. We wish to calculate β3(N).
1. As calculated in the previous two examples, defineI Lπ1 to be the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 andI Lπ2 to be the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
2. Then calculateL(N)
= Lπ1 ∩ Lπ2 = 〈e0, e1, e2, e3, e4, e5, e6 + e7 + e8〉.3. Finally, β3
(N)
is the number of good functions in L(N).
Hence β3(N)
= 2 · 36.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Two-Dimensional Example
Consider the subgroups N = 〈π1, π2〉. We wish to calculate β3(N).
1. As calculated in the previous two examples, defineI Lπ1 to be the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 andI Lπ2 to be the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
2. Then calculateL(N)
= Lπ1 ∩ Lπ2 = 〈e0, e1, e2, e3, e4, e5, e6 + e7 + e8〉.3. Finally, β3
(N)
is the number of good functions in L(N).
Hence β3(N)
= 2 · 36.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 64: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/64.jpg)
Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Two-Dimensional Example
Consider the subgroups N = 〈π1, π2〉. We wish to calculate β3(N).
1. As calculated in the previous two examples, defineI Lπ1 to be the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 andI Lπ2 to be the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
2. Then calculateL(N)
= Lπ1 ∩ Lπ2 = 〈e0, e1, e2, e3, e4, e5, e6 + e7 + e8〉.
3. Finally, β3(N)
is the number of good functions in L(N).
Hence β3(N)
= 2 · 36.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 65: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/65.jpg)
Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Two-Dimensional Example
Consider the subgroups N = 〈π1, π2〉. We wish to calculate β3(N).
1. As calculated in the previous two examples, defineI Lπ1 to be the full pre-image of ImΨπ1 ∩ F3 under Ψπ1 andI Lπ2 to be the full pre-image of ImΨπ2 ∩ F3 under Ψπ2 .
2. Then calculateL(N)
= Lπ1 ∩ Lπ2 = 〈e0, e1, e2, e3, e4, e5, e6 + e7 + e8〉.3. Finally, β3
(N)
is the number of good functions in L(N).
Hence β3(N)
= 2 · 36.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating α-values
For an integer j and a subgroup N of B define
αj
(N)
=∣∣∣µ ∈ B ∣∣∣ N = I
B
(µDj
)∣∣∣ .The next step in the algorithm is to calculate αj
(N)
for each
integer j and each subgroup N of B.
Define CN
to be the collection of all proper subgroups of B that
properly contain N. It then follows that
αj
(N)
= βj(N)−∑M∈C
N
αj
(M)
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating α-values
For an integer j and a subgroup N of B define
αj
(N)
=∣∣∣µ ∈ B ∣∣∣ N = I
B
(µDj
)∣∣∣ .The next step in the algorithm is to calculate αj
(N)
for each
integer j and each subgroup N of B.
Define CN
to be the collection of all proper subgroups of B that
properly contain N. It then follows that
αj
(N)
= βj(N)−∑M∈C
N
αj
(M)
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Matrix Representatives of Subgroups
We can identify a subgroup N of B with a matrix in Matp (Fp).To demonstrate this, observe the following example:
Consider the subgroup 〈π1, π2〉 = 〈(741) , (852)〉. Sinceη−1 (π1) = 0γ0 + 1γ1 + 0γ2 and η−1 (π2) = 0γ0 + 0γ1 + 1γ2 werepresent the subgroup 〈π1, π2〉 with the matrix0 1 0
0 0 10 0 0
.
We call (0, 1, 1) the pivot indicator of the above matrix.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Matrix Representatives of Subgroups
We can identify a subgroup N of B with a matrix in Matp (Fp).To demonstrate this, observe the following example:
Consider the subgroup 〈π1, π2〉 = 〈(741) , (852)〉. Sinceη−1 (π1) = 0γ0 + 1γ1 + 0γ2 and η−1 (π2) = 0γ0 + 0γ1 + 1γ2 werepresent the subgroup 〈π1, π2〉 with the matrix0 1 0
0 0 10 0 0
.
We call (0, 1, 1) the pivot indicator of the above matrix.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Matrix Representatives of Subgroups
We can identify a subgroup N of B with a matrix in Matp (Fp).To demonstrate this, observe the following example:
Consider the subgroup 〈π1, π2〉 = 〈(741) , (852)〉. Sinceη−1 (π1) = 0γ0 + 1γ1 + 0γ2 and η−1 (π2) = 0γ0 + 0γ1 + 1γ2 werepresent the subgroup 〈π1, π2〉 with the matrix0 1 0
0 0 10 0 0
.
We call (0, 1, 1) the pivot indicator of the above matrix.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating t-values
Define the set I to be the set of all p-tuples with entries in 0, 1.Then for ν ∈ I define the set M (ν) to be the set of all subgroupsof B whose matrix representative has pivot indicator ν.
For an integer j and a pivot indicator ν ∈ I we calculate
tj (ν) =∑
N∈M(ν)
αj
(N).
The next step in the algorithm is to calculate the value of tj (ν) foreach integer j and pivot indicator ν ∈ I.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating t-values
Define the set I to be the set of all p-tuples with entries in 0, 1.Then for ν ∈ I define the set M (ν) to be the set of all subgroupsof B whose matrix representative has pivot indicator ν.
For an integer j and a pivot indicator ν ∈ I we calculate
tj (ν) =∑
N∈M(ν)
αj
(N).
The next step in the algorithm is to calculate the value of tj (ν) foreach integer j and pivot indicator ν ∈ I.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating∣∣Bm
jk
∣∣-values
For integers k and m define Imk to be the set of all pivot indicatorsν ∈ I such that the first p − k entries in ν sum to m.
We can finally calculate ∣∣Bmjk ∣∣ =∑ν∈Imk
tj (ν)
for integers j , k , and m. Recall that∣∣Qm
jk
∣∣ is the number of faithful
irreducible characters of Hjk with degree pm+1
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
Calculating β-valuesCalculating α-valuesCalculating t-valuesCalculating |B|-values
Calculating∣∣Bm
jk
∣∣-values
For integers k and m define Imk to be the set of all pivot indicatorsν ∈ I such that the first p − k entries in ν sum to m.
We can finally calculate ∣∣Bmjk ∣∣ =∑ν∈Imk
tj (ν)
for integers j , k , and m. Recall that∣∣Qm
jk
∣∣ is the number of faithful
irreducible characters of Hjk with degree pm+1
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Comparison of Small Cases
1. When p = 3
I There are 26 non-trivial subgroups of B.I The algorithm can be (an has been) implemented through
hand calculations.
2. When p = 5I There are 42, 174 non-trivial sugroups of B.I The algorithm becomes too vast for hand calculations.
My project was then to write a computer program to implementthis algorithm in the case where p = 5.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Comparison of Small Cases
1. When p = 3I There are 26 non-trivial subgroups of B.I The algorithm can be (an has been) implemented through
hand calculations.
2. When p = 5I There are 42, 174 non-trivial sugroups of B.I The algorithm becomes too vast for hand calculations.
My project was then to write a computer program to implementthis algorithm in the case where p = 5.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 77: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/77.jpg)
Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Comparison of Small Cases
1. When p = 3I There are 26 non-trivial subgroups of B.I The algorithm can be (an has been) implemented through
hand calculations.
2. When p = 5
I There are 42, 174 non-trivial sugroups of B.I The algorithm becomes too vast for hand calculations.
My project was then to write a computer program to implementthis algorithm in the case where p = 5.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 78: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/78.jpg)
Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Comparison of Small Cases
1. When p = 3I There are 26 non-trivial subgroups of B.I The algorithm can be (an has been) implemented through
hand calculations.
2. When p = 5I There are 42, 174 non-trivial sugroups of B.I The algorithm becomes too vast for hand calculations.
My project was then to write a computer program to implementthis algorithm in the case where p = 5.
Dan Raies Faithful Irreducible Characters of the Wreath Product
![Page 79: Faithful Irreducible Characters of the Wreath Productpages.uoregon.edu/raies/LaTeX/Sample Beamer/defense.pdf · 2015-09-30 · Problem Statement Algorithm Implementation Group De](https://reader033.vdocuments.site/reader033/viewer/2022060508/5f2381e4f00bc922b015f1ae/html5/thumbnails/79.jpg)
Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Comparison of Small Cases
1. When p = 3I There are 26 non-trivial subgroups of B.I The algorithm can be (an has been) implemented through
hand calculations.
2. When p = 5I There are 42, 174 non-trivial sugroups of B.I The algorithm becomes too vast for hand calculations.
My project was then to write a computer program to implementthis algorithm in the case where p = 5.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Computational Difficulties
1. Memory management.
2. Basic computation and linear algebra over finite fields.
3. Computational subspace intersection.
4. Efficient looping.
5. Limits of primitive data structures.
6. Data verification.
Dan Raies Faithful Irreducible Characters of the Wreath Product
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Problem StatementAlgorithm
Implementation
p = 3 vs p = 5ProgrammingThe End
Thank You for Your Time
Any Questions?
Dan Raies Faithful Irreducible Characters of the Wreath Product