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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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