Automorphisms and Characters of Finite Groups Brittany Bianco, - - PowerPoint PPT Presentation

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

Automorphisms and Characters of Finite Groups

Brittany Bianco, Leigh Foster Mentor: Mandi A. Schaeffer Fry Metropolitan State University of Denver Nebraska Conference for Undergraduate Women in Mathematics January 26, 2019

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

BIG IDEA

Fixed Notation

◮ G = Sp4(q) where q is a power of an odd prime, p ◮ H = {diag(a, a−1, b, b−1) | a, b ∈ F∗ q} a subgroup of G ◮ ϕm p is a “field automorphism” of G ◮ σ is an automorphism of Q(e2πi/|G|)

Theorem

Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′(G) is also fixed by σ.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

Sp4(q) = {g is an invertible 4 × 4 matrix over Fq | gTJg = J} where J =     1 −1 1 −1    

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

◮ An identity element, e

∃ e ∈ G s.t. ∀ a ∈ G, a ⋆ e = e ⋆ a = a.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

◮ An identity element, e

∃ e ∈ G s.t. ∀ a ∈ G, a ⋆ e = e ⋆ a = a.

◮ An inverse for every group element

∀ a ∈ G, ∃ b ∈ G (or a−1) s.t. a ⋆ b = b ⋆ a = e

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

◮ An identity element, e

∃ e ∈ G s.t. ∀ a ∈ G, a ⋆ e = e ⋆ a = a.

◮ An inverse for every group element

∀ a ∈ G, ∃ b ∈ G (or a−1) s.t. a ⋆ b = b ⋆ a = e

under the binary operation ⋆

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

◮ An identity element, e

∃ e ∈ G s.t. ∀ a ∈ G, a ⋆ e = e ⋆ a = a.

◮ An inverse for every group element

∀ a ∈ G, ∃ b ∈ G (or a−1) s.t. a ⋆ b = b ⋆ a = e

under the binary operation ⋆ Example: Z12 under addition

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

By definition, a group (G, ⋆) has:

◮ Associativity

∀ a, b, c ∈ G, (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c)

◮ An identity element, e

∃ e ∈ G s.t. ∀ a ∈ G, a ⋆ e = e ⋆ a = a.

◮ An inverse for every group element

∀ a ∈ G, ∃ b ∈ G (or a−1) s.t. a ⋆ b = b ⋆ a = e

under the binary operation ⋆ Example: Z12 under addition Non-Example: Z12 under multiplication

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

So Sp4(q) is a group?

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

So Sp4(q) is a group? Recall Sp4(q) = {g is an invertible 4 × 4 matrix over Fq | gTJg = J}.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

So Sp4(q) is a group? Recall Sp4(q) = {g is an invertible 4 × 4 matrix over Fq | gTJg = J}.

◮ Associativity

Matrix multiplication is associative

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

So Sp4(q) is a group? Recall Sp4(q) = {g is an invertible 4 × 4 matrix over Fq | gTJg = J}.

◮ Associativity

Matrix multiplication is associative

◮ An identity element, e

e = I, the identity matrix since ITJI = J

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

So Sp4(q) is a group? Recall Sp4(q) = {g is an invertible 4 × 4 matrix over Fq | gTJg = J}.

◮ Associativity

Matrix multiplication is associative

◮ An identity element, e

e = I, the identity matrix since ITJI = J

◮ An inverse for every group element

Since g−1 also satisfies the group definition: (g−1)TJ(g−1) = J then every element has an inverse.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A subgroup H is a subset of group elements of a group G that is itself a group under the group operation.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A subgroup H is a subset of group elements of a group G that is itself a group under the group operation. Example: The evens mod 12 forms a subgroup of Z12 under addition.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A subgroup H is a subset of group elements of a group G that is itself a group under the group operation. Example: The evens mod 12 forms a subgroup of Z12 under addition. Non-Example: The odds mod 12 do not form a subgroup of Z12 under addition.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H ={diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A diagonal matrix has zeros everywhere except the main diagonal.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H ={diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A diagonal matrix has zeros everywhere except the main diagonal. So diag(a, a−1, b, b−1) is the diagonal matrix     a a−1 b b−1     With entries a, b ∈ F∗

q

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

But why is H a subgroup of G? If we let g =

    a a−1 b b−1     and J =     1 −1 1 −1     then

gTJg =    

a a−1 b b−1

   

1 −1 1 −1

     

a a−1 b b−1

  =  

a −a−1 b −b−1

   

a a−1 b b−1

  = J

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

But why is H a subgroup of G? Let A =  

a1 a−1

1

b1 b−1

1

  and B =  

a2 a−1

2

b2 b−1

2

 , then AB =  

a1a2 (a1a2)−1 b1b2 (b1b2)−1

  Thus H is closed under the group operation from G, so H is a subgroup of G.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A homomorphism is a function of one group to another that preserves the group operation. So for groups (G, ⋆) and (¯ G, ∗), then for any g1, g2 ∈ G f(g1 ⋆ g2) = f(g1) ∗ f(g2)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

A homomorphism is a function of one group to another that preserves the group operation. So for groups (G, ⋆) and (¯ G, ∗), then for any g1, g2 ∈ G f(g1 ⋆ g2) = f(g1) ∗ f(g2) An automorphism is a bijective homomorphism from a group G

• nto itself.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

ϕm

p is an automorphism of Sp4(q) that raises all entries of its

• perand to the power pm.

Example: let B3(i, s) ∈ G such that B3(i, s) =

    γi γ−i γs γ−s    

(where γ is a q − 1 root of 1 in F∗

q)

(as defined in Srinivasan [3, Srinivasan 1968].)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Fixed Notation

• G = Sp4(q) where q is a power of an odd prime, p
• H = {diag(a, a−1, b, b−1) | a, b ∈ F∗

q } a subgroup of G

• ϕm

p is a “field automorphism” of G

• σ is an automorphism of Q(e2πi/|G|)

ϕm

p is an automorphism of Sp4(q) that raises all entries of its

• perand to the power pm.

Example: let B3(i, s) ∈ G such that B3(i, s) =

    γi γ−i γs γ−s    

(where γ is a q − 1 root of 1 in F∗

q)

(as defined in Srinivasan [3, Srinivasan 1968].)

Observe ϕp(B3(i, s)) : ϕp    

γi γ−i γs γ−s

    =  

γip γ−ip γsp γ−sp

  = B3(ip, sp)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

REPRESENTATION

A representation is a homomorphism ρ from a group G into a group of n × n invertible matrices with entries in C. ρ : G → GLn(C) such that ρ(gh) = ρ(g)ρ(h) for all g, h ∈ G (where GLn(C) is the group of n × n invertible matrices with entries in C)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

TRACE

The trace of a matrix is the sum of its diagonal entries. So Tr(An×n) = a11 + a22 + . . . + ann.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

TRACE

The trace of a matrix is the sum of its diagonal entries. So Tr(An×n) = a11 + a22 + . . . + ann. So if h =     a a−1 b b−1     Then Tr(h) = a + a−1 + b + b−1.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

CHARACTER

A character χ is the composition of the trace function with the representation of a group element. χ = Tr ◦ρ χ(g) = Tr(ρ(g))

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

RECALL...

Theorem

Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′(G) is also fixed by σ.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

χ is irreducible if χ = χ1 + χ2 for characters χ1, χ2 Irr(H) is the set of irreducible characters of H.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

χ is irreducible if χ = χ1 + χ2 for characters χ1, χ2 Irr(H) is the set of irreducible characters of H. Irr(q−1)′(G) is the set of irreducible characters of G such that n is relatively prime to the quantity (q − 1).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Given the automorphism of ϕm

p of G and χ ∈ Irr(G), we can

• btain a new irreducible character ϕm

p χ via

ϕm

p χ(g) = χ(ϕm p (g))

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

ϕm

p is an automorphism of Sp4(q) that raises all entries of its operand

to the power pm. So ϕm

p (χ(g)) = χ(ϕm p (g)).

Looking at χ8(k), we claim that ϕp(χ8(k)) = χ8(kp).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

ϕm

p is an automorphism of Sp4(q) that raises all entries of its operand

to the power pm. So ϕm

p (χ(g)) = χ(ϕm p (g)).

Looking at χ8(k), we claim that ϕp(χ8(k)) = χ8(kp). For example, we know that ϕp(B3(i, s)) = B3(ip, sp). Now let χ = χ8(k) and g = B3(i, s), where χ8 is a character of G.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

ϕm

p is an automorphism of Sp4(q) that raises all entries of its operand

to the power pm. So ϕm

p (χ(g)) = χ(ϕm p (g)).

Looking at χ8(k), we claim that ϕp(χ8(k)) = χ8(kp). For example, we know that ϕp(B3(i, s)) = B3(ip, sp). Now let χ = χ8(k) and g = B3(i, s), where χ8 is a character of G. Notice that χ(g) = (˜ γik + ˜ γ−ik)(˜ γsk + ˜ γ−sk) (where ˜ γ is a q − 1 root of 1 in C.) [3]

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

ϕm

p is an automorphism of Sp4(q) that raises all entries of its operand

to the power pm. So ϕm

p (χ(g)) = χ(ϕm p (g)).

Looking at χ8(k), we claim that ϕp(χ8(k)) = χ8(kp). For example, we know that ϕp(B3(i, s)) = B3(ip, sp). Now let χ = χ8(k) and g = B3(i, s), where χ8 is a character of G. Notice that χ(g) = (˜ γik + ˜ γ−ik)(˜ γsk + ˜ γ−sk) (where ˜ γ is a q − 1 root of 1 in C.) [3] Consider ϕp(χ(g)) = χ(ϕp(g)) = (˜ γikp + ˜ γ−ikp)(˜ γskp + ˜ γ−skp)

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

ϕm

p is an automorphism of Sp4(q) that raises all entries of its operand

to the power pm. So ϕm

p (χ(g)) = χ(ϕm p (g)).

Looking at χ8(k), we claim that ϕp(χ8(k)) = χ8(kp). For example, we know that ϕp(B3(i, s)) = B3(ip, sp). Now let χ = χ8(k) and g = B3(i, s), where χ8 is a character of G. Notice that χ(g) = (˜ γik + ˜ γ−ik)(˜ γsk + ˜ γ−sk) (where ˜ γ is a q − 1 root of 1 in C.) [3] Consider ϕp(χ(g)) = χ(ϕp(g)) = (˜ γikp + ˜ γ−ikp)(˜ γskp + ˜ γ−skp) So ϕp(χ8(k)(g)) = χ8(kp)(g).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

A ϕm

p -invariant character is one which can go through ϕm p and

come out equal to itself as before the operation. So if χ is ϕm

p -invariant, then ϕm p (χ) = χ.

For example, if χ8(k) is fixed by ϕm

p , then its values are

Q-combinations of pm − 1 roots of unity.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Recall Q(e2πi/|G|).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Recall Q(e2πi/|G|). That is, the rational numbers plus the |G|th-roots of unity.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Recall Q(e2πi/|G|). That is, the rational numbers plus the |G|th-roots of unity. Fun Fact: Although all of our characters do live in C, we can actually restrict that to Q(e2πi/|G|).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Given an automorphism σ of Q(e2πi/|G|) and an irreducible character χ of G, we have another irreducible character (σχ) given by (σχ)(g) = σ(χ(g))

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Given an automorphism σ of Q(e2πi/|G|) and an irreducible character χ of G, we have another irreducible character (σχ) given by (σχ)(g) = σ(χ(g)) Recall (ϕm

p χ)(g) = χ(ϕm p (g))

Note that applying σ to χ behaves differently than applying ϕm

p

to χ.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

WE MADE IT!

Theorem

Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′(G) is also fixed by σ.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

THE LOCAL SIDE

Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Xk : F∗

q → C∗ is an irreducible representation of F∗ q, where

Xk(γ) = ˜ γk, where γ is a q − 1 root of 1 in F∗

q.

Lemma

If ϕm

p fixes Xk then ˜

γk is a pm − 1 root of 1.

(where ˜ γ is a q − 1 root of 1 in C.)

All characters of H can be obtained from those of the form Xk.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

THE LOCAL SIDE

Theorem (⋆) Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Lemma

Under assumption (⋆), then every pm − 1 root of 1 is σ-fixed.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

THE GLOBAL SIDE

Theorem Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Assume a character of G is fixed by ϕm

p .

Consider χ8: Recall that when χ8(k) is fixed by ϕm

p , then its values are

Q-combinations of pm − 1 roots of unity.

Lemma

Then its values are in Q(e2πi/(pm−1)).

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

THE GLOBAL SIDE

Theorem (⋆) Assume every ϕm

p -invariant member of Irr(H) is also fixed by σ.

Then every ϕm

p -invariant member of Irr(q−1)′ (G) is also fixed by σ.

Theorem

Under assumption (⋆), if χ8(k) is fixed by ϕm

p then χ8(k) is also fixed

by σ. This, with the previous lemmas, proves our theorem for χ8(k); the proofs for the other members of Irr(q−1)′(G) are similar.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

FUTURE DIRECTION

Conjecture

Let ℓ be an odd prime and let P be a Sylow ℓ-subgroup of G such that ϕp(g) ∈ P for each g ∈ P. Let m be a positive integer and assume every ϕm

p -invariant member of Irr(P) is also fixed by σℓ. Then every

ϕm

p -invariant member of Irrℓ′(G) is also fixed by σℓ. (Here σℓ is a

specific automorphism of Q(e2πi/|G|) depending on ℓ.) [3][1][2]

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

REFERENCES

Joseph A. Gallian. Contemporary Abstract Algebra. Houghton Miffllin, Boston, Massachusetts, 2002. Gordon James and Martin Liebeck. Representations and characters of groups. Cambridge University Press, New York, second edition, 2001. Bhama Srinivasan. The characters of the finite symplectic group Sp(4, q).

• Trans. Amer. Math. Soc., 131:488–525, 1968.

BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES

ACKNOWLEDGEMENTS

◮ National Science Foundation (Award No. DMS-1801156) ◮ Metropolitan State University of Denver ◮ Dr. Diane Davis ◮ Dr. Mandi A. Schaeffer Fry ◮ NCUWM Conference