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BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
Automorphisms and Characters of FiniteGroups
Brittany Bianco, Leigh FosterMentor: Mandi A. Schaeffer Fry
Metropolitan State University of Denver
Nebraska Conference for Undergraduate Women inMathematics
January 26, 2019
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
BIG IDEA
Fixed Notation
I G = Sp4(q) where q is a power of an odd prime, pI H = {diag(a, a−1, b, b−1) | a, b ∈ F∗q} a subgroup of GI ϕm
p is a “field automorphism” of G
I σ is an automorphism of Q(e2πi/|G|)
TheoremAssume 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 =
0 1 0 0−1 0 0 00 0 0 10 0 −1 0
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:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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|)
By definition, a group (G, ?) has:
I Associativity∀ a, b, c ∈ G, (a ? b) ? c = a ? (b ? c)
I An identity element, e∃ e ∈ G s.t. ∀ a ∈ G, a ? e = e ? a = a.
I 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 additionNon-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?
RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
I AssociativityMatrix multiplication is associative
I An identity element, ee = I, the identity matrix since ITJI = J
I An inverse for every group elementSince g−1 also satisfies the group definition: (g−1)TJ(g−1) = Jthen 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|)
So Sp4(q) is a group?RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
I AssociativityMatrix multiplication is associative
I An identity element, ee = I, the identity matrix since ITJI = J
I An inverse for every group elementSince g−1 also satisfies the group definition: (g−1)TJ(g−1) = Jthen 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|)
So Sp4(q) is a group?RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
I AssociativityMatrix multiplication is associative
I An identity element, ee = I, the identity matrix since ITJI = J
I An inverse for every group elementSince g−1 also satisfies the group definition: (g−1)TJ(g−1) = Jthen 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|)
So Sp4(q) is a group?RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
I AssociativityMatrix multiplication is associative
I An identity element, ee = I, the identity matrix since ITJI = J
I An inverse for every group elementSince g−1 also satisfies the group definition: (g−1)TJ(g−1) = Jthen 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|)
So Sp4(q) is a group?RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
I AssociativityMatrix multiplication is associative
I An identity element, ee = I, the identity matrix since ITJI = J
I An inverse for every group elementSince g−1 also satisfies the group definition: (g−1)TJ(g−1) = Jthen 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 isitself a group under the group operation.
Example: The evens mod 12 forms a subgroup of Z12 underaddition.Non-Example: The odds mod 12 do not form a subgroup of Z12under 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 isitself a group under the group operation.
Example: The evens mod 12 forms a subgroup of Z12 underaddition.
Non-Example: The odds mod 12 do not form a subgroup of Z12under 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 isitself a group under the group operation.
Example: The evens mod 12 forms a subgroup of Z12 underaddition.Non-Example: The odds mod 12 do not form a subgroup of Z12under 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 maindiagonal.
So diag(a, a−1, b, b−1) is the diagonal matrixa 0 0 00 a−1 0 00 0 b 00 0 0 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|)
A diagonal matrix has zeros everywhere except the maindiagonal.So diag(a, a−1, b, b−1) is the diagonal matrix
a 0 0 00 a−1 0 00 0 b 00 0 0 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 0 0 00 a−1 0 00 0 b 00 0 0 b−1
and J =
0 1 0 0−1 0 0 00 0 0 10 0 −1 0
then
gTJg =
a 0 0 00 a−1 0 00 0 b 00 0 0 b−1
0 1 0 0−1 0 0 00 0 0 10 0 −1 0
a 0 0 00 a−1 0 00 0 b 00 0 0 b−1
=
0 a 0 0−a−1 0 0 0
0 0 0 b0 0 −b−1 0
a 0 0 00 a−1 0 00 0 b 00 0 0 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 0 0 00 a−1
1 0 00 0 b1 00 0 0 b−1
1
and B =
a2 0 0 00 a−1
2 0 00 0 b2 00 0 0 b−1
2
, then
AB =
a1a2 0 0 00 (a1a2)
−1 0 00 0 b1b2 00 0 0 (b1b2)
−1
Thus H is closed under the group operation from G, so H is asubgroup 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 thatpreserves 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 Gonto 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|)
A homomorphism is a function of one group to another thatpreserves 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 Gonto 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|)
ϕmp is an automorphism of Sp4(q) that raises all entries of its
operand to the power pm.
Example: let B3(i, s) ∈ G such that B3(i, s) =
γi 0 0 00 γ−i 0 00 0 γs 00 0 0 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 0 0 00 γ−i 0 00 0 γs 00 0 0 γ−s
=
γip 0 0 00 γ−ip 0 00 0 γsp 00 0 0 γ−sp
= B3(ip, sp)
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|)
ϕmp is an automorphism of Sp4(q) that raises all entries of its
operand to the power pm.
Example: let B3(i, s) ∈ G such that B3(i, s) =
γi 0 0 00 γ−i 0 00 0 γs 00 0 0 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 0 0 00 γ−i 0 00 0 γs 00 0 0 γ−s
=
γip 0 0 00 γ−ip 0 00 0 γsp 00 0 0 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 agroup of n× n invertible matrices with entries in C.
ρ : G→ GLn(C) such thatρ(gh) = ρ(g)ρ(h) for all g, h ∈ G
(where GLn(C) is the group of n× n invertible matrices withentries 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.
So if h =
a 0 0 00 a−1 0 00 0 b 00 0 0 b−1
Then Tr(h) = a + a−1 + b + b−1.
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 0 0 00 a−1 0 00 0 b 00 0 0 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 therepresentation of a group element.
χ = Tr ◦ρχ(g) = Tr(ρ(g))
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
RECALL...
TheoremAssume 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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
χ is irreducible if χ 6= 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 isrelatively prime to the quantity (q− 1).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
χ is irreducible if χ 6= 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 isrelatively prime to the quantity (q− 1).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
Given the automorphism of ϕmp of G and χ ∈ Irr(G), we can
obtain a new irreducible character ϕmp χ via
ϕmp χ(g) = χ(ϕm
p (g))
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
ϕmp is an automorphism of Sp4(q) that raises all entries of its operand
to the power pm. So ϕmp (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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
ϕmp is an automorphism of Sp4(q) that raises all entries of its operand
to the power pm. So ϕmp (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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
ϕmp is an automorphism of Sp4(q) that raises all entries of its operand
to the power pm. So ϕmp (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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
ϕmp is an automorphism of Sp4(q) that raises all entries of its operand
to the power pm. So ϕmp (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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
ϕmp is an automorphism of Sp4(q) that raises all entries of its operand
to the power pm. So ϕmp (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
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
A ϕmp -invariant character is one which can go through ϕm
p andcome out equal to itself as before the operation.
So if χ is ϕmp -invariant, then ϕm
p (χ) = χ.
For example, if χ8(k) is fixed by ϕmp , then its values are
Q-combinations of pm − 1 roots of unity.
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -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 canactually restrict that to Q(e2πi/|G|).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -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 canactually restrict that to Q(e2πi/|G|).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -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 canactually restrict that to Q(e2πi/|G|).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
Given an automorphism σ of Q(e2πi/|G|) and an irreduciblecharacter χ of G,we have another irreducible character (σχ) given by
(σχ)(g) = σ(χ(g))
Recall(ϕm
p χ)(g) = χ(ϕmp (g))
Note that applying σ to χ behaves differently than applying ϕmp
to χ.
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
TheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
Given an automorphism σ of Q(e2πi/|G|) and an irreduciblecharacter χ of G,we have another irreducible character (σχ) given by
(σχ)(g) = σ(χ(g))
Recall(ϕm
p χ)(g) = χ(ϕmp (g))
Note that applying σ to χ behaves differently than applying ϕmp
to χ.
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
WE MADE IT!
TheoremAssume 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 SIDETheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
Xk : F∗q → C∗ is an irreducible representation of F∗q , whereXk(γ) = γk, where γ is a q− 1 root of 1 in F∗
q .
LemmaIf 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 SIDETheorem(?) Assume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
LemmaUnder assumption (?), then every pm − 1 root of 1 is σ-fixed.
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
THE GLOBAL SIDETheoremAssume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
Assume a character of G is fixed by ϕmp .
Consider χ8:Recall that when χ8(k) is fixed by ϕm
p , then its values areQ-combinations of pm − 1 roots of unity.
LemmaThen its values are in Q(e2πi/(pm−1)).
BACKGROUND: NOTATION PRE-THEOREM BACKGROUND: THEOREM RESULTS REFERENCES
THE GLOBAL SIDETheorem(?) Assume every ϕm
p -invariant member of Irr(H) is also fixed by σ.
Then every ϕmp -invariant member of Irr(q−1)′ (G) is also fixed by σ.
TheoremUnder assumption (?), if χ8(k) is fixed by ϕm
p then χ8(k) is also fixedby 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 ` be an odd prime and let P be a Sylow `-subgroup of G such thatϕp(g) ∈ P for each g ∈ P. Let m be a positive integer and assumeevery ϕm
p -invariant member of Irr(P) is also fixed by σ`. Then everyϕm
p -invariant member of Irr`′(G) is also fixed by σ`. (Here σ` is aspecific automorphism of Q(e2πi/|G|) depending on `.)
[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
I National Science Foundation (Award No. DMS-1801156)I Metropolitan State University of DenverI Dr. Diane DavisI Dr. Mandi A. Schaeffer FryI NCUWM Conference
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