Automorphisms and Characters of Finite GroupsAutomorphisms and Characters of Finite Groups Brittany Bianco, Leigh Foster Mentor: Mandi A. Schaeffer Fry Metropolitan State University

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


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.


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


• 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





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


















































Example: Z12 under addition

Non-Example: Z12 under multiplication















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.





So Sp4(q) is a group?RecallSp4(q) = {g is an invertible 4× 4 matrix over Fq | gTJg = J}.
































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.






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.






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.


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



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


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



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





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





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.





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.





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.





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)





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)


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)


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.


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.


CHARACTER

A character χ is the composition of the trace function with therepresentation of a group element.

χ = Tr ◦ρχ(g) = Tr(ρ(g))


RECALL...






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).





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).





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))





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).







p (g)).







So ϕp(χ8(k)(g)) = χ8(kp)(g).







p (g)).







So ϕp(χ8(k)(g)) = χ8(kp)(g).







p (g)).







So ϕp(χ8(k)(g)) = χ8(kp)(g).







p (g)).







So ϕp(χ8(k)(g)) = χ8(kp)(g).





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.





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|).



















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.





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.


WE MADE IT!





THE LOCAL SIDETheoremAssume every ϕm



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.


THE LOCAL SIDETheorem(?) Assume every ϕm



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


THE GLOBAL SIDETheoremAssume every ϕm



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)).


THE GLOBAL SIDETheorem(?) Assume every ϕm



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.


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]


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.


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

Documents

Automorphisms and Characters of Finite GroupsAutomorphisms and Characters of Finite Groups Brittany Bianco, Leigh Foster Mentor: Mandi A. Schaeffer Fry Metropolitan State University