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Representation Theory 2016-2017 Example Sheet 3

Module

Representation Theory

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Academic year: 2016/2017
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PART II REPRESENTATION THEORY

SHEET 3

Unless otherwise stated, all groups here are finite, and all vector spaces are finite-dimensional over a fieldFof characteristic zero, usuallyC.

1 Recall the character table ofS 4 from Sheet 2. Find all the characters ofS 5 induced from the irreducible characters ofS 4. Hence find the complete character table ofS 5. Repeat, replacingS 4 by the subgroup〈(12345),(2354)〉of order 20 inS 5.

2 Recall the construction of the character table of the dihedral groupD 10 of order 10 from Sheet 2. (a) Use induction from the subgroupD 10 ofA 5 toA 5 to obtain the character table of A 5. (b) LetGbe the subgroup of SL 2 (F 5 ) consisting of upper triangular matrices. Compute the character table ofG. Hint: bear in mind that there is an isomorphismG/Z→D 10.

3 LetH be a subgroup of the groupG. Show that for every irreducible representation ρforGthere is an irreducible representationρ′forHwithρa component of the induced representation IndGHρ′. Prove that ifAis an abelian subgroup ofGthen every irreducible representation ofG has dimension at most|G:A|.

4 Obtain the character table of the dihedral groupD 2 mof order 2m, by using induction from the cyclic subgroupCm. [Hint: consider the casesmodd andmeven separately, as for meven there are two conjugacy classes of reflections, whereasformodd there is only one.]

5 Prove the transitivity of induction: ifH < K < Gthen

IndGKIndKHρ∼=IndGHρ

for any representationρofH.

6 (a) LetV =U⊕W be a direct sum ofCG-modules. Prove that both the symmetric square and the exterior square ofV have submodules isomorphic toU⊗W. (b) CalculateχΛ 2 ρandχS 2 ρ, whereρis the irreducible representation of dimension 2 of D 8 ; repeat this forQ 8. Which of these characters contains the trivial character inthe two cases?

7 Letρ:G→GL(V) be a representation ofGof dimensiond. (a) Compute the dimension ofSnV and ΛnV for alln. (b) Letg∈Gand letλ 1 ,...,λdbe the eigenvalues ofgonV. What are the eigenvalues ofgonSnV and ΛnV? (c) Letf(t) = det(g−tI) be the characteristic polynomial ofρ(g). What is the rela- tionship between the coefficients offandχΛnV? (d) Find a relationship betweenχSnV andf.

1

2 PART II REPRESENTATION THEORY SHEET 3

8 LetGbe the symmetric groupSnacting naturally on the setX={ 1 ,...,n}. For any integerr 6 n 2 , writeXrfor the set of allr-element subsets ofX, and letπrbe the permutation

character of the action ofGonXr. Observeπr(1) =|Xr|=

(n r

)

. If 0 6 ℓ 6 k 6 n/2, show that 〈πk,πℓ〉=ℓ+ 1.

Letm =n/2 ifnis even, and m = (n−1)/2 ifnis odd. Deduce thatSn has distinct irreducible charactersχ(n)= 1G,χ(n− 1 ,1),χ(n− 2 ,2),...,χ(n−m,m)such that for allr 6 m,

πr=χ(n)+χ(n− 1 ,1)+χ(n− 2 ,2)+· · ·+χ(n−r,r).

In particular the class functionsπr−πr− 1 are irreducible characters ofSnfor 1 6 r 6 n/ 2 and equal toχ(n−r,r).

9 Letρ:G→GL(V) be a complex representation forGaffording the characterχ. Give the characters of the representationsV⊗V,S 2 V and Λ 2 V in terms ofχ. (i) LetWbe another finite-dimensional representation with characterψ. Show that

dimWG=

1

|G|

∑

g∈G

ψ(g)

whereWG={w∈W:gw=wfor allg∈G}. (ii) Prove that ifVis irreducible,V⊗V contains the trivial representation at most once. (iii) Given any irreducible characterχofG, theindicatorιχofχis defined by

ιχ=

1

|G|

∑

x∈G

χ(x 2 ).

By using the decompositionV⊗V =S 2 V⊕Λ 2 V, deduce that

ιχ=

{

0 , ifχis not real-valued ± 1 , ifχis real-valued.

Deduce that if|G|is odd thenGhas only one real-valued irreducible character. [Remark. The sign +, resp.−, indicates whetherρ(G) preserves an orthogonal, respectively symplectic form onV, and whether or not the representation can be realised over the reals. You can read about it in Ch. 23 of James and Liebeck.]

10 Suppose thatGis a Frobenius group with Frobenius kernelK. Show that (i)CG(k) 6 Kfor all 1 6 =k∈K. (ii) ifχis a non-trivial irreducible character ofKthen IndGKχis also irreducible withK not lying in its kernel. Hence explain how to construct the character table ofG, given the character tables ofKandG/K. [Hints for (ii): (a) First, show each element ofG\K permutes the conjugacy classes inK, and fixes only the identity. (b) Deduce that each element ofG\Kfixes only the trivial character ofK. (c) Use the Orbit-Stabilizer theorem to deduce that ifχ is a non-trivial irreducible character ofKthen the number of distinct conjugates ofχis|G:K|. (d) Use Frobenius reciprocity to show that ifχ is as above and φis an irreducible constituent of IndGKχ, then all|G :K|conjugates ofχare constituents of ResGKφ. Finally compare degrees to get the result.]

11 Construct the character table of the symmetric groupS 6. Identify which of your char- acters are equal to the charactersχ(6),χ(5,1),χ(4,2),χ(3,3)constructed in question 8.

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Representation Theory 2016-2017 Example Sheet 3

Module: Representation Theory

8 Documents
Students shared 8 documents in this course

University: University of Cambridge

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PART II REPRESENTATION THEORY
SHEET 3
Unless otherwise stated, all groups here are finite, and all vector spaces are finite-dimensional
over a field Fof characteristic zero, usually C.
1Recall the character table of S4from Sheet 2. Find all the characters of S5induced from
the irreducible characters of S4. Hence find the complete character table of S5.
Repeat, replacing S4by the subgroup h(12345),(2354)iof order 20 in S5.
2Recall the construction of the character table of the dihedral group D10 of order 10 from
Sheet 2.
(a) Use induction from the subgroup D10 of A5to A5to obtain the character table of
A5.
(b) Let Gbe the subgroup of SL2(F5) consisting of upper triangular matrices. Compute
the character table of G.
Hint: bear in mind that there is an isomorphism G/Z →D10.
3Let Hbe a subgroup of the group G. Show that for every irreducible representation
ρfor Gthere is an irreducible representation ρ′for Hwith ρa component of the induced
representation IndG
Hρ′.
Prove that if Ais an abelian subgroup of Gthen every irreducible representation of G
has dimension at most |G:A|.
4Obtain the character table of the dihedral group D2mof order 2m, by using induction
from the cyclic subgroup Cm. [Hint: consider the cases modd and meven separately, as for
meven there are two conjugacy classes of reflections, whereas for modd there is only one.]
5Prove the transitivity of induction: if H < K < G then
IndG
KIndK
Hρ∼
=IndG
Hρ
for any representation ρof H.
6(a) Let V=U⊕Wbe a direct sum of CG-modules. Prove that both the symmetric
square and the exterior square of Vhave submodules isomorphic to U⊗W.
(b) Calculate χΛ2ρand χS2ρ, where ρis the irreducible representation of dimension 2 of
D8; repeat this for Q8. Which of these characters contains the trivial character in the two
cases?
7Let ρ:G→GL(V) be a representation of Gof dimension d.
(a) Compute the dimension of SnVand ΛnVfor all n.
(b) Let g∈Gand let λ1, . . . , λdbe the eigenvalues of gon V. What are the eigenvalues
of gon SnVand ΛnV?
(c) Let f(t) = det(g−tI) be the characteristic polynomial of ρ(g). What is the rela-
tionship between the coefficients of fand χΛnV?
(d) Find a relationship between χSnVand f.
1

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