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Representation Theory 2016-2017 Example Sheet 4

Module

Representation Theory

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Academic year: 2016/2017
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PART II REPRESENTATION THEORY

SHEET 4

Unless otherwise stated, all vector spaces are finite-dimensional overC. In the first seven questions we letG= SU(2). Questions 9 onwards deal with a variety of topics at Tripos standard.

1 LetVn be the vector space of complex homogeneous polynomials of degreenin the variablesxandy. Describe a representationρnofGonVnand show that it is irreducible. What is its character? Show thatVnis isomorphic to its dualVn∗.

2 Decompose the representationV 4 ⊗V 3 into irreducibleG-spaces (that is, find a direct sum of irreducible representations which is isomorphic toV 4 ⊗V 3 ; in this and the following questions, you are not being asked to find such an isomorphismexplicitly). DecomposeV 1 ⊗n into irreducibles.

3 Determine the character ofSnV 1 forn>1. DecomposeS 2 Vnand Λ 2 Vninto irreducibles forn>1. DecomposeS 3 V 2 into irreducibles.

4 LetGact on the space M 3 (C) of 3×3 complex matrices, by conjugation:

A: X 7→ A 1 XA− 11 ,

whereA 1 is the 3×3 block diagonal matrix with block diagonal entriesA,1. Show that this gives a representation ofGand decompose it into irreducibles.

5 Letχnbe the character of the irreducible representationρn ofGon Vn of dimension n+ 1. Show that 1 2 π

∫ 2 π

0

K(z)χnχmdθ=δnm,

wherez=eiθandK(z) = 12 (z−z− 1 )(z− 1 −z). [ Note that all you need to know about integrating on the circleis orthogonality of characters: 1 2 π

∫ 2 π 0 z

ndθ=δn, 0. This is really a question about Laurent polynomials. ]

6 Check that the usual formula for integrating functions defined onS 3 ⊆R 4 defines a G-invariant inner product on the vector space of integrable functions on

G= SU(2) =

{(

a b − ̄b a ̄

)

:aa ̄+b ̄b= 1

}

,

and normalize it so that the integral over the group is one.

7 Compute the character of the representationSnV 2 of G for any n > 0. Calculate dimC(SnV 2 )G(by which we mean the subspace ofSnV 2 whereGacts trivially). Deduce that the ring of complex polynomials in three variablesx,y,zwhich are invariant under the action of SO(3) is a polynomial ring. Find a generator for this polynomial ring.

1

2 PART II REPRESENTATION THEORY SHEET 4

8 (a) LetGbe a compact group. Show that there is a continuous group homomorphism ρ:G→O(n) if and only ifGhas ann-dimensional representation overR. Here O(n) denotes the subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form. (b) Explicitly construct such a representationρ: SU(2)→SO(3) by showing that SU(2) acts on the vector space of matrices of the form { A=

(

a b c −a

)

∈M 2 (C) :A+At= 0

}

by conjugation. Show that this subspace is isomorphic toR 3 , that (A,B)7→ −tr(AB) is an invariant positive definite symmetric bilinear form, and thatρis surjective with kernel{±I}.

9 TheHeisenberg groupof orderp 3 is the (non-abelian) subgroup

G=

1 a x 0 1 b 0 0 1

:a,b,x∈Fp

.

of matrices over the finite fieldFp(pprime). LetHbe the subgroup ofGcomprising matrices witha= 0 andZbe the subgroup ofGof matrices witha=b= 0. (a) Show thatZ=Z(G), the centre ofG, and thatG/Z =F 2 p. Note that this implies that the derived subgroupG′is contained inZ. [You can check by explicit computation that it equalsZ, or you can deduce this from the list of irreducible representations found in (d) below.] (b) Find all 1-dimensional representations ofG. (c) Letψ:Fp→C×be a non-trivial 1-dimensional representation of the cyclic group Fp=Z/p, and define a 1-dimensional representationρψofHby

ρψ

1 0 x 0 1 b 0 0 1

=ψ(x).

Show that IndGHρψis an irreducible representation ofG. (d) Prove that the collection of representations constructed in (b) and (c) gives a com- plete list of all irreducible representations. (e) Determine the character of the irreducible representation IndGHρψ.

10 Recall the character table ofG = PSL 2 (7) from Sheet 2, q. Identify the columns corresponding to the elementsxandywherexis an element of order 7 (eg the unitriangular matrix with 1 above the diagonal) andyis an element of order 3 (eg the diagonal matrix with entries 4 and 2). The groupGacts as a permutation group of degree 8 on the set of Sylow 7-subgroups (or the set of 1-dimensional subspaces of the vector space (F 7 ) 2 ). Obtain the permutation character of this action and decompose it into irreducible characters. *(Harder) Show that the groupGis generated by an element of order 2 and an element of order 3 whose product has order 7. [Hint: for the last part use the formula that the number of pairs of elements conjugate tox andyrespectively, whose product is conjugate tot, equalsc

∑

χ(x)χ(y)χ(t− 1 )/χ(1),where the sum runs over all the irreducible characters ofG, andc=|G| 2 (|CG(x)||CG(y)||CG(t)|)− 1 .]

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Representation Theory 2016-2017 Example Sheet 4

Module: Representation Theory

8 Documents
Students shared 8 documents in this course

University: University of Cambridge

Was this document helpful?
PART II REPRESENTATION THEORY
SHEET 4
Unless otherwise stated, all vector spaces are finite-dimensional over C. In the first seven
questions we let G= SU(2).Questions 9 onwards deal with a variety of topics at Tripos
standard.
1Let Vnbe the vector space of complex homogeneous polynomials of degree nin the
variables xand y. Describe a representation ρnof Gon Vnand show that it is irreducible.
What is its character? Show that Vnis isomorphic to its dual V∗
n.
2Decompose the representation V4⊗V3into irreducible G-spaces (that is, find a direct
sum of irreducible representations which is isomorphic to V4⊗V3; in this and the following
questions, you are not being asked to find such an isomorphism explicitly). Decompose V⊗n
1
into irreducibles.
3Determine the character of SnV1for n>1.
Decompose S2Vnand Λ2Vninto irreducibles for n>1.
Decompose S3V2into irreducibles.
4Let Gact on the space M3(C) of 3 ×3 complex matrices, by conjugation:
A:X7→ A1XA−1
1,
where A1is the 3 ×3 block diagonal matrix with block diagonal entries A, 1. Show that this
gives a representation of Gand decompose it into irreducibles.
5Let χnbe the character of the irreducible representation ρnof Gon Vnof dimension
n+ 1.
Show that 1
2πZ2π
0
K(z)χnχmdθ =δnm,
where z=eiθ and K(z) = 1
2(z−z−1)(z−1−z).
[ Note that all you need to know about integrating on the circle is orthogonality of characters:
1
2πR2π
0zndθ =δn,0. This is really a question about Laurent polynomials. ]
6Check that the usual formula for integrating functions defined on S3⊆R4defines a
G-invariant inner product on the vector space of integrable functions on
G= SU(2) =  a b
−¯
b¯a:a¯a+b¯
b= 1,
and normalize it so that the integral over the group is one.
7Compute the character of the representation SnV2of Gfor any n>0. Calculate
dimC(SnV2)G(by which we mean the subspace of SnV2where Gacts trivially).
Deduce that the ring of complex polynomials in three variables x, y, z which are invariant
under the action of SO(3) is a polynomial ring. Find a generator for this polynomial ring.
1

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