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Representation Theory 2016-2017 Example Sheet 1

Module

Representation Theory

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Academic year: 2016/2017
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PART II REPRESENTATION THEORY

SHEET 1

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

1 Letρbe a representation of the groupG. (a) Show thatδ:g7→detρ(g) is a 1-dimensional representation ofG. (b) Prove thatG/kerδis abelian. (c) Assume thatδ(g) =−1 for someg∈G. Show thatGhas a normal subgroup of index 2.

2 Letθ :G →F×be a 1-dimensional representation of the groupG, and letρ: G→ GL(V) be another representation. Show thatθ⊗ρ:G→GL(V) given byθ⊗ρ:g7→θ(g)·ρ(g) is a representation ofG, and that it is irreducible if and only ifρis irreducible.

3 Find an example of a representation of some finite group over some field of charac- teristicp, which is not completely reducible. Find an example of such a representation in characteristic 0 for an infinite group.

4 LetN be a normal subgroup of the groupG. Given a representation of the quotient G/N, use it to obtain a representation ofG. Which representations ofGdo you get this way? Recall that the derived subgroupG′ofGis the unique smallest normal subgroup ofG such thatG/G′ is abelian. Show that the 1-dimensional complex representations ofGare precisely those obtained fromG/G′.

5 Describe Weyl’s unitary trick. LetGbe a finite group acting on a complex vector spaceV, and let〈, 〉:V ×V →C be a skew-symmetric form, i.〈y, x〉=−〈x, y〉for allx, yinV. Show that the form (x, y) =|G 1 |

∑

〈gx, gy〉, where the sum is over all elementsg∈G, is

aG-invariant skew-symmetric form. Does this imply that every finite subgroup of GL 2 m(C) is conjugate to a subgroup of the symplectic group 1 Sp 2 m(C)?

6 LetG=〈g〉be a cyclic group of ordern. (i)Gacts onR 2 as symmetries of the regularn-gon. Choose a basis ofR 2 , and write the matrixR(g) representing the action of a generatorgin this basis. Is this an irreducible representation? (ii) Now regardR(g) above as a complex matrix, so that we get a representation ofG onC 2. DecomposeC 2 into its irreducible summands.

7 LetGbe a cyclic group of ordern. Explicitly decompose the complex regular represen- tation ofGas a direct sum of 1-dimensional representations, by givingthe matrix of change of coordinates from the natural basis{eg}g∈Gto a basis where the group action is diagonal.

1 the group of all linear transformations of a 2m-dimensional vector space overCthat preserve a non-

degenerate, skew-symmetric, bilinear form. 1

2 PART II REPRESENTATION THEORY SHEET 1

8 LetGbe the dihedral groupD 10 of order 10,

D 10 =〈x, y : x 5 = 1 =y 2 , yxy− 1 =x− 1 〉.

Show thatGhas precisely two 1-dimensional representations. By considering the effect ofy on an eigenvector ofxshow that any complex irreducible representation ofGof dimension at least 2 is isomorphic to one of two representations of dimension 2. Show that all these representations can be realised overR.

9 LetGbe the quaternion groupQ 8 of order 8,

Q 8 =〈x, y|x 4 = 1, y 2 =x 2 , yxy− 1 =x− 1 〉.

By considering the effect ofyon an eigenvector of xshow that any complex irreducible representation ofGof dimension at least 2 is isomorphic to the standard representation of Q 8 of dimension 2. Show that this 2-dimensional representation cannot be realised overR; that is,Q 8 is not a subgroup of GL 2 (R).

10 Suppose thatF is algebraically closed. Using Schur’s lemma, show that ifGis a finite group with trivial centre andHis a subgroup ofGwith non-trivial centre, then any faithful representation ofGis reducible on restriction toH. What happens forF=R?

11 LetGbe a subgroup of order 18 of the symmetric groupS 6 given by

G=〈(123),(456),(23)(56)〉.

Show thatGhas a normal subgroup of order 9 and four normal subgroups of order 3. By considering quotients, show thatGhas two representations of degree 1 and four inequivalent irreducible representations of degree 2. Deduce thatGhas no faithful irreducible representa- tions.

12 Show that ifρis a homomorphism from the finite groupGto GLn(R), then there is a matrixP∈GLn(R) such thatP ρ(g)P− 1 is an orthogonal matrix for eachg∈G. (Recall that the real matrixAis orthogonal ifAtA=I.) Determine all finite groups which have a faithful 2-dimensional representation overR.

SM, Lent Term 2017 Comments on and corrections to this sheet may be emailed tosm@dpmms.cam.ac

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Representation Theory 2016-2017 Example Sheet 1

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 1
Unless otherwise stated, all groups here are finite, and all vector spaces are finite-dimensional
over a field Fof characteristic zero, usually C.
1Let ρbe a representation of the group G.
(a) Show that δ:g7→ det ρ(g) is a 1-dimensional representation of G.
(b) Prove that G/ ker δis abelian.
(c) Assume that δ(g) = −1 for some g∈G. Show that Ghas a normal subgroup of
index 2.
2Let θ:G→F×be a 1-dimensional representation of the group G, and let ρ:G→
GL(V) be another representation. Show that θ⊗ρ:G→GL(V) given by θ⊗ρ:g7→ θ(g)·ρ(g)
is a representation of G, and that it is irreducible if and only if ρis irreducible.
3Find an example of a representation of some finite group over some field of charac-
teristic p, which is not completely reducible. Find an example of such a representation in
characteristic 0 for an infinite group.
4Let Nbe a normal subgroup of the group G. Given a representation of the quotient
G/N, use it to obtain a representation of G. Which representations of Gdo you get this way?
Recall that the derived subgroup G′of Gis the unique smallest normal subgroup of G
such that G/G′is abelian. Show that the 1-dimensional complex representations of Gare
precisely those obtained from G/G′.
5Describe Weyl’s unitary trick.
Let Gbe a finite group acting on a complex vector space V, and let h,i:V×V→C
be a skew-symmetric form, i.e. hy, x i=−h x, y ifor all x, y in V.
Show that the form (x, y) = 1
|G|Phgx, gyi, where the sum is over all elements g∈G, is
aG-invariant skew-symmetric form.
Does this imply that every finite subgroup of GL2m(C) is conjugate to a subgroup of the
symplectic group1Sp2m(C)?
6Let G=hgibe a cyclic group of order n.
(i) Gacts on R2as symmetries of the regular n-gon. Choose a basis of R2, and write
the matrix R(g) representing the action of a generator gin this basis. Is this an irreducible
representation?
(ii) Now regard R(g) above as a complex matrix, so that we get a representation of G
on C2. Decompose C2into its irreducible summands.
7Let Gbe a cyclic group of order n. Explicitly decompose the complex regular represen-
tation of Gas a direct sum of 1-dimensional representations, by giving the matrix of change
of coordinates from the natural basis {eg}g∈Gto a basis where the group action is diagonal.
1the group of all linear transformations of a 2m-dimensional vector space over Cthat preserve a non-
degenerate, skew-symmetric, bilinear form.
1

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