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Representation Theory 2016-2017 Example Sheet 2

Module

Representation Theory

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Academic year: 2016/2017
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PART II REPRESENTATION THEORY

SHEET 2

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

1 Letρ:G→GL(V) be a representation ofGof dimensiond, and affording characterχ. Show that kerρ={g∈G|χ(g) =d}. Show further that|χ(g)| 6 dfor allg∈G, with equality if and only ifρ(g) =λI, a scalar multiple of the identity, for some root of unityλ.

2 Letχbe the character of a representationV ofGand letgbe an element ofG. Ifgis an involution (i.e 2 = 1 6 =g), show thatχ(g) is an integer andχ(g)≡χ(1) mod 2. IfGis simple (but notC 2 ), show that in factχ(g)≡χ(1) mod 4. (Hint: consider the determinant ofgacting onV.) Ifghas order 3 and is conjugate tog− 1 , show thatχ(g)≡χ(1) mod 3.

3 Construct the character table of the dihedral groupD 8 and of the quaternion groupQ 8. You should notice something interesting.

4 Construct the character table of the dihedral groupD 10. Each irreducible representation ofD 10 may be regarded as a representation of the cyclic subgroupC 5. Determine how each irreducible representation ofD 10 decomposes into irre- ducible representations ofC 5. Repeat forD 12 ∼=S 3 ×C 2 and the cyclic subgroupC 6 ofD 12.

5 Construct the character tables ofA 4 ,S 4 ,S 5 , andA 5. The groupSnacts by conjugation on the set of elements ofAn. This induces an action on the set of conjugacy classes and on the set of irreducible characters ofAn. Describe the actions in the cases wheren= 4 andn= 5.

6 A certain group of order 720 has 11 conjugacy classes. Two representations of this group are known and have corresponding charactersαandβ. The table below gives the sizes of the conjugacy classes in the group and the values whichαandβtake on them.

1 15 40 90 45 120 144 120 90 15 40 α 6 2 0 0 2 2 1 1 0 −2 3 β 21 1 − 3 −1 1 1 1 0 − 1 −3 0

Prove that the group has an irreducible representation of degree 16 and write down the corresponding character on the conjugacy classes.

7 The table below is a part of the character table of a certain finite group, with some of the rows missing. The columns are labelled by the sizes of the conjugacy classes, and γ= (−1 +i

√

7)/2,ζ= (−1 +i

√

3)/2. Complete the character table. Describe the group in terms of generators and relations.

1 3 3 7 7 χ 1 1 1 1 ζ ζ ̄ χ 2 3 γ ̄γ 0 0

1

2 PART II REPRESENTATION THEORY SHEET 2

8 Letxbe an element of ordernin a finite groupG. Say, without detailed proof, why (a) ifχis a character ofG, thenχ(x) is a sum ofnth roots of unity; (b)τ(x) is real for every characterτ ofGif and only ifxis conjugate tox− 1 ; (c)xandx− 1 have the same number of conjugates inG. Prove that the number of irreducible characters ofGwhich take only real values (so- calledreal characters) is equal to the number of self-inverse conjugacy classes (so-calledreal classes).

9 A group of order 168 has 6 conjugacy classes. Three representations of this group are known and have corresponding charactersα,βandγ. The table below gives the sizes of the conjugacy classes and the valuesα,βandγtake on them.

1 21 42 56 24 24 α 14 2 0 −1 0 0 β 15 − 1 −1 0 1 1 γ 16 0 0 −2 2 2

Construct the character table of the group. [You may assume, if needed, the fact that

√

7 is not in the fieldQ(ζ), whereζis a primitive 7th root of unity.] The character table thus obtained is in fact the character table of the groupG= PSL 2 (7) of 2×2 matrices with determinant 1 over the fieldF 7 (of seven elements) modulo the two scalar matrices. Deduce directly from the character table thatGis simple 1.

10 The groupM 9 is a certain subgroup of the symmetric groupS 9 generated by the two elements (1, 4 , 9 ,8)(2, 5 , 3 ,6) and (1, 6 , 5 ,2)(3, 7 , 9 ,8). You are given the following facts about M 9 : - there are six conjugacy classes: —C 1 contains the identity. — For 2 6 i 6 4,|Ci|= 18 andCicontainsgi, whereg 2 = (2, 3 , 8 ,6)(4, 7 , 5 ,9),g 3 = (2, 4 , 8 ,5)(3, 9 , 6 ,7) andg 4 = (2, 7 , 8 ,9)(3, 4 , 6 ,5). —|C 5 |= 9, andC 5 containsg 5 = (2,8)(3,6)(4,5)(7,9) —|C 6 |= 8, andC 6 containsg 6 = (1, 2 ,8)(3, 9 ,4)(5, 7 ,6). - every element ofM 9 is conjugate to its inverse. Calculate the character table ofM 9. [Hint: You may find it helpful to notice that g 22 =g 23 =g 42 =g 5 .]

11 Let a finite groupGact on itself by conjugation. Find the character of the corresponding permutation representation.

1 It is known that there are precisely five non-abelian simple groups of order less than 1000. The smallest

of these isA 5 ∼=PSL 2 (5), whileGis the second smallest. The others areA 6 , PSL 2 (8) and PSL 2 (11). It is also known that forp>5, PSL 2 (p) is simple.

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Representation Theory 2016-2017 Example Sheet 2

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 2
Unless otherwise stated, all groups here are finite, and all vector spaces are finite-dimensional
over a field Fof characteristic zero, usually C.
1Let ρ:G→GL(V) be a representation of Gof dimension d, and affording character χ.
Show that ker ρ={g∈G|χ(g) = d}. Show further that |χ(g)|6dfor all g∈G, with
equality if and only if ρ(g) = λI, a scalar multiple of the identity, for some root of unity λ.
2Let χbe the character of a representation Vof Gand let gbe an element of G. If gis
an involution (i.e. g2= 1 6=g), show that χ(g) is an integer and χ(g)≡χ(1) mod 2. If Gis
simple (but not C2), show that in fact χ(g)≡χ(1) mod 4. (Hint: consider the determinant
of gacting on V.) If ghas order 3 and is conjugate to g−1, show that χ(g)≡χ(1) mod 3.
3Construct the character table of the dihedral group D8and of the quaternion group Q8.
You should notice something interesting.
4Construct the character table of the dihedral group D10.
Each irreducible representation of D10 may be regarded as a representation of the cyclic
subgroup C5. Determine how each irreducible representation of D10 decomposes into irre-
ducible representations of C5.
Repeat for D12 ∼
=S3×C2and the cyclic subgroup C6of D12.
5Construct the character tables of A4,S4,S5, and A5.
The group Snacts by conjugation on the set of elements of An. This induces an action
on the set of conjugacy classes and on the set of irreducible characters of An. Describe the
actions in the cases where n= 4 and n= 5.
6A certain group of order 720 has 11 conjugacy classes. Two representations of this group
are known and have corresponding characters αand β. The table below gives the sizes of the
conjugacy classes in the group and the values which αand βtake on them.
1 15 40 90 45 120 144 120 90 15 40
α6 2 0 0 2 2 1 1 0 −2 3
β21 1 −3−1 1 1 1 0 −1−3 0
Prove that the group has an irreducible representation of degree 16 and write down the
corresponding character on the conjugacy classes.
7The table below is a part of the character table of a certain finite group, with some
of the rows missing. The columns are labelled by the sizes of the conjugacy classes, and
γ= (−1 + i√7)/2, ζ= (−1 + i√3)/2. Complete the character table. Describe the group in
terms of generators and relations.
1 3 3 7 7
χ11 1 1 ζ¯
ζ
χ23γ¯γ0 0
1

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