MA4142 REPRESENTATION THEORY OF FINITE

MA4142 REPRESENTATION THEORY OF FINITE GROUPS PROBLEM SHEET 1 This is the rst problems sheet. The deadline for submitting solu- tions is 5th November, 10.00. Total marks of correct solutions is 20 marks. 1) Let G be the group of order 16 dened in terms of generators and relations G =< x; y : x4 = 1 = y4; yx = x3y > Find all 1-dimensional representations of G. 2) i) Prove that there is a unique representation 1 : D8 ! GL2(C) for which Here D6 = f1; a; a2; a3; a4; a5; a6; a7; b; ab; a2b; a3b; a4b; a5b; a6b; a7bg is the dihedral group, where a8 = 1 = b2 and bab = a??1. Is 1 an irreducible representation? Compute the character of 1. ii) Let = p 2 2 (1 + i) and = p 2 2 (1 ?? i): Prove that there is a unique representation 2 : D4 ! GL2(C) 1 2 PROBLEM SHEET 1 for which 2(a) = C and 2(b) = D, where C = 0 0 ;D = 0 1 1 0 : Compute the character of 2. iii) Find an invertible matrix P 2 GL2C such that for any x 2 D4 one has P2(x)P??1 = 1(x): 3) i) Let C3 = f1; t; t2g; t3 = 1 be the cyclic group of order three. Let V be the 2-dimensional vector space with basis e1 and e2. Prove that there exist a unique C3-module structure on V such that te1 = e2; and te2 = e1 ?? e2 Describe the corresponding representation and decompose it as a direct sum of irreducible representations. 4) i) Let C6 be a cyclic group of order 6 with a generator t. Describe all 1-dimensional representations of C6. ii) Let V be a C6-module. For each 6-th root of unity , we let V = fv 2 V jtv = vg: Prove that V = Mi=5 i=0 Vi Here = 1 2 + p 3 2 i is the primitive root of unity. iii) Let C6 be a C6-module, where the action of t on V is given by t(1; 2; 3; 4; 5; 6) = (6; 1; 2; 3; 4; 5): Consider a = (1; 2; 3; 1; 2; 3). Describe the smallest submodul W V such that a 2 W. What is the dimension of W? Find a C6-submodule U such that U W. Decompose V and W as the direct sum of irre- ducible representations. 5) Let G = D8 be the dihedral group. Thus G =< x; y : x8 = 1 = y2; xy = yx??1 >. Prove that there exist a unique representation : G ! GL2(C) MA4142 REPRESENTATION THEORY OF FINITE GROUPS 3 for which (x) = ??7 10 ??5 7 and (y) = ??5 6 ??4 5 nd the character of . Is an irreducible representation? 6) Let C12 be the cyclic group of order 12, with a generator t and let e1; ; e12 be a standard basis of C12, which is considered as a C10-module via the action tei = ( ei+1 i < 12 e1; i = 12: Find the dimension of a minimal submodule which contains the element e1 ?? e2 + e3 ?? 34 + e5 ?? e6 + e7 ?? e8 + e9 ?? e10 + e11 ?? e12.