- Consider the group G= {e,a,b,c} with the following multiplication table: e|ab|c eeabe aaecb bbce a cbae Let V be the regular representation of G, with basis vectors fe, fa, fb, fe and the group action 8-fh=fg+h (a) Write down an explicit form of the matrices Me,Ma, Mb, Mc E GL(V) for this representation. After- words, perform an explicit check that Ma Mb=Mc- C (b) Check that the subspace W spanned by vectors v = (1,0, 1,0) and v = (0,1,0,1) is invariant under this action. (c) Using the method discussed in the lectures, find the representation complement W' of W. (d) Find a basis of W' and write an explicit form of the matrices MMMM encoding the regular representation action on W' in this basis. (e) Calculate the character of W' and determine whether W' is reducible. (f) Using the characters table for this group: e a b 1 1 1 1 1 1-1-1 XR XR XR 1-1 1-1 XR 1 -1 -1 where R, R2, R3, R are the four irreducible representations of G, find the decomposition of W' as the direct sum of irreducible representations.