- Consider the group G= {e,a,b,c} with the following multiplication table: e|ab|c eeabe aaecb bbce a...

 

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- 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 v₁ = (1,0, 1,0) and v₂ = (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 MM₁M²M 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 с 1 1 1 1 1 1-1-1 XR₁ XR₂ XR 1-1 1-1 XR₂ 1 -1 -1 where R₁, R2, R3, R₁ are the four irreducible representations of G, find the decomposition of W' as the direct sum of irreducible representations. - 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 v₁ = (1,0, 1,0) and v₂ = (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 MM₁M²M 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 с 1 1 1 1 1 1-1-1 XR₁ XR₂ XR 1-1 1-1 XR₂ 1 -1 -1 where R₁, R2, R3, R₁ are the four irreducible representations of G, find the decomposition of W' as the direct sum of irreducible representations.