Please help answer the following algebra questions. Provide step-by-step solutions. (i) Now let G be a group with the presentation
G=
You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation
p:G –> GL(3,C)
withp(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(b) = (0, 1, 0; 0, 0, 1; 1, 0, 0) (ii) For the representation p:G –> GL(3,C) defined in (i) above, let p*:G–>GL(3,C) be the representation given by p*(g)=p(g^(-1))^T for all gEG. Write down p*(a) and p*(b). Are p and p* equivalent representations?See the attachment for the full solution. (i) Now let G be a group with the presentation
G=
You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation
p:G –> GL(3,C)
withp(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(b) = (0, 1, 0; 0, 0, 1; 1, 0, 0) Solution:
If ρ:G→GL(n,C) is a representation, how is the character of ρ defined?
Prove that equivalent representations have equal characters.The character of ρ is defined as a mapping G C sending each g to the trace of ρ(g).If ρ and ρ are equivalent, then there exists an invertible matrix M such that ρ(g)=Mρ^ (g) M^(-1),for all g. This implies that matrices ρ(g) and ρ(g) have the same traces.(ii) For the representation p:G –> GL(3,C) defined in (i) above, let p*:G–>GL(3,C) be the representation given by p*(g)=p(g^(-1))^T for all gEG. Write down p*(a) and p*(b). Are p and p* equivalent representations? Suppose that ρ:G→GL(n,C) is a representation. Show that
ρ^*:G→GL(n,C)
ρ^* (g)=ρ(g^(-1) )^T
is also a representation of G. [Here if A is a matrix then A^T is the transpose of A.]Let us show that ρ^* is a homomorphism. We have
ρ^* (gg^ )=ρ(〖(gg^)〗^(-1) )^T=〖ρ((〖g〗^(-1) g^(-1)))〗^T= 〖(ρ(〖g^〗^(-1) )ρ(g^(-1) ))〗^T=〖ρ(g^(-1) )〗^T 〖ρ(g^(-1) )〗^T=ρ^* (g) ρ^* (g).




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