Exercise [13.32]

yoavs · 02 Dec 2008, 15:01

A feature of groups is the product relation: C=AB

. What this actually means is that A "maps" B to C.
Using this understanding, and the hint given by Penrose, the matrix representation of T(k) (e,g, ${T_m}^n$ , m and n are the indices that represent the corresponding group elements) should be 1 if the group product $m=k \cdot n$ is fulfilled.

The reason is that $T(C)=T(A \cdot B) = T(A) \cdot T(B)$ . Writing this in matrix form gives (using Einstein convention):
${C_q}^j = {A_q}^p \cdot {B_p}^j$ . Now if B maps elements j to p and c maps j to q than A needs to map p to q and should therefore be non zero only if Q = AP

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Exercise [13.32]

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