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I am reading James and Liebeck's book on Representations and Characters of Groups.
Exercise 1 of Chapter 3 reads as follows:
Let G be the cyclic group of order m, say G = < a : [itex] a^m = 1 [/itex] >.
Suppose that A [itex] \in GL(n \mathbb{C} ) [/itex], and define [itex] \rho : G \rightarrow GL(n \mathbb{C} ) [/itex] by
[itex] \rho : a^r \rightarrow A^r \ \ (0 \leq r \leq m-1 ) [/itex]
Show that [itex] \rho [/itex] is a representation of G over [itex] \mathbb{C} [/itex] iff and only if [itex] A^m = I [/itex]
The solution given is as follows (functions are applied from the right)
Suppose [itex] \rho [/itex] is a representation of G. Then
[itex] I = 1 \rho = ( a^m ) \rho = {(a \rho)}^m = A^m [/itex]
Conversely assume that [itex] A^m = I [/itex]. Then [itex] ( a^i ) \rho = A^i [/itex] for all integers i.
Therefore for all integers i, j
[itex] ( a^i a^j ) \rho \ = \ ( a^{i+j} ) \rho \ = \ A^{i+j} \ = \ A^i A^j \ = \ ( a^i \rho ) a^j \rho ) [/itex]
and so [itex] \rho [/itex] is a representation.
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That is all fine but I am just getting into representations and wish to get an intuitive understanding of what is happening.
In the above I am suprised that we have no explicit form for A.
Does this mean we have many representations for G in this case - that is any and every matrix A for which [itex] A^m = I [/itex]
Can someone please confirm this?
Exercise 1 of Chapter 3 reads as follows:
Let G be the cyclic group of order m, say G = < a : [itex] a^m = 1 [/itex] >.
Suppose that A [itex] \in GL(n \mathbb{C} ) [/itex], and define [itex] \rho : G \rightarrow GL(n \mathbb{C} ) [/itex] by
[itex] \rho : a^r \rightarrow A^r \ \ (0 \leq r \leq m-1 ) [/itex]
Show that [itex] \rho [/itex] is a representation of G over [itex] \mathbb{C} [/itex] iff and only if [itex] A^m = I [/itex]
The solution given is as follows (functions are applied from the right)
Suppose [itex] \rho [/itex] is a representation of G. Then
[itex] I = 1 \rho = ( a^m ) \rho = {(a \rho)}^m = A^m [/itex]
Conversely assume that [itex] A^m = I [/itex]. Then [itex] ( a^i ) \rho = A^i [/itex] for all integers i.
Therefore for all integers i, j
[itex] ( a^i a^j ) \rho \ = \ ( a^{i+j} ) \rho \ = \ A^{i+j} \ = \ A^i A^j \ = \ ( a^i \rho ) a^j \rho ) [/itex]
and so [itex] \rho [/itex] is a representation.
===================================================================
That is all fine but I am just getting into representations and wish to get an intuitive understanding of what is happening.
In the above I am suprised that we have no explicit form for A.
Does this mean we have many representations for G in this case - that is any and every matrix A for which [itex] A^m = I [/itex]
Can someone please confirm this?