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I'm given the following in a homework question:

Let $G$ be a group and $k$ an algebraically closed field.

(a) Show that the action of $G \times G$ on $C_k (G)$ defined by $$ (g_1, g_2) \varphi (x) = \varphi(g_1^{-1} x g_2) \hspace{1cm} g_1, g_2 , x \in G , \varphi \in C_k(G)$$ defines a representation $\pi$ of $G \times G$

Let $\rho : G \rightarrow GL(E) $ be a finite dimensional irreducible representation. Let $ M(\rho) = \{$ span of the matrix coefficients of $ \rho \} \subset C_k(G)$

(b) Show that $M(\rho)$ is a subrepresentation of $\pi$.

My question(s):

1) what is "span of the matrix coefficients"?

2) I need to show that for $m \in M(\rho)$: $\pi (g_1, g_2, m) \in M(\rho)$ $\forall g_1, g_2 \in G$. Can I write "let $M := \rho$", the matrix representation of $\rho$ and then $M(\rho) = \{cM | c \in k\}$?

3) And am I right in assuming that $E$ has to be a vector space over $k$?

Many thanks for your help.

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you should consider a different title that gives an idea of what this question is about. – BBischof Mar 28 '11 at 15:08
What is $C_k(G)$? – Arturo Magidin Mar 28 '11 at 15:19
@BBischof: Sure, what would you suggest? – Rudy the Reindeer Mar 28 '11 at 17:20
@Arturo: the set of functions from $G$, a group, to $k$, a field. – Rudy the Reindeer Mar 28 '11 at 17:21
@Arturo: confusingly not the set of continuous functions, even though $C$ is used. But I don't think it's a typo in the script, the script is rather typo-free. – Rudy the Reindeer Mar 28 '11 at 17:22
up vote 1 down vote accepted

$E$ is usually a finite dimensional $k$-vector space in this context; one can do more general things, but it does not hurt to start small!

You have a map $\rho:G\to\operatorname{GL}(E)$. On the other hand, if $E$ is a finite dimensional vector space, then picking one of its bases we obtain an isomorphism $\phi:\operatorname{GL}(E)\cong\operatorname{GL}(n,k)$ to the group of invertible $n\times n$ matrices with coefficients in $k$. Finally, for each $i$, $j\in\{1,\dots,n\}$ we can consider the function $p_{i,j}:\operatorname{GL}(n,k)\to k$ which maps each matrix to its $(i,j)$th entry.

The space $M(\rho)$ is the $k$-vector subspace of the spaces $C_k(G)$ of all functions $G\to k$ spanned by the set of functions $$\bigl\{p_{i,j}\circ\phi\circ\rho:i,j\in\{1,\dots,n\}\bigr\}.$$ The space $M(\rho)$ does not depend on the choice of basis we did, as you should check.

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Many thanks! I have another question: what is the exact difference between $GL(E)$, the isomorphisms form $E$ to $E$ and $GL(n,k)$, the invertible $n \times n$ matrices with coefficients in $k$? (assuming $E$ is a vector space over $k$). Aren't invertible matrices automorphisms? And can I not write any linear automorphism as a matrix? – Rudy the Reindeer Mar 30 '11 at 12:51
@Matt: $GL(E)$ is the set of functions $E\to E$ whch are linear isomorphisms; on the other hand, $GL(n,k)$ is the set of $n\times n$ matrices with non-zero determinant. Since the two sets have completely different elements, they are clearly different! – Mariano Suárez-Alvarez Mar 30 '11 at 16:14

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