I'm reading some notes on Representation Theory. The chapter on orthogonality relations starts with stating Schur's Lemma:
Let $V,W$ be irreducible $\mathbb{C}G$-modules. Then $Hom_{\mathbb{C}G}(V,W)\cong\mathbb{C}$ if $V \cong W$ and is $\{0\}$ otherwise.
The remark on the matrix consequences of this Lemma follows:
If $\sigma$ and $\tau$ are irreducible complex matrix representations of respective dimensions $m,n$ of the finite group $G$, then, for any $m \times n$ matrix $X$, we have $\displaystyle\sum_{g \in G} \sigma (g)X\tau(g^{-1}) = [0]_{m \times n}$ $(1)$ if $\sigma$ and $\tau$ are not equivalent, while $\displaystyle\sum_{g \in G} \sigma (g)X\sigma(g^{-1}) = \frac{|G|}{n}Trace(X)I_{n \times n}$ $(2)$. There is more that follows, but this is the part I don't understand.
From the Schur's Lemma above we know that if $\sigma$ and $\tau$ are not equivalent then $Hom_{\mathbb{C}G}(V,W)=\{0\}$, but I'm still not sure where $(1)$ and $(2)$ come from. I'd appreciate some help.
Best Answer
Suppose that $(V, \sigma)$ and $(W, \tau)$ are complex representations of $G$. Then the vector space $\operatorname{Hom}_{\mathbb{C}}(V, W)$ of all linear maps from $V$ to $W$ is also a representation, with action given on an $f: V \to W$ by the formula $g \cdot f = \tau(g) \circ f \circ \sigma(g^{-1})$.
Now, the fixed-point subspace $$\operatorname{Hom}_{\mathbb{C}}(V, W)^G = \{f \in \operatorname{Hom}_{\mathbb{C}}(V, W) \mid g \cdot f = f \text{ for all } g \in G\}$$ is actually equal to the space of $G$-equivariant maps $\operatorname{Hom}_{\mathbb{C}G}(V, W)$, since by the formula above we would have for any fixed-point in that space, that $\tau(g) \circ f = f \circ \sigma(g)$.
Lastly, for any representation $(U, \rho)$, the projection $U \to U^G$ is given by $u \mapsto \frac{1}{\lvert G \rvert} \sum_{g \in G} g \cdot u$.
Now let's suppose that $(V, \sigma)$ and $(W, \tau)$ are irreducible representations, and $X: W \to V$ is any linear map between them. $X$ belongs to the space $\operatorname{Hom}_{\mathbb{C}}(W, V)$, but using the projection to the fixed-point space, we have that $\sum_{g \in G} g \cdot X = \sum_{g \in G} \sigma(g) X \tau(g^{-1})$ is a $G$-equivariant map, in the space $\operatorname{Hom}_{\mathbb{C}G}(W, V)$. Since $V$ and $W$ are irreducible, by Schur's lemma either $\dim V \neq \dim W$ and this map is zero, or $\dim V = \dim W$ and the map is an isomorphism or zero.
To prove the last part, about exactly what multiple of the identity we should get, note that since we already know that $\sum_g \sigma(g) X \sigma(g^{-1})$ needs to be a multiple of the identity, it suffices to know its trace: $$\operatorname{tr}\left(\sum_g \sigma(g) X \sigma(g^{-1})\right) = \sum_g \operatorname{tr}\left( X \sigma(g^{-1}) \sigma(g) \right) = |G| \operatorname{tr}(X)$$ while the trace of the identity is $n$, the dimension of the space.