I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be brilliant.
I have a finite group $G$ and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let $L:V\rightarrow W$ be a linear map and define $p:V\rightarrow W$ by $p(x)=\sum_{g\in G}g^{-1}Lg(x)$.
Prove that $p$ is a homomorphism of $\mathbb{C}G$-modules.
I don't understand the what the definition of $p$ means.
Thanks,
Andy.
Best Answer
The notation here is a little awkward, so I'm going to adjust it a bit. You have that $V$ is a $\mathbb{C}G$-module, so for each $v\in V$ and $g\in G$, you should know what $g.v$ means (and similarly for $g.w$, $w\in W$).
Now, starting from a linear transformation $L:V\to W$ and $x\in V$ we can compute $g^{-1}.L(g.x)$ (that is, act on $x$ with $g$, map it over to $W$ using $L$, and then act on that by $g^{-1}$). This produces a vector in $W$.
We can do this for each $g\in G$, and since $W$ is a vector space, it makes sense to add all these elements together. In the end, we get a map $$p(x)=\sum_{g\in G} g^{-1}.L(g.x).$$ It is claimed that this is a $G$-map, so you need to prove that $p(h.x)=h.p(x)$ for every $h\in G$.
Remark about notation: Saying that $V$ and $W$ are $\mathbb{C}G$-modules is the same as having homomorphisms $$\pi_V:\mathbb{C}G\to \mathrm{End}(V)\;\;\mbox{ and }\;\;\pi_W:\mathbb{C}G\to \mathrm{End}(W)$$ Therefore, $\pi_V(g):V\to V$ is a linear map, as is $\pi_W(g):W\to W$. If we abuse notation and identify $g$ with its image under these maps, then we can interpret $g^{-1}Lg(x)$ as the composition of maps $$ V\longrightarrow^{\pi_V(g)}V\longrightarrow^LW\longrightarrow^{\pi_W(g^{-1})}W $$