Projections from endomorphism algebra of $G$-equivariant maps

Question

Let $(V,\rho)$ be a representation of finite group $G$. $V$ can be decomposed to the direct sum of isotypic components $$V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_m,$$ where, for each $i$, $V_i$ is $d_i$ copy of one irreducible representation of $G$ ($V_i=W_i^{d_i}$). Suppose the character of $W_i$ is denoted by $\chi_i$. Then $$p_i=\frac{\dim(W_i)}{|G|}\sum_{g \in G} \chi_i^*(g)\rho_g: V \to V$$ is a projection of $V$ onto $V_i$.

Now, I am looking for the projection of $\text{End}_G(V)$ onto $\text{End}_G(V_i)$.

So far, I know $$\text{End}_G(V)\cong \bigoplus_i \text{End}_G(V_i).$$

I also consider $\pi_i$ and $\eta_i$ to be the natural projection and injection between $V$ and $V_i$. Then $p_i=\eta_i \circ \pi_i$, and, for every $\tau \in \text{End}_G(V)$, $$\tau \mapsto \pi_i \tau \eta_i$$ is a mapping from $\text{End}_G(V)$ to $\text{End}_G(V_i)$, but it seems this map is not a ring isomorphism (or is it?)

So would you help me find a projection from $\text{End}_G(V)$ to $\text{End}_G(V_i)$ based on $p_i$?

Answer 1

I think I have found my answer. Let $p_i: V \to V$ be the projection whose image is $V_i$. Then, the mapping $\tau \mapsto p_i \tau$ is the projection from $\text{End}_G(V)$ onto $\text{End}_G(V)$ whose image is $\text{End}_G(V_i)$.

Proof:

It is obvious that the mapping is closed under addition and is idempotent. It is also closed under ring multiplication since $p_i$ are in the centre of $\text{End}_G(V)$, which, in turn, implies $$p_i \tau_1 \tau_2 =p_i ^2\tau_1 \tau_2 =p_i \tau_1 p_i\tau_2 .$$

Answer 2

More generally, let $i\colon V\hookrightarrow W$ and $p\colon W\twoheadrightarrow V$ be $G$-equivariant homomorphisms such that $p\circ i=\mathrm{id}_V$ (i.e., $p$ is a section of $i$). Then the following is a (non-unital) ring homomorphism: $$i\circ -\circ p\colon \mathrm{End}_G(V)\to\mathrm{End}_G(W).$$ Indeed, it is clearly additive. Moreover, it is multiplicative: $$\begin{align*} (i\circ F\circ p)\circ (i\circ G\circ p)&=i\circ F\circ \mathrm{id}_V\circ G\circ p\\ &=i\circ F\circ G\circ p. \end{align*}$$