The group algebra as “universal endomorphisms”

My lecture notes on representation theory now include some discussion of the group algebra (though I haven’t yet covered this in class). The motivation I am emphasizing (though the emphasis comes from the unorthodox practice of making it the topic of Exercise 3.1.5 in Chapter 3…) is that the elements of the group algebra $k(G)$ correspond exactly, through their action, to what might be called “universal” endomorphisms of representations (though “functorial” is the more proper term).

More precisely, a “universal endomorphism” is a collection $\varepsilon_{\rho}$ of linear maps
$\varepsilon_{\rho}\,:\, E\longrightarrow E$
defined for all representations
$\rho\,:\, G\longrightarrow \mathrm{GL}(E),$
in such a way that for any $G$ -homomorphism
$\Phi\,:\, E\longrightarrow F,$
we have
$\Phi\circ \varepsilon_{\rho}=\varepsilon_{\tau}\circ \Phi$
(which means that the obvious square diagram, which I am unable to reproduce with the LaTeX plugin of this blog, commutes). The elementary result of the exercise is that, for any such collection, there is a unique element $a\in k(G)$ such that
$\varepsilon_{\rho}(v)=\rho(a)v$
for all representations $\rho$ , and all vectors $v$ in the space of $\rho$ . (Hint: the element $a$ is simply $\varepsilon_{k(G)}(1)$ — where $k(G)$ has of course the left multiplication action –, and to show that it has the required property, one first deals with $\rho=k(G)$ itself using the $G$ -homomorphisms given by right-multiplication by some element $b$ , etc.)

The reason I like this viewpoint (besides the fact that it is not discussed this way in the books I know, and one likes to be at least infinitesimally original) is that it makes it clear that there must be some nice formula for the projectors on isotypic components for finite groups (in good characteristic): indeed, it is easy to check that, for a fixed irreducible representation $\pi$ , the family
$\varepsilon_{\rho}=(\text{projector on the }\pi-\text{isotypic component of }\rho)$
satisfies the functoriality property above. Thus this must be given by the action of some element of the group algebra, and by inspection, one is led to the “right” one (using the orthogonality of characters, in my notes, though maybe there is an even more intrinsic reason?) To me, this seems to demystify the result: it is not a question of checking a miraculous formula, but of using some reasonable enough reasoning to find the right one.

(Note: the isotypic projectors are also themselves $G$ -homomorphisms; it is of course not difficult to check that a “universal” endomorphism is a universal $G$ -homomorphism if and only if the corresponding element $a$ is in the center of the group algebra.)

The group algebra as “universal endomorphisms”

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Kowalski