Making use of extra symmetries; more examples?

Question

TL; DR.

In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I wonder if there are more such examples.

Formal description

To be more precise, let $R$ be an associative unital complex algebra and $M$ a finite dimensional left $R$-module. Let further $G$ be a finite group with a complex-linear action on $M$, such that both actions commute. Then any $G$-isotypic component $M_i$ of $M$ is a left $R$-module, yielding a decomposition of $R$-module

$$ M = M_1 \oplus \ldots \oplus M_l.$$

The proof of it lies in the fact that the projection to $\chi$'th $G$-isotypic component is realizable as an element in the group algebra $A = \mathbb{C}[G]$

$$ \frac{1}{|G|} \Sigma_{g\in G} \overline{\chi(g)} g.$$

Examples arise in the field of modular forms (diamond operator.. etc) [1] and geometric representation theory [2, theorem 8.1.16] (classification of the representations of the affine Hecke algebra, extra symmetries from the stabilizer of a chosen point in the nilpotent cone.. etc).

However, this seems to rely on the fact that the projection is realizable as an element. Moreover, the element in this case seems exclusive for topological compact groups.. therefore the questions:

Questions

1. What else algebra $A$ has an element that serves as the projection to a given isotypic component for any of its left modules?

2. More such examples as in extra compatible symmetries reducing the complexity of certain mathematical object?

Reference

• [1] A First Course in Modular Forms-[Diamond and Shurman]
• [2] Representation Theory and Complex Geometry-[Chriss and Ginzburg]

Answer 1

Your setup is much more specific than necessary. All you need is two rings $R, S$ with commuting actions on an abelian group $M$ (which is therefore an $(R, S^{op})$-bimodule) such that $M$ is semisimple as an $S$-module. Then, as an $S$-module, $M$ decomposes canonically into a direct sum of isotypic components, and since by hypothesis $R$ acts by $S$-module endomorphisms it preserves this decomposition. Of course this setup is easiest to guarantee if $S$ is semisimple and group algebras are a common source of semisimple rings.

This setup occurs, for example, in the double commutant theorem, which is famously applied to prove Schur-Weyl duality. Here $M = V^{\otimes n}$ where $V$ is a finite-dimensional vector space over a field of characteristic zero (to be safe), $R = k[GL(V)]$, and $S = k[S_n]$.

Also, you ask:

What else algebra $A$ has an element that serves as the projection to a given isotypic component for any of its left modules?

Every semisimple ring has this property. By the Artin-Wedderburn theorem, a semisimple ring $R$ is a finite direct product $\prod_i M_{n_i}(D_i)$ of matrix rings over division rings, one for every simple module. The projection to the isotypic component given by the $i^{th}$ simple module is the primitive central idempotent given by the element of this direct product equal to the identity matrix in the $i^{th}$ factor and zero otherwise. The interesting thing about the group algebra case is that we don't need to write out the Artin-Wedderburn decomposition to write down the idempotent.

More such examples as in extra compatible symmetries reducing the complexity of certain mathematical object?

This is a maybe overly broad question. Using symmetry is a common and ubiquitous strategy in mathematics.