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