I have some finite set $S$ and a collection of complex vector spaces $\{W^{s}\}_{s\in S}$ which are distinguished from one another. I also have a finitely generated group $G$ and some left action of $G$ on $S$ (given some $g\in G$ and $s\in S$, I'll denote said action by $g(s)$ for shorthand). I then have $|S|$ unitary representations of $G$ that behave as follows:

$\rho_{s}:G\to{\rm Hom}\big(W^{s},\bigoplus_{g\in G} W^{g(s)}\big)$ for each $s\in S$, where

$\rho_{s}(g): W^{s}\to W^{g(s)}$

Of course, $\dim(W^{g(s)})=\dim(W^{s})$ for all $g\in G, s\in S$ (hence admitting such unitary representations). For instance, given $g,h\in G$, then e.g.



Has the above setup been studied? What is the appropriate nomenclature?


Upon reflection, this seems to be some deconstruction of the representation of a groupoid.


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