Consider a finitely generated (in fact finite) commutative algebra $A$ generated by projectors $P$ and (invertible) torsion elements $I$ with extra relations of the form $P*I = P$ for certain $P$ and certain $I$.
Is there a nice way to understand permutation representations of this algebra?
For instance, an interesting representation for me is the action of $A$ on $A \times A$. Is there an intelligent way to decompose this representation into orbits?
I thought about how the projectors give a filtration of any representation $R$, meaning that for each set of projectors $U$, there is a subset $F_U \subset R$ of mutual fixed points of all the projectors in $U$. Of course for $U \subset V$, $F_V \subset F_U$.
Further, at each level, there is a set of invertibles $G_U$ (forming a finite abelian group) such that for no $I \in G_U$ and $P \in F_U$ do we have $P*I = P$. $F_U$ is a representation of $G_U$. However, it seems that even in an irreducible representation of $A$, $F_U$ need not be an irrep of $G_U$. Are there any restrictions at all?
Here is an interesting example of such an algebra:
Fix a finite abelian group $G$ and form an algebra generated by symbols $P_H$ for each subgroup $H < G$ and $I_{\chi}$ for each 1d (linear) irrep of $G$.
Let the $P_H$'s be projectors such that $P_H * P_{H'} = P_{H''}$ where $H''$ is the smallest subgroup contained in both $H$ and $H'$.
Further, let $I_\chi * I_{\chi'} = I_{\chi \otimes \chi'}$ (so these are the invertibles) and include the relations $P_H*I_\chi = P_H$ whenever the representation $\chi$ restricted to the subgroup $H$ is trivial.
What are the representations of this algebra?