Stanley–Reisner rings for symmetric simplicial complexes, $G$-semimatroids and Abelian arrangements

We extend the notion of face rings simplicial complexes and posets to case finite-length (possibly infinite) with a group action. The action on complex induces an ring, we prove that ring invariants is isomorphic quotient poset under mild condition action.We also identify class actions preserve homotopical Cohen–Macaulay property quotients. When acted-upon independence semimatroid, h-polynomial can be read off Tutte polynomial associated Moreover, in this additional ensures characteristic 0 every does not divide explicitly computable number. This implies same for Stanley–Reisner rings. In particular, holds toric, elliptic and, more generally, $(p,q)$-arrangements. As byproduct, connected components (also known as layers) such arrangements are characteristic.