Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials

TL;DR

This paper develops a graded character formula for finite group actions on Stanley--Reisner rings of complete simplicial fans, linking algebraic, geometric, and combinatorial invariants.

Contribution

It introduces an equivariant h-polynomial and a hybrid fan tool to compute Poincaré polynomials of invariants and quotients in toric geometry.

Findings

Derived a graded character formula for group actions on Stanley--Reisner rings.

Computed Poincaré polynomials of invariants under reflection group actions.

Showed the Poincaré polynomial of a quotient toric orbifold matches that of an associated hybrid fan.

Abstract

We derive a graded character formula for the action of any finite group on the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan, which is given by an equivariant version of the classical h-polynomial. This gives the graded character formula for the representation of the group on the cohomology of the associated toric variety when the fan is rational. As an application, we use a navel tool, which we called hybrid fan, to compute the Poincar\'e polynomial of the invariants of the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan under a finite reflection group action. This implies that the Poincar\'e polynomial of the quotient of a compact toric orbifold by any finite reflection group is equal to the Poincar\'e polynomial of the compact toric orbifold associated to the hybrid fan.