Representations of cyclic groups acting on complete simplicial fans

TL;DR

This paper proves that when a cyclic group acts properly on a complete simplicial fan, the induced representation on the associated toric variety's cohomology is a permutation representation, revealing symmetry properties.

Contribution

It establishes that the group action on the cohomology of the toric variety is always a permutation representation under the given conditions.

Findings

Representation is permutation type

Group acts properly on fan

Cohomology reflects fan symmetry

Abstract

Let $sigma$ be a complete simplicial fan in finite dimensional real Euclidean space $V$, and let $G$ be a cyclic subgroup of $GL(V)$ which acts properly on $\sigma$. We show that the representation of $G$ carried by the cohomology of $X_{sigma}$, the toric variety associated to $sigma$, is a permutation representation.