Representations of cyclic groups acting on complete simplicial fans

Let L be a lattice in d-dimensional real Euclidean space V , and let σ be a polyhedral decomposition of V . If the cones of σ are generated by vectors in L, σ is called a (complete) fan in L (and we say σ is integral with respect to L). If each i-dimensional cone of σ is generated by i vectors, we say σ is simplicial. Associated to any fan σ in V is a toric variety, Xσ. In [7], Stembridge showed that when σR is the simplicial fan defined by the hyperplanes orthogonal to the roots of a crystallographic root system R, and G is the Weyl group of R, the representation of G carried by H(XσR ,Q) is isomorphic to a permutation representation of G (ignoring the grading in the case that G is not of type A or B). The main purpose of this paper is to show an analogous result for cyclic groups G acting appropriately on arbitrary fans. Suppose G is a finite subgroup of GL(V ) that acts as a group of homomorphisms of L and additionally as a set of automorphisms of some complete simplicial fan σ. Then G acts on the toric variety Xσ. We say that the action of G is proper if whenever an element of G fixes a cone in σ, then it fixes that cone pointwise.