Toric surfaces with symmetries by reflections

Abstract

Let $W$ be a reflection group in a plane and $P$ a rational polygon that is invariant under the $W$-action. The action of $W$ on $P$ induces a $W$-action on the toric variety $X_P$ associated with $P$. In this paper, we study the $W$-representation on the cohomology $H^\ast(X_P)$ and show that the invariant subring $H^\ast(X_P)^W$ is isomorphic to the cohomology ring of the toric variety associated with the \emph{fundamental region}~$P/W$. As an example, we provide an explicit description of the main result for the case of the toric variety associated with the fan of Weyl chambers of type $G_2$.