Toric surfaces with symmetries by reflections

TL;DR

This paper investigates how reflection groups act on toric surfaces derived from rational polygons, revealing that the invariant cohomology corresponds to the cohomology of the fundamental region’s toric variety, with explicit examples for Weyl chambers.

Contribution

It establishes an isomorphism between the invariant cohomology of toric surfaces under reflection group actions and the cohomology of the associated fundamental region, providing explicit descriptions for specific cases.

Findings

Invariant subring of cohomology is isomorphic to the cohomology of the fundamental region.

Explicit example for Weyl chambers of type G2.

Demonstrates the structure of cohomology under reflection symmetries.

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 $G_2$.