Geometric representations of finite groups on real toric spaces

TL;DR

This paper introduces a framework for constructing geometric representations of finite groups via real toric spaces, providing combinatorial descriptions and explicit computations of group actions on homology.

Contribution

It develops a new combinatorial approach to analyze group actions on real toric spaces and computes specific representations related to Weyl groups and Foulkes representations.

Findings

Explicit Weyl group representations on homology of real toric varieties.

Connection between homology of real toric varieties and topology of posets.

Geometric realization of Foulkes representations.

Abstract

We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type $A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.