The equivariant Euler characteristic of real Coxeter toric varieties

TL;DR

This paper derives a formula for the equivariant Euler characteristic of real Coxeter toric varieties, revealing connections to symmetric group characters and Euler numbers, thus advancing understanding of their topological and algebraic properties.

Contribution

It provides a general formula for the equivariant Euler characteristic of real Coxeter toric varieties as a generalized character of the Weyl group, including explicit results for type A.

Findings

Formula for equivariant Euler characteristic as a generalized character.

In type A, the character degree relates to Euler numbers.

Connections between topological invariants and algebraic group actions.

Abstract

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\CT_W(\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\Sym_n$ whose degree is (up to sign) the $n^{\text{th}}$ Euler number.