Rational cohomology of the real Coxeter toric variety of type A

TL;DR

This paper computes the Betti numbers and symmetric group representations of the rational cohomology of the real Coxeter toric variety of type A, revealing its cohomological structure and non-generation in degree 1.

Contribution

It provides explicit formulas for Betti numbers and symmetric group actions on cohomology, connecting Coxeter toric varieties with poset homology and cohomological properties.

Findings

Betti numbers of the real Coxeter toric variety are explicitly calculated.

Symmetric group representations on cohomology are characterized.

The rational cohomology ring is not generated in degree 1.

Abstract

The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.