The cohomology rings of real permutohedral varieties

TL;DR

This paper explicitly describes the cohomology ring of real permutohedral varieties, including its multiplicative structure, expanding understanding of their topological invariants in real algebraic geometry.

Contribution

It provides the first explicit description of the cohomology ring and its multiplicative structure for real permutohedral varieties, building on prior Betti number computations.

Findings

Explicit cohomology ring structure described

Multiplicative structure characterized via alternating permutations

Advances understanding of topological invariants in real algebraic geometry

Abstract

A permutohedral variety is a remarkable object in various areas of mathematics, and its topological invariants are widely recognized. However, only little is known about a real permutohedral variety, that is, the real locus of a permutohedral variety. The rational Betti numbers of real permutohedral varieties were computed in terms of alternating permutations in 2012. In this paper, we provide explicit descriptions of the cohomology ring of real permutohedral varieties. In particular, we describe the multiplicative structure in terms of alternating permutations.