Cohomology of type $B$ real permutohedral varieties

TL;DR

This paper explicitly describes the cohomology ring structure of type B real permutohedral varieties using combinatorial objects called B-snakes, extending previous knowledge beyond Betti numbers.

Contribution

It provides the first explicit description of the multiplicative cohomology structure of type B real permutohedral varieties in terms of B-snakes.

Findings

Cohomology rings of type B real permutohedral varieties are described explicitly.

The multiplicative structure is characterized using B-snakes.

This advances understanding of the topology of real permutohedral varieties.

Abstract

Type $A$ and type $B$ permutohedral varieties are classic examples of mathematics, and their topological invariants are well known. This naturally leads to the investigation of the topology of their real loci, known as type $A$ and type $B$ real permutohedral varieties. The rational cohomology rings of type $A$ real permutohedral varieties are fully described in terms of alternating permutations. Until now, only rational Betti numbers of type $B$ real permutohedral varieties have been described in terms of $B$-snakes. In this paper, we explicitly describe the multiplicative structure of the cohomology rings of type $B$ real permutohedral varieties in terms of $B$-snakes.