Cohomology of type B$B$ real permutohedral varieties

Abstract Type and type 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 and type real permutohedral varieties. The rational cohomology rings of type real permutohedral varieties are fully described in terms of alternating permutations. Until now, only rational Betti numbers of type real permutohedral varieties have been described in terms of ‐snakes. In this paper, we explicitly describe the multiplicative structure of the cohomology rings of type real permutohedral varieties in terms of ‐snakes.