The cohomology rings of real permutohedral varieties

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.