Combinatorics Seminar: Combinatorics of Stembridge codes, permutahedral varieties, and their extensions

Abstract: It is well-known that the cohomology of the permutahedral variety has the classical Eulerian polynomial as its Hilbert series. A less well-known extension of this result is that the Hilbert series of the stellahedral variety is the binomial Eulerian polynomial. In this talk, we study these cohomologies and the $\mathfrak{S}_n$-representations they carry using the theory of Chow ring of matroids introduced by Feichtner and Yuzvinsky, and popularized by Adiprasito, Huh, and Katz. We answer a question of Stembridge on finding a permutation basis for the cohomology of the permutahedral variety. We find such a basis is given by the Feichtner-Yuzvinsky basis of the Chow ring of the Boolean matroid and construct and $\mathfrak{S}_n$-equivariant bijection between the basis and the codes introduced by Stembridge. Next, we consider a parallel story for the stellahedral variety by applying Braden, Huh, Matherne, Proudfoot, and Wang's augmented Chow ring in the case of the Boolean matroid. Finding a permutation basis in this case involves generalizing the graph associahedron construction of the stellohedron and proving that the augmented Chow ring is Feichtner-Yuzvinsky's Chow ring with respect to a certain building set. If time permits, we will talk about some new combinatorial interpretations of the Eulerian quasisymmetric functions, which are symmetric function analogs of the Eulerian and binomial Eulerian polynomials.

Host: Martha Precup