Combinatorics Seminar -- Stembridge codes, Chow ring of the permutahedral variety, and their extensions

It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutahedral variety. In this talk, we answer a question of Stembridge on finding a permutation basis for the permutation representation this cohomology carries. Our method involves an S_n-equivariant bijection between a basis for the Chow ring of the Boolean matroid and codes introduced by Stembridge. There is a parallel binomial Eulerian story related to the stellahedral variety. We also find a permutation basis for the permutation representation on the cohomology of the stellahedral variety. This involves the augmented Chow ring of a matroid introduced by Braden, Huh, Matherne, Proudfoot and Wang. Along the way, we also obtain some new results on augmented Chow rings.