Stembridge codes, permutahedral varieties, and their extensions

It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutahedral variety. Stanley obtained a formula showing that the cohomology carries a permutation representation of $\mathfrak{S}_n$. We answer a question of Stembridge on finding an explicit permutation basis of this cohomology. We observe that the Feichtner-Yuzvinsky basis for the Chow ring of the Boolean matroid is such a permutation basis, and then we construct an $\mathfrak{S}_n$-equivariant bijection between this basis and codes introduced by Stembridge, thereby giving a combinatorial proof of Stanley's formula. We obtain an analogous result for the stellahedral variety. We find a permutation basis of the permutation representation its cohomology carries. This involves the augmented Chow ring of a matroid introduced by Braden, Huh, Matherne, Proudfoot and Wang. Along the way, we obtain a general result on augmented Chow rings (which was also independently obtained by Eur) asserting that augmented Chow rings of matroids are actually Chow rings in the sense of Feichtner and Yuzvinsky. In the last part of the paper, we study enumerative aspects of the permutahedra and the stellohedra related to these permutation bases.