Stembridge codes and Chow rings

2022/12/10 by Hsin-Chieh Liao, Liao, Hsin-Chieh
Computer Science · Mathematics · #05E10 #05E14 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2212.05362

openalex publication_date 2022/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutohedral variety. We answer a question of Stembridge on finding a geometric explanation of the permutation representation this cohomology carries. Our explanation involves an \mathfrakS_n-equivariant bijection between a basis for the Chow ring of the Boolean matroid and codes introduced by Stembridge. There are analogous results for the stellohedral variety. We provide a geometric explanation of the permutation representation that its cohomology carries. 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.

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