Decompositions of Chow rings of direct sums of matroids

TL;DR

This paper establishes recursive decompositions of Chow rings of direct sums of matroids, leading to new formulas for Eulerian numbers and generalizations to posets, advancing algebraic combinatorics.

Contribution

It introduces dual recursive decompositions of Chow rings for matroid direct sums and extends these results to augmented rings and posets, providing new structural insights.

Findings

Decomposition of Chow rings into irreducible modules

Recursive formula for Eulerian numbers

Generalization to Chow polynomials of posets

Abstract

We prove two dual recursive decompositions as a graded $\underline{\mathrm{CH}}(M)\otimes \underline{\mathrm{CH}}(N)$-module of the Chow ring $\underline{\mathrm{CH}}(M\oplus N)$ of the direct sum of matroids. We use this to obtain a decomposition of $\underline{\mathrm{CH}}(M\oplus N)$ into irreducible $\underline{\mathrm{CH}}(M) \otimes \underline{\mathrm{CH}}(N)$-modules. The result implies a new recursive formula for the Eulerian numbers. Similarly, we find a recursive decomposition of the augmented Chow ring $\mathrm{CH}(M \oplus N)$ into $\mathrm{CH}(M)\otimes \mathrm{CH}(N)$-modules, generalizing some of the results of arXiv:2002.03341. We prove analogous decompositions of (augmented) Chow polynomials of weakly ranked posets in the sense of arXiv:2411.04070.