Decompositions of Chow rings of direct sums of matroids
Paweł Pielasa
TL;DR
This work develops two dual recursive decompositions for the Chow ring $\underline{\mathrm{CH}}(M\oplus N)$ of the direct sum of matroids, presenting a detailed module decomposition over $\underline{\mathrm{CH}}(M)\otimes\underline{\mathrm{CH}}(N)$ that includes a sum over nonempty flats and a shifted degree $[-1]$ component. The authors extend these decompositions to the augmented Chow ring and generalize the framework to weakly ranked posets via Chow polynomials, leveraging the KLS theory to translate between combinatorial invariants and Hilbert functions of Chow rings. A key methodological highlight is the construction of explicit injection maps using toric geometry of Bergman fans (and their stars) to realize the decompositions, together with dual perspectives that recover irreducible components and yield a new recurrence for Eulerian numbers. The results unify and extend prior decompositions for matroids, provide new recurrences for classical combinatorial sequences, and connect Chow-theoretic decompositions with poset Chow polynomials, offering a robust toolkit for analyzing direct sums of matroids and related combinatorial structures. The work has potential implications for understanding the Kähler package in broader non-realizable settings and for exploring polynomial invariants in related poset frameworks.
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.