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.
Categories: math.AG math.CO