Dual Chow polynomials of matroids and posets

Abstract

We introduce and study dual Chow functions associated to kernels in incidence algebras of weakly ranked posets. Given a kernel, its dual Chow function is defined as the Chow function associated to the sign-twisted reverse kernel. For kernels satisfying a natural skew-symmetry condition, such as the Eulerian kernel of an Eulerian poset or the kernel given by R-polynomials on Bruhat intervals, this construction recovers the ordinary Chow function. In contrast, when this skew-symmetry fails, the dual Chow function gives a genuinely different invariant. The main example considered in this paper is the dual Chow function associated to the characteristic function. We develop the basic theory of these dual Chow functions, with particular emphasis on posets arising from matroids. We prove chain formulas, unimodality and gamma-positivity results, formulas under standard poset operations, and deletion formulas for matroids. Along the way, we also obtain a general deletion formula for the ab-index of matroids, which leads to new formulas for extended ab-indices and, in turn, specializes to several deletion formulas appearing in the literature.