Valuative Invariants of Catalan Matroids

Abstract

We decompose the indicator function of each $(a, b)$-Catalan matroid polytope as a weighted sum of indicator function of matroid polytopes that correspond to direct sums of uniform matroids. Catalan matroids lie in the interior of the convex hull of direct sums of uniform matroids in the polytope of all matroids introduced by Ferroni and Fink. Moreover, we describe combinatorially the coefficients of the convex combination of direct sums of uniform matroid corresponding to an $(a,b)$-Catalan matroid. In particular, this allows us to derive explicit formulas for arbitrary valuative invariants of $(a, b)$-Catalan matroids. Among other applications, we prove that $(a,b)$-Catalan matroids are Ehrhart positive, and we find formulas for the Kazhdan--Lusztig invariants of these matroids.