Categorical valuative invariants of polyhedra and matroids

TL;DR

This paper introduces a new categorical framework for valuative invariants of polyhedra and matroids, enabling more sophisticated calculations and extensions of classical invariants within an additive category.

Contribution

It provides categorical lifts of key matroid invariants, such as the Poincare and Kazhdan--Lusztig polynomials, facilitating equivariant computations and new theoretical insights.

Findings

Categorical lifts of classical matroid invariants are constructed.

The framework allows equivariant calculations respecting automorphism groups.

New extensions of the Kazhdan--Lusztig polynomial are developed.

Abstract

We introduce the notion of a categorical valuative invariant of polyhedra or matroids, in which alternating sums of numerical invariants are replaced by split exact sequences in an additive category. We provide categorical lifts of a number of valuative invariants of matroids, including the Poincare polynomial, the Chow and augmented Chow polynomials, and certain two-variable extensions of the Kazhdan--Lusztig polynomial and Z-polynomial. These lifts allow us to perform calculations equivariantly with respect to automorphism groups of matroids.