The Universal Valuation of Coxeter Matroids

TL;DR

This paper introduces the universal valuation for Coxeter matroids, extending valuation theory to a broad class of combinatorial and geometric structures related to Lie groups and flag varieties.

Contribution

It computes the universal valuative invariant for Coxeter matroids and related polyhedral objects, advancing the understanding of their combinatorial and geometric properties.

Findings

Computed the universal valuation of Coxeter matroids.

Analyzed Coxeter Schubert matroids and their relation to flag varieties.

Extended valuation theory to generalized Coxeter permutohedra.

Abstract

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.