Quadratic exchange equations for Coxeter matroids

TL;DR

This paper introduces quadratic equations derived from minuscule representations that, when tropicalized, precisely characterize Coxeter matroids satisfying the strong exchange property, linking algebraic and combinatorial structures.

Contribution

It establishes a new set of quadratic equations for Coxeter matroids via tropicalisation of minuscule representation orbits, connecting algebraic geometry with combinatorics.

Findings

Quadratic equations characterize Coxeter matroids with the strong exchange property.

Tropicalisation links algebraic equations to combinatorial properties.

New algebraic tools for studying Coxeter matroids and their properties.

Abstract

Tropicalisation (with trivial coefficients) is a process that turns a polynomial equation into a combinatorial predicate on subsets of the set of variables. We show that for each minuscule representation of a simple reductive group, there is a set of quadratic equations cutting out the orbit of the highest weight vector whose tropicalisation characterises the set of Coxeter matroids for that representation which satisfy the strong exchange property.