Quadratic exchange equations for Coxeter matroids

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.

49 pages, 7 figures. Added Remark 4.12, a discussion of type B and D equations derived in the literature. Moved proof of Theorem 4.8 to appendix