Grothendieck weights and K-theoretic positivity for matroids

Abstract

We introduce a method for studying $K$-theoretic positivity on permutohedral toric varieties through the topology of spaces arising in tropical geometry. The key ingredient is the theory of Grothendieck weights developed by the author. We prove two positivity results using this method. The first result is the positivity of the Euler characteristics of tautological bundles associated with an arbitrary matroid and twisted by a nef line bundle. This gives numerical evidence for a conjectural vanishing theorem. The second result generalizes the external activity complex of Berget--Fink, originally defined for a pair of matroids, to the case of any tuple of matroids with no common loop. We deduce a formula for its graded $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of the matroids. After a change of variables, its coefficients alternate in sign. We also prove the Cohen--Macaulayness of each such complex using the vanishing theorems for combinatorial geometries developed by Eur--Fink--Larson. This proof is new even in the case of a pair of matroids. As an application, we interpret certain Chern numbers of tautological quotient classes as counts of facets, partially answering a question of Berget--Eur--Spink--Tseng.