Underlying Flag Polymatroids

TL;DR

This paper explores the geometric and combinatorial structures of flag polymatroids, generalizing underlying flag matroids, and connects them to toric varieties and Young tableaux enumeration.

Contribution

It introduces a geometric framework linking matroids and flag polymatroids via monotone path polytopes, extending the theory to polymatroids and their associated toric varieties.

Findings

Polytopes of underlying flag polymatroids are simple and normally equivalent to nestohedra.

Polymatroids from subspace arrangements produce smooth toric varieties in flag varieties.

Monotone paths on polymatroid polytopes relate to Young tableaux enumeration.

Abstract

We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying flag matroids to polymatroids. We show that the polytopes associated to underlying flag polymatroid are simple by proving that they are normally equivalent to certain nestohedra. We use this to show that polymatroids realized by subspace arrangements give rise to smooth toric varieties in flag varieties and we interpret our construction in terms of toric quotients. We give various examples that illustrate the rich combinatorial structure of flag polymatroids. Finally, we study general monotone paths on polymatroid polytopes, that relate to the enumeration of certain Young tableaux.