Polymatroid Schubert varieties

Abstract: The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "matroid Schubert variety". In contrast, the lattice of flats of a polymatroid exhibits many combinatorial pathologies and admits no similar geometric model. We address this situation by defining the lattice $\mathcal L_P$ of "combinatorial flats" of a polymatroid $P$. Combinatorially, $\mathcal L_P$ exhibits good behavior analogous to that of $\mathcal L_M$: it is graded, determines $P$ when $P$ is simple, and is top-heavy. When $P$ is realizable over a field of characteristic 0, we show that $\mathcal L_P$ is modelled by a "polymatroid Schubert variety". Our work generalizes a number of results of Ardila-Boocher and Huh-Wang on matroid Schubert varieties; however, the geometry of polymatroid Schubert varieties is noticeably more complicated than that of matroid Schubert varieties. Many natural questions remain open.