Toric matrix Schubert varieties and their polytopes

Abstract

Abstract. Given a matrix Schubert variety Xπ, it can be written as Xπ = Yπ × C q (where q is maximal possible). We characterize when Yπ is toric (with respect to a (C * ) 2n−1 -action) and study the associated polytope Φ(P(Yπ)) of its projectivization. We construct regular triangulations of Φ(P(Yπ)) which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

Complexity of the usual torus action on Kazhdan-Lusztig varieties

“…Given w ∈ S n there exists an affine variety Y w such that X w = Y w × C d where d is as large as possible. Let us describe Y w and d more precisely, following [EM16] with the change of conventions we explain in Remark 3.9. We remark that since the connected components of D…”

“…Remark 3.9. The matrix Schubert varieties appearing in [EM16,Por20] are defined using a (B − × B)-action on C n×n where B − is the group of lower diagonal matrices. This changes the conventions, e.g.…”

“…The Y w of complexity-0, i.e. toric varieties, have been classified in [EM16]. A hook with corner (i, j) consists of boxes (i , j ) such that j = j and i ≥ i or i = i and j ≥ j .…”

“…We show in Section 3.2 that the weight cone of the usual torus action on Y w is the edge cone of an acyclic bipartite graph. In [EM16] the authors give a combinatorial criterion for Y w to be toric. By simple arguments using graphs to investigate the dimension of the weight cone, we hoped to classify complexity-1 matrix Schubert varieties, but surprisingly it turns out that there are none, while all other complexities can be obtained.…”

From generalized permutahedra to Grothendieck polynomials via flow polytopes

“…The flow polytope F G associated to a directed acyclic graph G is the set of all flows f : E(G) → R 0 of size one. Flow polytopes are fundamental objects in combinatorial optimization [17], and in the past decade they were also uncovered in representation theory [1,12], the study of the space of diagonal harmonics [8,13], and the study of Schubert and Grothendieck polynomials [4,5]. A natural way to analyze a convex polytope is to dissect it into simplices.…”