Subword complexes via triangulations of root polytopes

Abstract

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials.

Toric matrix Schubert varieties and their polytopes

“…Since the appearance of Knutson's and Miller's work in [KM04,KM05] there has been a flurry of research into the geometric realization of subword complexes with progress in realizing families of subword complexes homeomorhpic to spheres [Stu11,Ceb12,PP12,PS12,SS12,CLS14,BCL15]. The first paper which tackled realizing a large family of subword complexes which are homeomorphic to balls is [EM15]. Our current paper is further progress in this direction.…”

“…For a definition of a reduced pipe dream, also called RC-graphs, as well as further background on it see [BB93,FK94,KM04] or [EM15].…”

“…In Section 4 we define subword complexes and show that the aftermentioned triangulations of Φ(P(Y π )) are a geometric realization of a family of subword complexes homeomorphic to balls. Finally, in Section 5 we show how to view the results of [EM15] in terms of canonical triangulations of Φ(P(Y π )).…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…. , 1), namely, w (k) n,p = 1 k ×w (p,n) , we treat only the case k = 1, see also [39]. In the case k ≥ 2 one comes to a task to count and find a lattice path type interpretation for the number of k-pgulations of a convex n-gon that is the number of partitioning of a convex n-gon on parts which are all equal to a convex (p + 2)-gon, by a (maximal) family of diagonals such that each diagonal has at most k internal intersections with the members of a family of diagonals selected.…”

“…39 Known also as Orlik-Terao algebra. 40 For the readers convenience we recall definitions of statistics inv(F) and the major index maj(F).…”