Root polytopes, triangulations, and the subdivision algebra. I

Abstract

Abstract. The type A n root polytope P(A + n ) is the convex hull in R n+1 of the origin and the points e i − e j for 1 ≤ i < j ≤ n + 1. Given a tree T on the vertex set [n + 1], the associated root polytope P(T ) is the intersection of P(A + n ) with the cone generated by the vectors e i − e j , where (i, j) ∈ E(T ), i < j. The reduced forms of a certain monomial m[T ] in commuting variables x ij under the reduction x ij x jk → x ik x ij + x jk x ik + βx ik can be interpreted as triangulations of P(T ). Using these triangulations, the volume and Ehrhart polynomial of P(T ) are obtained. If we allow variables x ij and x kl to commute only when i, j, k, l are distinct, then the reduced form of m[T ] is unique and yields a canonical triangulation of P(T ) in which each simplex corresponds to a noncrossing alternating forest. Most generally, in the noncommutative case, which was introduced in the form of a noncommutative quadratic algebra by Kirillov, the reduced forms of all monomials are unique.

Root Polytopes and Borel Subalgebras

“…Some authors (see in particular [16] and [17]), intend by root polytope the convex hull of the positive roots together with the origin, first introduced in [9] for the root system of type A n . We call this the positive root polytope and, if confusion may arise, we call P Φ the complete root polytope.…”

Subword complexes via triangulations of root polytopes

“…In this paper we restrict ourselves to a class of root polytopes including P(A + n ), which have subdivision algebras as defined in [12]. We discuss subdivision algebras in relation to Grothendieck polynomials in Section 5.…”

“…, 2] and whose work served as the stepping stone for the present project. In the papers [12,14,15,16] Mészáros studied triangulations of root polytopes that we utilize in this work (some of the mentioned papers are in the language of flow polytopes, but in view of [17,Section 4] some of their content can also be understood in the language of root polytopes).…”

Toric matrix Schubert varieties and their polytopes

“…The main tool developed in [Mész11] which is used to construct the canonical triangulation of Theorem 5.1 is the subdivision algebra. Subdivision algebras have since been utilized in solving various problems in [Mész14a, Mész14b, Mész15a, Mész15b, Mész15c, MM15].…”