Root polytopes, triangulations, and the subdivision algebra, II

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.