Root polytopes, triangulations, and the subdivision algebra, I

TL;DR

This paper explores the geometric and algebraic properties of root polytopes associated with trees, establishing connections between triangulations, volume, Ehrhart polynomials, and algebraic reductions, with implications for combinatorial and geometric understanding.

Contribution

It introduces a novel interpretation of monomial reductions as triangulations of root polytopes and derives volume and Ehrhart polynomial formulas using these triangulations.

Findings

Reduced forms correspond to triangulations of root polytopes.

Volume and Ehrhart polynomial of P(T) are explicitly obtained.

Unique reduced forms in noncommutative case yield canonical triangulations.

Abstract

The type A_n full root polytope 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 the full root polytope with the cone generated by the vectors e_i-e_j, where (i, j) is an edge of 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}+\beta x_{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, the reduced forms of all monomials in the noncommutative case are unique.