Root polytopes, triangulations, and the subdivision algebra. I

The type $A_{n}$ root polytope $\mathcal{P}(A_{n}^+)$ is the convex hull in $\mathbb{R}^{n+1}$ of the origin and the points $e_i-e_j$ for $1\leq i<j \leq n+1$. Given a tree $T$ on the vertex set $[n+1]$, the associated root polytope $\mathcal{P}(T)$ is the intersection of $\mathcal{P}(A_{n}^+)$ with the cone generated by the vectors $e_i-e_j$, where $(i, j) \in E(T)$, $i<j$. The reduced forms of a certain monomial $m[T]$ in commuting variables $x_{ij}$ under the reduction $x_{ij}x_{jk} \rightarrow x_{ik}x_{ij}+x_{jk}x_{ik}+\beta x_{ik}$ can be interpreted as triangulations of $\mathcal{P}(T)$. Using these triangulations, the volume and Ehrhart polynomial of $\mathcal{P}(T)$ 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 $\mathcal{P}(T)$ 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.