Root polytopes, triangulations, and the subdivision algebra, II

Abstract:

The type $C_{n}$ root polytope $\mathcal {P}(C_{n}^+)$ is the convex hull in $\mathbb {R}^{n}$ of the origin and the points $e_i-e_j, e_i+e_j, 2e_k$ for $1\leq i<j \leq n, k \in [n]$. Given a graph $G$, with edges labeled positive or negative, associate to each edge $e$ of $G$ a vector v$(e)$ which is $e_i-e_j$ if $e=(i, j)$, $i<j$, is labeled negative and $e_i+e_j$ if it is labeled positive. For such a signed graph $G$, the associated root polytope $\mathcal {P}(G)$ is the intersection of $\mathcal {P}(C_{n}^+)$ with the cone generated by the vectors v$(e)$, for edges $e$ in $G$. The reduced forms of a certain monomial $m[G]$ in commuting variables $x_{ij}, y_{ij}, z_k$ under reductions derived from the relations of a bracket algebra of type $C_n$, can be interpreted as triangulations of $\mathcal {P}(G)$. Using these triangulations, the volume of $\mathcal {P}(G)$ can be calculated. If we allow variables to commute only when all their indices are distinct, then we prove that the reduced form of $m[G]$, for “good” graphs $G$, is unique and yields a canonical triangulation of $\mathcal {P}(G)$ in which each simplex corresponds to a noncrossing alternating graph in a type $C$ sense. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type $C_n$. We also study the bracket algebra of type $D_n$ and show that a family of monomials has unique reduced forms in it. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type $D_n$.