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Root polytopes, triangulations, and the subdivision algebra. II. (English) Zbl 1233.05216
Summary: 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 $$D_n$$.

(For part I, see [ibid. 363, No. 8, 4359–4382 (2011; Zbl 1233.05215)].)

##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 16S99 Associative rings and algebras arising under various constructions 51M25 Length, area and volume in real or complex geometry 52B11 $$n$$-dimensional polytopes
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