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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)].)

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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