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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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[1] Matthias Beck and Sinai Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. · Zbl 1114.52013
[2] Emeric Deutsch, Dyck path enumeration, Discrete Math. 204 (1999), no. 1-3, 167 – 202. · Zbl 0932.05006 · doi:10.1016/S0012-365X(98)00371-9 · doi.org
[3] Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147 – 182. · Zbl 0940.05070
[4] W. Fong, Triangulations and Combinatorial Properties of Convex Polytopes, Ph.D. Thesis, 2000.
[5] Israel M. Gelfand, Mark I. Graev, and Alexander Postnikov, Combinatorics of hypergeometric functions associated with positive roots, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 205 – 221. · Zbl 0876.33011 · doi:10.1007/978-1-4612-4122-5_10 · doi.org
[6] Edward L. Green, Noncommutative Gröbner bases, and projective resolutions, Computational methods for representations of groups and algebras (Essen, 1997) Progr. Math., vol. 173, Birkhäuser, Basel, 1999, pp. 29 – 60. · Zbl 0957.16033
[7] A. N. Kirillov, On some quadratic algebras, L. D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, vol. 201, Amer. Math. Soc., Providence, RI, 2000, pp. 91 – 113. · Zbl 0966.05080 · doi:10.1090/trans2/201/07 · doi.org
[8] A. N. Kirillov, personal communication, 2007.
[9] K. Mészáros, Root polytopes, triangulations, and the subdivision algebra, I, http:// arxiv.org/abs/0904.2194. · Zbl 1233.05215
[10] Alexander Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN 6 (2009), 1026 – 1106. · Zbl 1162.52007 · doi:10.1093/imrn/rnn153 · doi.org
[11] V. Reiner, Quotients of Coxeter complexes and P-Partitions, Ph.D. Thesis, 1990. · Zbl 0751.06002
[12] Victor Reiner, Signed posets, J. Combin. Theory Ser. A 62 (1993), no. 2, 324 – 360. · Zbl 0773.06008 · doi:10.1016/0097-3165(93)90052-A · doi.org
[13] Richard P. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333 – 342. Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978). · Zbl 0812.52012
[14] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. · Zbl 0928.05001
[15] Thomas Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47 – 74. , https://doi.org/10.1016/0166-218X(82)90033-6 Thomas Zaslavsky, Erratum: ”Signed graphs”, Discrete Appl. Math. 5 (1983), no. 2, 248. · Zbl 0503.05060 · doi:10.1016/0166-218X(83)90047-1 · doi.org
[16] Thomas Zaslavsky, Orientation of signed graphs, European J. Combin. 12 (1991), no. 4, 361 – 375. · Zbl 0761.05095 · doi:10.1016/S0195-6698(13)80118-7 · doi.org
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