Root polytopes, triangulations, and the subdivision algebra, II

Author:
Karola Mészáros

Journal:
Trans. Amer. Math. Soc. **363** (2011), 6111-6141

MSC (2010):
Primary 05E15, 16S99, 51M25, 52B11

Published electronically:
April 28, 2011

MathSciNet review:
2817421

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Abstract: The type root polytope is the convex hull in of the origin and the points for . Given a graph , with edges labeled positive or negative, associate to each edge of a vector v which is if , , is labeled negative and if it is labeled positive. For such a signed graph , the associated root polytope is the intersection of with the cone generated by the vectors v, for edges in . The reduced forms of a certain monomial in commuting variables under reductions derived from the relations of a bracket algebra of type , can be interpreted as triangulations of . Using these triangulations, the volume of can be calculated. If we allow variables to commute only when all their indices are distinct, then we prove that the reduced form of , for ``good'' graphs , is unique and yields a canonical triangulation of in which each simplex corresponds to a noncrossing alternating graph in a type 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 . We also study the bracket algebra of type 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 .

**[BR]**Matthias Beck and Sinai Robins,*Computing the continuous discretely*, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. MR**2271992****[D]**Emeric Deutsch,*Dyck path enumeration*, Discrete Math.**204**(1999), no. 1-3, 167–202. MR**1691869**, 10.1016/S0012-365X(98)00371-9**[FK]**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. MR**1667680****[F]**W. Fong,*Triangulations and Combinatorial Properties of Convex Polytopes*, Ph.D. Thesis, 2000.**[GGP]**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. MR**1429893**, 10.1007/978-1-4612-4122-5_10**[G]**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. MR**1714602****[K1]**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. MR**1772287****[K2]**A. N. Kirillov, personal communication, 2007.**[M]**K. Mészáros, Root polytopes, triangulations, and the subdivision algebra, I,`http://``arxiv.org/abs/0904.2194`.**[P]**Alexander Postnikov,*Permutohedra, associahedra, and beyond*, Int. Math. Res. Not. IMRN**6**(2009), 1026–1106. MR**2487491**, 10.1093/imrn/rnn153**[R1]**V. Reiner,*Quotients of Coxeter complexes and P-Partitions*, Ph.D. Thesis, 1990.**[R2]**Victor Reiner,*Signed posets*, J. Combin. Theory Ser. A**62**(1993), no. 2, 324–360. MR**1207741**, 10.1016/0097-3165(93)90052-A**[S1]**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). MR**593545****[S2]**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. MR**1676282****[Z1]**Thomas Zaslavsky,*Signed graphs*, Discrete Appl. Math.**4**(1982), no. 1, 47–74. MR**676405**, 10.1016/0166-218X(82)90033-6**[Z2]**Thomas Zaslavsky,*Orientation of signed graphs*, European J. Combin.**12**(1991), no. 4, 361–375. MR**1120422**, 10.1016/S0195-6698(13)80118-7

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Additional Information

**Karola Mészáros**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI:
https://doi.org/10.1090/S0002-9947-2011-05371-7

Keywords:
Root polytope,
type $C_{n}$,
type $D_{n}$,
triangulation,
volume,
Ehrhart polynomial,
noncrossing alternating graph,
subdivision algebra,
bracket algebra,
reduced form,
noncommutative Gröbner basis

Received by editor(s):
October 6, 2009

Received by editor(s) in revised form:
April 17, 2010

Published electronically:
April 28, 2011

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.