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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $ 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$.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05371-7
PII: S 0002-9947(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.