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Root polytopes, triangulations, and the subdivision algebra. I

Author: Karola Mészáros
Journal: Trans. Amer. Math. Soc. 363 (2011), 4359-4382
MSC (2010): Primary 05E15, 16S99, 52B11, 52B22, 51M25
Published electronically: March 16, 2011
MathSciNet review: 2817421
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Abstract: The type $ A_{n}$ root polytope $ \mathcal{P}(A_{n}^+)$ is the convex hull in $ \mathbb{R}^{n+1}$ of the origin and the points $ e_i-e_j$ for $ 1\leq i<j \leq n+1$. Given a tree $ T$ on the vertex set $ [n+1]$, the associated root polytope $ \mathcal{P}(T)$ is the intersection of $ \mathcal{P}(A_{n}^+)$ with the cone generated by the vectors $ e_i-e_j$, where $ (i, j) \in E(T)$, $ i<j$. The reduced forms of a certain monomial $ m[T]$ in commuting variables $ x_{ij}$ under the reduction $ x_{ij}x_{jk} \rightarrow x_{ik}x_{ij}+x_{jk}x_{ik}+\beta x_{ik}$ can be interpreted as triangulations of $ \mathcal{P}(T)$. Using these triangulations, the volume and Ehrhart polynomial of $ \mathcal{P}(T)$ are obtained. If we allow variables $ x_{ij}$ and $ x_{kl}$ to commute only when $ i, j, k, l$ are distinct, then the reduced form of $ m[T]$ is unique and yields a canonical triangulation of $ \mathcal{P}(T)$ in which each simplex corresponds to a noncrossing alternating forest. Most generally, in the noncommutative case, which was introduced in the form of a noncommutative quadratic algebra by Kirillov, the reduced forms of all monomials are unique.

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Additional Information

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

Keywords: Root polytope, triangulation, volume, Ehrhart polynomial, subdivision algebra, quasi-classical Yang-Baxter algebra, reduced form, noncrossing alternating tree, shelling, noncommutative Gröbner basis
Received by editor(s): October 6, 2009
Received by editor(s) in revised form: December 7, 2009
Published electronically: March 16, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.