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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Root polytopes, triangulations, and the subdivision algebra. I
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Trans. Amer. Math. Soc. 363 (2011), 4359-4382 Request permission


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
  • MR Author ID: 823389
  • Received by editor(s): October 6, 2009
  • Received by editor(s) in revised form: December 7, 2009
  • Published electronically: March 16, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4359-4382
  • MSC (2010): Primary 05E15, 16S99, 52B11, 52B22, 51M25
  • DOI:
  • MathSciNet review: 2792991