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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05265-7

Published electronically:
March 16, 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 tree on the vertex set , the associated root polytope is the intersection of with the cone generated by the vectors , where , . The reduced forms of a certain monomial in commuting variables under the reduction can be interpreted as triangulations of . Using these triangulations, the volume and Ehrhart polynomial of are obtained. If we allow variables and to commute only when are distinct, then the reduced form of is unique and yields a canonical triangulation of 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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05265-7

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.