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

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

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 .

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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.