Mathematics Colloquia and Seminars

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A type A_{n-1} root polytope is the convex hull in R^n of the origin and a subset of the points e_i-e_j, 1\leq i< j \leq n. A collection of triangulations of these polytopes can be described by reduced forms of monomials in the subdivision algebra, which is a commutative algebra generated by n^2 variables x_{ij}, for 1\leq i< j \leq n. In a closely related noncommutative algebra, the reduced forms of monomials are unique, and correspond to shellable triangulations whose simplices are indexed by noncrossing alternating trees. Using these triangulations Ehrhart polynomials are computed. The results are extended to a more general family of polytopes.