Triangulations of root polytopes
and Kirillov's conjectures

A type A_n-1 root polytope is the convex hull in Rⁿ of the origin and a subset of the points e_i-e_j, 1≤ i< j≤ n. A collection of triangulations of these polytopes can be described by reduced forms of monomials in an algebra generated by n^2 variables x_ij, for 1≤ i< j≤ 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 we compute Ehrhart polynomials of a family of root polytopes. We extend the above results to more general families of polytopes and algebras of types C_n and D_n. Special cases of our results prove several conjectures of Kirillov regarding these algebras.