Normalized volumes of configurations related with root systems and complete bipartite graphs

Let Phi subset of Z(n) denote one of the classical irreducible root systems A(n-1), B-n, C-n and D-n, and write Phi((+)) for the configuration consisting of all positive roots of Phi together with the origin of R-n. Gelfand, Graev and Postnikov, in: V.I. Arnold, I.M. Gelfand, M. Smirnov, V.S. Retakh (Eds.), Arnold-Gelfand, Mathematics Seminars, Geometry and Singularity Theory, Birkhauser, Boston, 1997, pp. 205-221 showed that by constructing an explicit unimodular triangulation, the normalized volume of the convex hull of A(n-1)((+)) is equal to the Catalan number. On the other hand, Fong (Triangulations and Combinatorial Properties of Convex Polytopes, Dissertation, MIT Press, Cambridge, MA, 2000) computed the normalized volume of the convex hull of each of the configurations B-n((+)), C-n((+)) and D-n((+)). Moreover, the normalized volume of the convex hull of the subconfiguration of A(n-1)((+)) arising from a complete bipartite graph was computed by Ohsugi and Hibi (Illinois J. Math. 44 (2000) 391) and Fong. The purpose of the present paper is, via the theory of Grobner bases of toric ideals and triangulations, to compute the normalized volume of the convex hull of each of the subconfigurations of B-n((+)), C-n((+)) and D-n((+)) arising from a complete bipartite graph. (C) 2002 Elsevier Science B.V. All rights reserved.