Chopped and sliced cones and representations of Kac–Moody algebras

Abstract We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove a structure theorem expressing the number of integral points in a slice of such a cone by means of a vector partition function. We observe that this notion applies to weight multiplicities of Kac–Moody algebras and to Clebsch–Gordan coefficients for semisimple Lie algebras. This has algorithmic applications, as we demonstrate computing some explicit examples.

[1]  Jesús A. De Loera,et al.  On the Computation of Clebsch–Gordan Coefficients and the Dilation Effect , 2006, Exp. Math..

[2]  Peter Littelmann,et al.  Paths and root operators in representation theory , 1995 .

[3]  Gert Heckman,et al.  Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups , 1982 .

[4]  Thomas Bliem.  Weight Multiplicities for so5(C) , 2008, ITSL.

[5]  Michele Vergne,et al.  Residue formulae for vector partitions and Euler-MacLaurin sums , 2003, Adv. Appl. Math..

[6]  Bernd Sturmfels.  Note on Vector Partition Functions , 2004 .

[7]  Jesús A. De Loera,et al.  Counting Integer Flows in Networks , 2003, Found. Comput. Math..

[8]  Charles Cochet,et al.  Volume Computation for Polytopes and Partition Functions for Classical Root Systems , 2006, Discret. Comput. Geom..

[9]  Peter Littelmann,et al.  Cones, crystals, and patterns , 1998 .

[10]  I. Stewart,et al.  Infinite-dimensional Lie algebras , 1974 .

[11]  Etienne Rassart,et al.  A vector partition function for the multiplicities of sl_k(C) , 2003, math/0307227.

[12]  Etienne Rassart.  A polynomiality property for Littlewood-Richardson coefficients , 2004, J. Comb. Theory, Ser. A.

[13]  G. R. Blakley.  Combinatorial remarks on partitions of a multipartite number , 1964 .

[14]  Bernd Sturmfels,et al.  On Vector Partition Functions , 1995, J. Comb. Theory, Ser. A.

[15]  Y. Tschinkel,et al.  Geometric methods in algebra and number theory , 2005 .

[16]  P. Tirao,et al.  A closed formula for weight multiplicities of representations of , 2004 .

[17]  Corrado De Concini,et al.  Nested sets and Jeffrey-Kirwan residues , 2005 .

[18]  Andrei Zelevinsky,et al.  Tensor product multiplicities, canonical bases and totally positive varieties , 1999, math/9912012.

[19]  Israel M. Gelfand,et al.  Finite-dimensional representations of the group of unimodular matrices , 1950 .

[20]  Wolfgang Dahmen,et al.  The number of solutions to linear diophantine equations and multivariate splines , 1988 .