On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Ja n 20 05 The computation of Kostka numbers and Littlewood - Richardson coefficients is # P - complete

Kostka numbers and Littlewood-Richardson coefficients play an essential role in the representation theory of the symmetric groups and the special linear groups. There has been a significant amount of interest in their computation ([1], [10], [11], [2], [3]). The issue of their computational complexity has been a question of folklore, but was asked explicitly by E. Rassart [10]. We prove that th...

Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients

Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers Kλβ and Littlewood-Richardson coefficients cλμ (the type A weight multiplicities and Clebsch-Gordan coefficients). We show that both are given by piecewise polynomial functions in the entries of the partitions and compositions parametrizing them, and that the domains of polynomia...

An Elementary Method for Computing the Kostka Coefficients

A Generalization of the Kostka-foulkes Polynomials

Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials and twocolumn Macdonald-Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincaré polynomials of isotypic components of certain graded GL(n)-modules supported in a nilpotent conjugacy class closure i...

Finding polynomials to count lattice points; Computer explorations with MuPAD-Combinat

We are interested in Algebraic Combinatorics, a subject we give an overview, and in Symbolic Computation. In this paper we describe a problem we recently encountered in some experiments we are still carrying on. This problem is about generating polynomials counting lattice points in certain particular types of convex polytopes. The lattice points numbers are interpreted as Kostka numbers and Li...