Etienne Rassart

Abstract:   The Kostka numbers K_l,m appear in combinatorics when expressing the Schur functions in terms of the monomial symmetric functions, as K_l,m counts the number of semistandard Young tableaux of shape l and content m. They also appear in representation theory as the multiplicities of weights in the irreducible representations of type A.

     Using a variety of tools from representation theory (Gelfand-Tsetlin diagrams), convex geometry (vector partition functions), symplectic geometry (Duistermaat-Heckman measure) and combinatorics (hyperplane arrangements), we show that the Kostka numbers are given by polynomials in the cells of a complex of cones. For fixed l, the nonzero K_l,m consist of the lattice points inside a permutahedron. By relating the complex of cones to a family of hyperplane arrangements, we provide an explanation for why the polynomials giving the Kostka numbers exhibit interesting factorization patterns in the boundary regions of the permutahedron. We will consider A₂ and A₃ (partitions with at most three and four parts) as running examples, with lots of pictures.

     I will also say a few words as to how some of the techniques used generalize to the case of Littlewood-Richardson coefficients.

     This is joint work with Sara Billey and Victor Guillemin.

Here are some directions to Northeastern University. Lake Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street.