Polynomiality properties of Kostka numbers and Littlewood-Richardson
coefficients.
Etienne Rassart, (MIT)
Abstract: The Kostka numbers $K_{\lambda\mu}$ appear in combinatorics when
expressing the Schur functions in terms of the monomial symmetric
functions, as $K_{\lambda\mu}$ counts the number of semistandard Young
tableaux of shape $\lambda$ and content $\mu$. 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 $\lambda$, the nonzero
$K_{\lambda\mu}$ 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 $A_2$ and $A_3$ (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.
Applied Algebra seminar home