Polynomiality properties of Kostka numbers and LittlewoodRichardson coefficientsEtienne RassartMIT
October 22,

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 (GelfandTsetlin diagrams), convex geometry (vector partition functions), symplectic geometry (DuistermaatHeckman 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 LittlewoodRichardson coefficients. This is joint work with Sara Billey and Victor Guillemin. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

