A polynomiality property for Littlewood-Richardson coefficients

Abstract

We present a polynomiality property of the Littlewood-Richardson coefficients c_{\lambda\mu}^{\nu}. The coefficients are shown to be given by polynomials in \lambda, \mu and \nu on the cones of the chamber complex of a vector partition function. We give bounds on the degree of the polynomials depending on the maximum allowed number of parts of the partitions \lambda, \mu and \nu. We first express the Littlewood-Richardson coefficients as a vector partition function. We then define a hyperplane arrangement from Steinberg's formula, over whose regions the Littlewood-Richardson coefficients are given by polynomials, and relate this arrangement to the chamber complex of the partition function. As an easy consequence, we get a new proof of the fact that c_{N\lambda N\mu}^{N\nu} is given by a polynomial in N, which partially establishes the conjecture of King, Tollu and Toumazet that c_{N\lambda N\mu}^{N\nu} is a polynomial in N with nonnegative rational coefficients.