A polynomiality property for Littlewood–Richardson coefficients

We present a polynomiality property of the Littlewood–Richardson coefficients cλμν. The coefficients are shown to be given by polynomials in λ, μ and ν 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 λ, μ and ν. We first express the Littlewood–Richardson coefficients as a vector partition function. We then define a hyperplane arrangement from Steinbergʹs 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 cNλ NμNν is given by a polynomial in N, which partially establishes the conjecture of King et al. (CRM Proceedings and Lecture Notes, Vol. 34, 2003) that cNλ NμNν is a polynomial in N with nonnegative rational coefficients.