A polynomiality property for Littlewood-Richardson coefficients

  • Authors:

  • Etienne Rassart

  • Affiliations:

  • Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2004

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Abstract

We present a polynomiality property of the Littlewood-Richardson coefficients cλµv. The coefficients are shown to be given by polynomials in λ, µ and v 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 v. 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 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.