70 research outputs found
A polynomiality property for Littlewood-Richardson coefficients
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.Comment: 14 page
A combinatorial proof that Schubert vs. Schur coefficients are nonnegative
We give a combinatorial proof that the product of a Schubert polynomial by a
Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses
Assaf's theory of dual equivalence to show that a quasisymmetric function of
Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of
Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants
of the Grassmannian.Comment: 26 pages, several colored figure
Shifted symmetric functions and multirectangular coordinates of Young diagrams
In this paper, we study shifted Schur functions , as well as a
new family of shifted symmetric functions linked to Kostka
numbers. We prove that both are polynomials in multi-rectangular coordinates,
with nonnegative coefficients when written in terms of falling factorials. We
then propose a conjectural generalization to the Jack setting. This conjecture
is a lifting of Knop and Sahi's positivity result for usual Jack polynomials
and resembles recent conjectures of Lassalle. We prove our conjecture for
one-part partitions.Comment: 2nd version: minor modifications after referee comment
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