Littlewood-Richardson polynomials

A. I. Molev

Abstract

We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles, and the stability property can also be derived from the puzzle rule. As another application, we use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the (virtual) Casimir elements for the general linear Lie algebra in the basis of the (virtual) quantum immanants constructed by A. Okounkov and G. Olshanski.