## 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.

This paper is available as a