| The classical Littlewood-Richardson coefficients are remarkable nonnegative integers which occupy a prominent place in combinatorics, representation theory and geometry. We introduce a natural generalization of these coefficients called the Littlewood-Richardson polynomials and give a combinatorial rule for their calculation. We apply this rule to expand the product 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. The same rule yields a positive and stable formula for the product of equivariant Schubert classes on the Grassmannian. The first positive formula for such a product was given by A. Knutson and T. Tao by using combinatorics of puzzles although the stability property was not apparent from their rule. |