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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.
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