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Séminaire Lotharingien de Combinatoire, B42j (1999), 19 pp.

# Anatol N. Kirillov, Anne Schilling, Mark Shimozono

# Various Representations of the Generalized Kostka Polynomials

**Abstract.**
The generalized Kostka polynomials *K*_{l,R}(*q*) are labeled by a
partition *l* and a sequence of rectangles *R*. They are *q*-analogues
of multiplicities of the finite-dimensional irreducible representation
*W*(*l*) of *gl*(*n*) with highest weight *l* in the tensor product of the
*W*(*R*(*i*))'s. We review several representations
of the generalized Kostka polynomials, such as the charge, path space,
quasi-particle and bosonic representation. In addition we describe a
bijection between Littlewood-Richardson tableaux and rigged
configurations, and sketch a proof that it preserves the appropriate
statistics. This proves in particular the equality of the quasi-particle
and charge representation of the generalized Kostka polynomials.

Received: December 16, 1998; Accepted: February 15, 1999.

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