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 Kl,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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