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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