Séminaire Lotharingien de Combinatoire, B42j (1999), 19 pp.
Anatol N. Kirillov, Anne Schilling, Mark Shimozono
Various Representations of the Generalized Kostka Polynomials
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.
The following versions are available: