Geodesic
Various Representations of the Generalized Kostka Polynomials
Séminaire lotharingien de combinatoire, Tome 42 (1998-1999)

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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. 

@article{SLC_1998-1999_42_a10,
     author = {Anatol N. Kirillov and Anne Schilling and Mark Shimozono},
     title = {Various {Representations} of the {Generalized} {Kostka} {Polynomials}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {42},
     year = {1998-1999},
     url = {http://geodesic.mathdoc.fr/item/SLC_1998-1999_42_a10/}
}
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Anatol N. Kirillov; Anne Schilling; Mark Shimozono. Various Representations of the Generalized Kostka Polynomials. Séminaire lotharingien de combinatoire, Tome 42 (1998-1999). http://geodesic.mathdoc.fr/item/SLC_1998-1999_42_a10/