1999/12/12 by Anatol N. Kirillov, Kirillov, Anatol N. · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #math.CO #math.QA
paper · pdf · doi:10.48550/arxiv.math/9912094
LaTeX, 104 pages, revised version, many new exercises added (about 35 pages), and some typos are corrected
arxiv created 2000/11/27 · arxiv updated 2009/11/30
We report about results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call \it Liskova semigroup. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular q-Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized q-Gaussian polynomials; parabolic Kostant partition function and its q-analog; certain generating functions on the set of transportation matrices. In each case we apply rigged configurations technique to obtain some interesting and new information about Kostka-Foulkes and parabolic Kostka polynomials, Kostant partition function, MacMahon, Gelfand-Tsetlin and Chan-Robbins polytopes. We describe certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polynomials; domino tableaux and rigged configurations. We study also some properties of l-restricted generalized exponents and the stable behaviour of certain Kostka-Foulkes polynomials.