Ubiquity of Kostka polynomials
Abstract
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 -Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized -Gaussian polynomials; parabolic Kostant partition function and its -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 -restricted generalized exponents and the stable behaviour of certain Kostka-Foulkes polynomials.
Cite
@article{arxiv.math/9912094,
title = {Ubiquity of Kostka polynomials},
author = {Anatol N. Kirillov},
journal= {arXiv preprint arXiv:math/9912094},
year = {2007}
}
Comments
LaTeX, 104 pages, revised version, many new exercises added (about 35 pages), and some typos are corrected