Ubiquity of Kostka Polynomials

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 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 polyno-mials; 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, MacMa-hon, Gelfand–Tsetlin and Chan–Robbins polytopes. We describe certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polyno-mials; domino tableaux and rigged configurations. We study also some properties of l–restricted generalized exponents and the stable behaviour of certain Kostka–Foulkes polynomials.