Ubiquity of Kostka polynomials (Q2751975)

In this paper some results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with representation theory and combinatorics are reported. The set of all parabolic Kostka polynomials forms a semigroup, called here Liskova semigroup. It is shown that polynomials frequently appearing in representation theory and combinatorics belong to the Liskova semigroup. Among such polynomials are studied rectangular \(q\)-Catalan numbers, generalized exponent polynomials, principal specializations of the internal product of Schur functions, generalized \(q\)-Gaussian polynomials, the parabolic Kostant partition function and its \(q\)-analog, and certain generating functions on the set of transportation matrices. In each case rigged configurations technique is applied to obtain some interesting and new information about Kostka-Foulkes and parabolic Kostka polynomials, the Kostant partition function, MacMahon, Gelfand-Tsetlin and Chan-Robbins polytopes. Certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polynomials and between domino tableaux and rigged configurations are described. Some properties of \(l\)-restricted generalized exponents and the stable behaviour of certain Kostka-Foulkes polynomials are studied. Many open problems and conjectures are proposed.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00054].