Ubiquity of Kostka polynomials
type: Item
instance of: scholarly article
zbMATH Open document ID: 0982.05105
publication date: 2002-03-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9912094
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1665294
zbMATH Keywords: Kostka polynomial
zbMATH Keywords: Liskova semigroup
zbMATH Keywords: \(q\)-Catalan number
zbMATH Keywords: Schur function
zbMATH Keywords: generalized \(q\)-Gaussian polynomial
zbMATH Keywords: Kostant partition function
zbMATH Keywords: MacMahon, Gelfand-Tsetlin and Chan-Robbins polytopes
author: Anatol N. Kirillov
MaRDI profile type: Publication
arXiv ID: math/9912094
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title: Ubiquity of Kostka polynomials
review text: 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].
reviewed by: Ioan Tomescu
description: scientific article; zbMATH DE number 1665294