Ubiquity of Kostka polynomials
Ubiquity of Kostka polynomials
复制标题
Kostka 多项式的普遍性
DOI：
10.1142/9789812810199_0006
复制
发表时间：
1999
期刊：
影响因子：
--
通讯作者：
A. Kirillov
中科院分区：
文献类型：
--
作者：
A. Kirillov
我们报告围绕Kostka-Foulkes和抛物Kostka多项式及其与表示论和组合学的联系的结果。所有抛物型Kostka多项式的集合形成一个半群，我们称之为Liskova半群。我们证明了在表示论和组合数学中经常出现的多项式属于Liskova半群。在这些多项式中，我们研究矩形$q$-Catalan数;广义指数多项式; Schur函数内积的主要特化;广义$q$-高斯多项式;抛物型Kostant配分函数及其$q$-模拟;某些生成函数的运输矩阵集。在每种情况下，我们应用操纵配置技术，以获得一些有趣的和新的信息，Kostka-Foulkes和抛物Kostka多项式，Kostant配分函数，MacMahon，Gelfand-Tsetlin和Chan-Robbins多面体。我们描述了广义饱和度和富尔顿的代数和抛物线Kostka多项式之间的某些连接;多米诺骨牌tableaux和操纵配置。我们还研究了$l$-限制广义指数的一些性质和某些Kostka-Foulkes多项式的稳定性。
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.