On a family of symmetric rational functions

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• 作者：Alexei Borodin^a ; ^b ; ^{borodin@math.mit.edu}
• 关键词：Symmetric rational functions
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：973-1018
• 全文大小：798 K
• 卷排序：306

文摘

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall–Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to the P and Q Hall–Littlewood polynomials. We establish (a) a combinatorial formula that represents our functions as partition functions for certain path ensembles in the square grid; (b) symmetrization formulas for non-skew functions; (c) identities of Cauchy and Pieri type; (d) explicit formulas for principal specializations; (e) two types of orthogonality relations for non-skew functions.Our construction is closely related to the half-infinite volume, finite magnon sector limit of the higher spin six-vertex (or XXZ) model, with both sets of functions representing higher spin six-vertex partition functions and/or transfer-matrices for certain domains.