arXiv math/00081632000-08-22math.QAmath.CO

带状算子与Hall-Littlewood对称函数

Ribbon Operators and Hall-Littlewood Symmetric Functions

Mike Zabrocki

备注：: 10 pages; to appear Advances in Mathematics
发表信息：: Advances in Mathematics, 156, No. 1 (2000), pp.33-43

摘要：

给定分区$\la = (\la_1, \la_2, ... \la_k)$，设$\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$。很容易看出，图$\la\slash \la^{rc}$是连接的，没有$2 \times 2$子图，我们将其称为带状图。对于每个功能区$R$，我们关联一个对称函数运算符$S^R$。我们可以将带状物$maj(R)$的主索引定义为适合带状物的任何排列的主索引。本文讨论了算子$H_{1^k}^q = \sum_R q^{maj(R)} S^R$，其中和是在所有大小为$k$的$2^{k-1}$带上的.我们在这里表明，$H_{1^k}^q$具有真正显着的性能，特别是它是一个罗德里格斯运营商，增加了一列的霍尔-利特尔伍德对称功能。我们相信，我们在这里介绍的一些工具来证明我们的结果也应该是独立的利益，并可能是有用的，以建立进一步的对称函数的身份。

英文摘要：

Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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