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A research on hook length posets from combinatorics and representation theory

组合数学和表示论对钩长度偏序集的研究

基本信息

批准号：
15540028
负责人：
TAGAWA Hiroyuki
金额：
$ 2.37万
依托单位：
WAKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540028/
关键词：
hook length poset hook length formula d-complete poset Coxeter group

项目摘要

The purpose of this research is a strict classification of hook length posets, a clarification of combinatorial and representative structure of hook length posets. Chiefly, the following results were obtained.1. We introduced a poset (called a leaf poset) which is an extension of the d-complete poset, and we showed that all leaf posets are hook length posets. As a corollary of the above result, we found many identities with respect to Schur functions, which are analogues or extensions of Cauchy's identity. Also, we proved that all leaf posets are multivaliable hook length posets. Here, we call a hook length poset a multivaliable hook length poset if its hook length formula can be extended to a multivaliable formula. Moreover, we found a composition method of a hook length poset by using a known (multivaliable) hook length poset. Any hook length poset with at most seven elements is constructed by this composition method.2. We proved several identities of Cauchy-type determinant and Schur-type Pfaffian, which was conjectured by Soichi Okada in 2003.3. It is known that a (lambda-) minuscule element of a Coxeter group is a fully commutative element, and a fully commutative element of a symmetric group is equal to a 321-avoiding permutation. For a Coxeter group, we introduced a fully covering element which was an extension of 321-avoiding permutations, and we proved that the Coxeter groups whose fully commutative elements coincide with their fully covering elements are the Coxeter groups of type A, D, E.

本文对钩长偏序集进行了严格的分类，阐明了钩长偏序集的组合结构和表示结构。结果表明：1.我们引入了一个偏序集（称为叶偏序集），它是d-完备偏序集的一个扩展，并且我们证明了所有的叶偏序集都是钩长偏序集。作为上述结果的推论，我们发现了许多关于Schur函数的恒等式，它们是Cauchy恒等式的类似或推广。证明了所有的叶偏序集都是多值钩长偏序集。这里，我们称一个钩长偏序集为多值钩长偏序集，如果它的钩长公式可以推广为多值公式。此外，我们发现了一个合成方法的钩长偏序集，通过使用已知的（多值）钩长偏序集。用这种合成方法构造了任意至多七个元素的钩长偏序集.证明了SoichiOkada在2003年3月提出的Cauchy型行列式和Schur型Pfronan的几个恒等式。已知Coxeter群的（λ-）极小元是全交换元，对称群的全交换元等于321-避免置换。对于Coxeter群，我们引入了一个全覆盖元，它是321-避免置换的推广，并证明了全交换元与全覆盖元重合的Coxeter群是A，D，E型Coxeter群.