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.

本研究的目的是对钩长偏序集进行严格分类，阐明钩长偏序集的组合结构和代表性结构。主要得到以下结果： 1．我们引入了一个偏序集（称为叶偏序集），它是 d-完全偏序集的扩展，并且我们证明所有叶偏序集都是钩长度偏序集。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，它们是柯西恒等式的类似物或扩展。此外，我们还证明了所有叶偏序集都是多值钩长偏序集。在这里，如果钩长度公式可以扩展到多值公式，我们将钩长度偏序集称为多值钩长度偏序集。此外，我们通过使用已知的（多值）钩长度偏序集找到了钩长度偏序集的合成方法。任意最多有7个元素的钩长偏序集都可以用这种组合方法构造。 2．我们证明了Soichi Okada在2003.3猜想的Cauchy型行列式和Schur型Pfaffian的几个恒等式。已知 Coxeter 群的 (lambda-) 小元素是完全交换元素，对称群的完全交换元素等于 321 避免排列。对于Coxeter群，我们引入了完全覆盖元素，它是321避免排列的扩展，并且证明了其完全交换元素与其完全覆盖元素一致的Coxeter群是A、D、E型Coxeter群。