A research on hook length posets from combinatorics and representation theory

本研究的目的是对钩长序集进行严格的分类，澄清钩长序集的组合结构和代表性结构。主要得到了以下结果：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舒尔函数的恒等式，它们是柯西恒等式的类似或扩展。同时证明了所有的叶序集都是多变量钩长序集。这里，如果钩子长度公式可以扩展为多变量公式，则称钩子长度偏序集为多变量钩子长度偏序集。此外，我们还利用已知的（多变量的）钩子长度偏序集，找到了钩子长度偏序集的复合方法。任何最多有7个元素的钩长序集都可以用这种组合方法构造。证明了冈田松一（Soichi Okada）在2003年3月提出的柯西型行列式和schur型Pfaffian的几个恒等式。已知Coxeter群的（λ -）极小元是一个完全可交换元，对称群的完全可交换元等于一个避免321的置换。对于Coxeter群，我们引入了一个全覆盖元，它是321避免置换的扩展，并证明了其全交换元与其全覆盖元一致的Coxeter群是a， D， E型Coxeter群。

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.