Relations between character formula of classical groups, and the generating functions of P-partitions of d-complete posets

经典群的特征公式与d-完全偏序集P-划分的生成函数之间的关系

• 批准号: 13640022
• 负责人: ISHIKAWA Masao
• 金额: $ 1.79万
• 依托单位: Tottori University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640022/
• 关键词: combinatorics; plane partitions; poset; Pfaffian; generating functions; Pfaffians; generating function; representation theory; Bruhat order; Coxeter groups

D-complete posets are defined by R. P. Proctor in relation with the generalized Weyl groups of Kac-Moody algebra. R. P. Proctor showed that d-complete posets are obtained from 15 irreducible d-complete ones. We studied the generating functions of (P. w)-partitions of 15 irreducible d-complete posets and showed that generating functions of any d-complete posets are obtained from those of irreducible ones. Further we also showed that the ordinary one variable generating functions can extended to certain multi-variable generating functions. Along the proof of those formulas, we find a lot of interesting determinants and Pfaffians which are extensions of classical ones. We also use (k, l)-hook Schur functions and its formulas to evaluate those determinants and Pfaffians. We also found that there are several evidences that we can expect that there must be a kind of k-rim hook tableaux for general d-complete posets and a kind of hook formula must hold for these rim hook tableaux. The simplest case corresponds to the classical formula, i.e., Young's lattice, which corresponds to the ordinary Young semi-standard tableaux. The second simplest case is the shifted shapes. There are famous symmetric functions, i.e., Schur Q-functions, associated with shifted shapes. It is also interesting theme to study similar generating functions of posets. We also study the (P. w)-partitions of height at most n, which can be considered as a generalization of ordinary plane partitions. Usually these generating functions do not have hook formulas, but the value of these generating functions at q=-1 might be very interesting. In this way we found a lot of interesting formulas and the study is still in progress.

D-完全偏序集是由R. P.普罗克特与Kac-Moody代数的广义Weyl群的关系. R. P.普罗克特证明了d-完全偏序集是由15个不可约的d-完全偏序集得到的。本文研究了15个不可约d-完全偏序集的（P，w）-划分的生成函数，证明了任何d-完全偏序集的生成函数都可以由不可约偏序集的生成函数得到。进一步证明了普通的一元母函数可以推广到某些多元母函数。沿着这些公式的证明，我们发现了许多有趣的行列式和Pfrons，它们是经典行列式和Pfrons的推广。我们还使用（k，l）-hook Schur函数及其公式来计算这些行列式和Pfuman。我们还发现，有几个证据表明，对于一般的d-完备偏序集，一定存在一种k-rim hook tableaux，并且对于这种k-rim hook tableaux，一定存在一种hook公式。最简单的情况对应于经典公式，即，杨氏晶格，它对应于普通的杨氏半标准表。第二种最简单的情况是变形。有著名的对称函数，即，Schur Q-函数，与移位的形状相关联。研究偏序集的相似生成函数也是一个有趣的课题。我们还研究了高度不超过n的（P，w）-分拆，它可以被认为是普通平面分拆的推广。通常这些生成函数没有钩子公式，但是这些生成函数在q=-1时的值可能非常有趣。通过这种方式，我们发现了很多有趣的公式，研究仍在进行中。