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

D-完全偏序集是由R.P.Proctor定义的，它与Kac-Moody代数的广义Weyl群有关。P.Proctor证明了由15个不可约d-完全偏序集得到d-完全偏序集。研究了15个不可约d-完全偏序集的(p，w)-分拆的生成函数，证明了任何d-完全偏序集的生成函数都是由不可约d-完全偏序集的生成函数得到的。此外，我们还证明了普通的一元母函数可以推广到某些多元母函数。随着这些公式的证明，我们发现了许多有趣的行列式和Pfaffian，它们是经典行列式和Pfaffian的推广。我们还使用(k，L)-钩舒尔函数及其公式来计算这些行列式和Pfaffian。我们还发现，有几个证据表明，对于一般的d-完全偏序集，一定存在一种k-rim钩子画面，并且对于这些边钩子画面，一定存在一种钩子公式。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画面。第二种最简单的情况是移位的形状。有一些著名的对称函数，即Schur Q-函数，与移位的形状有关。研究偏序集的相似生成函数也是一个有趣的主题。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处的值可能非常有趣。通过这种方式，我们发现了许多有趣的公式，研究仍在进行中。

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.