LITTLEWOOD TYPE FORMULA OF THE FINITE FORMULA OF THE CLASSICAL GROUPS

本文的第一个动机是得到Schur函数的某些Littlewood型公式及其B，C，D型扩张，作为本文所得到的Pfurans的小求和的应用。在我们的论文J.的Alg。我们证明了这类公式可以通过仅使用Binet-Cauchy公式得到，Binet-Cauchy公式是我们的Pfweans的次求和公式的一个简单特例。最近，我们得到了一些非常有趣的Plucker关系，如公式的Pfweans，也得到了一个简化和组合证明我们的小求和公式，这将出现在我们未来的文件。进一步我们发现，我们必须研究我们的公式的表示理论方面，作为一个例子，字符的plethisms，我们发现小求和公式是一个非常强大的和适用的工具的字符理论。我们还研究了d-完全偏序集的钩公式，并证明了这些钩公式中的大多数可以通过对Pfweans的某些行列式的求值来证明。我们证明了这些公式的偏序集称为鸟，插入，等等。这些d-完全偏序集是由普罗克特定义的，与广义Weyl群的简单laced Kac-Moody李代数。所以这个话题也和表征理论有关。这些天，我们也开始研究与正交多项式和Rogers-Ramanujan型恒等式的关系。所以我们的研究是非常有成果的。

The first motivation of our research is to obtain certain Littlewood type formulas of Schur functions and it's B, C, D type extensions as an application of the minor summations of Pfaffians obtained in our paper. In our paper in J. of Alg. we showed that these kinds of formulas are vastly obtained by using only the Binet-Cauchy formula, which is a simple special case of our minor summation formulas of Pfaffians. Recently we obtained some very interesting Plucker relation like formulas on Pfaffians and also obtained a simplified and combinatorial proof of our minor summation formula which will appear in our future paper. Further we found that we have to study the representation theoretical aspects of our formulas, as an examples, plethisms of characters, and we found the minor summation formulas is an very strong and applicable tool for the character theory. We also investigated the hook-formulas of d-complete posets and we showed that the most of those hook formulas can be proved by the evaluations of certain determinants of Pfaffians. We proved these formulas for the poset called birds, insets, and etc. These d-complete posets are defined by Proctor associated with the generalized Weyl group of Simply laced Kac-Moody Lie algebra. So this topic is also related to the representation theory. These days we also started to investigate the relations with the orthogonal polynomials and Rogers-Ramanujan type identities. So our research was very fruitful.