LITTLEWOOD TYPE FORMULA OF THE FINITE FORMULA OF THE CLASSICAL GROUPS

点击翻译按钮获取中文摘要

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.