LITTLEWOOD TYPE FORMULA OF THE FINITE FORMULA OF THE CLASSICAL GROUPS
Project/Area Number  09640037 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  TOTTORI UNIVERSITY 
Principal Investigator 
ISHIKAWA Masao TOTTORI UNIVERSITY, FACULTY OF EDUCATION AND REGIONAL SCIENCE, ASSOCIATE PROFESSOR, 教育地域科学部, 助教授 (40243373)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  COMBINATORICS / PFAFFIAN / CHARACTERS / PARTITIONS / PLANE PARTITIONS / SCHUR FUNCTIONS / POSETS / GENERATING FUNCTIONS / combinatorics / Pfaffians / Schur functions / character / partitions / plane partitions / generating functions / poset / Combinatories / characters / Pfaffian / Schur polynomials / partitpan 
Research Abstract 
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 BinetCauchy 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 hookformulas of dcomplete 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 dcomplete posets are defined by Proctor associated with the generalized Weyl group of Simply laced KacMoody 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 RogersRamanujan type identities. So our research was very fruitful.

Report
(5results)
Research Output
(23results)