From spin representations of the symmetric groups to Hirota equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05180
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Okayama University (2022); Kumamoto University (2017-2021)
• Principal Investigator: YAMADA HiroFumi 岡山大学, 自然科学研究科, 特命教授 (40192794)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: シューア函数 / 対称群 / ヴィラソロ代数

Outline of Final Research Achievements

I am studying combinatorial aspects of Hirota bilinear equations for Kdv and modified KdV hierarchies. Mikio Sato wrote a short notes on these objects in Japanese in 1980, and tabulated his experimental results. Only recently I have realized what Sato was thinking and saying. Schur functions and Schur's Q functions are essentially related to the subject.
And also modular representations of the symmetric groups at p=2 have a deep connection with thesehierarchies. I am also investigating certain representation of the Virasoro algebra, which, I think, is related to KdV. Three papers on the reduced Fock representation of the Virasoro algebra appeared which are written jointly with Kazuya Aokage and Eriko Shinkawa.
And one more paper with Naoki Chigira will appear, I hope, which is aninteresting combinatorics of partitions.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

非線型偏微分方程式の一つであるKdV方程式は，ソリトンと呼ばれる特別な解をもつ．自然現象の解明におおいに寄与するが，一方数学的にもその代数性等際立った性質を具備している．私は，佐藤幹夫による無限次元グラスマン多様体の理論を土台にしつつ，組合せ論的な側面に着目して研究を進めている．非線型現象を表す微分方程式が対称群という小さな代数系の組合せ論で制御されている，ということを明らかにしつつあるという意味で
学術的な意義は小さくないと信ずる．