| 研究実績の概要 |
I continue the study of combinatorial properties of certain polynomials which naturally appear in an investigation of some integrable systems such as Painleve V I, rational, trigonometric and elliptic Calogero-Moser type models and etc. I and S. Fomin discovered hidden algebraic structure behind these integrable systems (the so-called Fomin-Kirillov algebras(FK-algebras for short)) . The study algebraic and combinatorial properties of these algebras plays important role in the study of algebraic and combinatorial properties of the integrals of motion of the systems mentioned.The results of these investigations in the course of the preparation/writing books "Algebraic Bethe Ansatz"(jointly with R.Sakamoto), and "On some quadratic algebras".
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
It is really very surprising/mysterious fact that FK-algebras falls on the crossroad of many branches of Mathematics and Mathematical Physics such as Algebra,Geometry, Combinatorics, Special Functions,Low dimensional Topology,Integrable Systems and so on and so far.
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| 今後の研究の推進方策 |
I'm planning to continue my research concerning some quadratic algebras and related polynomials such as the Okamoto and Umemura ones, as well as the Schubert and Grtothendieck ones.
I'm planning to report on my results obtained in the body of the Project, during the Conference ECCOM 20, Bogota, Columbia, scheduled on June 15-26, 2021 instead of June 15-26, 2020, since coronavirus pandemics. By this reason I ask to extend my Kakenhi grant on the next Academic Year 2021.
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| 次年度使用額が生じた理由 |
My research of algebraic and combinatorial properties of the Okamoto, Umemura, Grothendfieck,.. polynomials touches a wide variety of fields in Mathematics such as Algebra, Algebraic Geometry and Combinatorics, Special Functions, Knot Theory, Integrable Systems and Probability Theory. To have further progress concerning the Project , I'm planning to visit (partly online) some leading specialists in the fields of Mathematics listed above in Japan, e.g. M.Noumi, Y.Yamada, S.Okada, M. Masuda, M. Yoshinaga, M. Yamazaki. M. Kapranov,..., as well as some oversea ones, including A. Borodin, M. Haiman, ....
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