Representation theoretical Research on Cluster algebras and crystal bases
| Project/Area Number |
15K04794
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Sophia University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
中筋 麻貴 上智大学, 理工学部, 准教授 (30609871)
五味 靖 上智大学, 理工学部, 准教授 (50276515)
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| Project Period (FY) |
2015-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 結晶基底 / クラスター代数 / 幾何結晶 / 熱帯化 / ２重Bruhat cell / potential / 2重Bruhat cell / 単項式表示 / double Bruhat cell / 一般小行列式 / Double Bruhat cell / tropicalization / 量子群 / Demazure結晶 / 単項式教示 / 一般化小行列式 |
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| Outline of Final Research Achievements |
We studied on the subject of the relation between cluster algebras and crystal bases by using the method of monomial realizations and polyhedral realizations among 2015~2020.
We succeeded in describing the initial cluster of the cluster algebras on double Bruhat cells in terms of monomial realizations of certain Demazure crystals. Moreover, for type A, we also succeeded in describing all cluster variables of the cluster algebras on the double Bruhat cell determined by Coxeter elements. We extend our interest to inducing geometric crystal structure on cluster varieties. All these results have been published in academic journals.
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| Academic Significance and Societal Importance of the Research Achievements |
結晶基底と現在、活発に研究が進んでいるクラスター代数との具体的な関係について明らかにした。まだまだ、わかっていないことが多いが、それだけに新しい発見が期待できる研究であるといえる。とくに、potential と呼ばれる多様体上の函数については有限次元リー代数の場合のみならず、無限次元のKac-Moody リー代数の場合にも構成ができると期待できる成果を得られた。
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Report
(7 results)
Research Products
(37 results)
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[Presentation] Polyhedral Realization of Crystal Bases2015
Author(s)
中島 俊樹
Organizer
South eastern Lie Theory workshop on Algebraic and Combinatorial Representation Theory
Place of Presentation
North Carolina State University,(USA)
Year and Date
2015-10-10
Related Report
Int'l Joint Research / Invited
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