| 研究課題/領域番号 |
21F20788
|
|---|
| 研究機関 | 千葉大学 |
|---|
研究代表者 |
山崎 玲 (井上玲) 千葉大学, 大学院理学研究院, 教授 (30431901)
|
|---|
| 研究分担者 |
PALLISTER JOE 千葉大学, 理学研究院, 外国人特別研究員
|
|---|
| 研究期間 (年度) |
2021-04-28 – 2023-03-31
|
| キーワード | cluster algebras / dynamical systems / friezes / T- and Y-systems / representation theory |
|---|
| 研究実績の概要 |
The research of the year April 2021 - March 2022 has resulted in the two papers
titled ``~A and ~D type cluster algebras: Triangulated surfaces and friezes'' and ``Period 2 quivers and their T- and Y-systems''. The first of those has been submitted to the Journal of Algebraic Combinatorics, and the corrections have been completed.
In the first paper we were able to classify cluster variables for cluster algebras of affine ~A and ~D type. This was done by looking at these cluster variables as arcs of triangulations for appropriate surfaces, a method pioneered in [Fomin, Shapiro and Thurston 2008]. By first identifying the T-system cluster variables and periodic quantities studied in [Fordy and Hone 2014] and [Pallister 2020] as arcs (cluster variables) we were able to identify all other cluster variables in terms of these. We were also able to prove that these cluster variables can be arranged as friezes. Our results agree with similar results found via the representation theory of these quivers. In the second paper we were interested in finding all period 2 quivers. These are quivers that, if mutated twice, look essentially the same. This turned out to be a difficult problem, instead we found all period two quivers with a low number of vertices, of which there are surprisingly many. We then used results of [Nakanishi 2011] to write the T- and Y- systems for these quivers, which here are systems of 2 recurrence relations. We looked at some interesting examples of these which had periodic quantities.
|
| 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
We finished two papers this year, and we think that the project is going well as planned. Especially, the second paper is based on a new problem we started for this project.
|
| 今後の研究の推進方策 |
In the above second paper, we identified period 2 quivers with a low number of vertices. We also noted the T-systems that have obvious periodic quantities. The next step of this work would be to look at the more complicated systems. In particular we remark that, in [Hone and Inoue 2014], links were made between systems from period 1 quivers and discrete Painleve equations, so it would be interesting to explore possible relations between our systems and discrete Painleve.
Since we have provided a reasonable assessment of the ~A and ~D cluster algebras in the above first paper, we could ask if something similar can be done for ~E type. The issue is that there is no triangulated surface for these, with the rules designated in [Fomin, Shapiro and Thurston 2008]. Possibly if we relax these rules we can obtain a surface for ~E type. Even if we cannot do this, it may be possible to identify the frieze structure for the cluster algebras by other means. At the very least this could be found by a computer, since there are only finitely many ~E type cluster algebras.
|
