2021 Fiscal Year Annual Research Report
Affine cluster algebras as dynamical systems, surface triangulations, quiver representations and friezes
| Project/Area Number |
21F20788
|
|---|
| Research Institution | Chiba University |
|---|
Principal Investigator |
山崎 玲 (井上玲) 千葉大学, 大学院理学研究院, 教授 (30431901)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
PALLISTER JOE 千葉大学, 理学研究院, 外国人特別研究員
|
|---|
| Project Period (FY) |
2021-04-28 – 2023-03-31
|
| Keywords | cluster algebras / dynamical systems / friezes / T- and Y-systems / representation theory |
|---|
| Outline of Annual Research Achievements |
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.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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.
|
| Strategy for Future Research Activity |
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.
|
