2021 Fiscal Year Annual Research Report
Supersymmetric rigged configurations and crystals of quantum affine supergroups
| Project/Area Number |
21F51028
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| Research Institution | Osaka City University |
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Principal Investigator |
尾角 正人 大阪市立大学, 大学院理学研究科, 教授 (70221843)
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| Co-Investigator(Kenkyū-buntansha) |
SCRIMSHAW TRAVIS 大阪市立大学, 大学院理学研究科, 外国人特別研究員
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| Project Period (FY) |
2021-11-18 – 2024-03-31
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| Keywords | Crystal basis / Lie superalgebra / skew Howe duality |
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| Outline of Annual Research Achievements |
I completed two projects related to (skew) Howe duality with collaborators that we were working on during the past two years. The papers have appeared on the arXiv preprint server. One is a rational lifting of a classical result to help us better understand the invariants. The other develops a new probabilistic model on partitions that results in a new link with coding theory through Krawtchouk polynomials. I and collaborators have also connected other probabilistic models and Schubert calculus. We are writing multiple papers based on these results. Two students I have been mentoring obtained results on a superalgebra analog of soliton cellular automata, which has been accepted for a highly competitive talk at the international conference "Formal Power Series and Algebraic Combinatorics."
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
The project I am working on relating geometry with probability theory using Fock spaces from physics seems poised to make a large impact based upon conversations I have had about the results due to the unexpected link. The project is taking longer than previously expected to finish as we continue to obtain new results as we are writing. While this has resulted in less time being able to make progress on my main result plan of developing a super analog of the X = M conjecture, some slow progress is still being made.
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| Strategy for Future Research Activity |
Using the results of the super box-ball system, I plan to develop a super analog of rigged configurations. I will also finish writing the papers connecting Schubert calculus and probability theory using tools from physics given the perceived importance of the results. There has been some recent progress on Lie superalgebras that I expect to be able to apply to this project, which I plan to learn more about. I will also write code in order to experiment with possible definitions of rigged configurations for Lie superalgebras.
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