2022 Fiscal Year Final Research Report
Cyclotomic KLR algebras in type C: cellularity and blocks
| Project/Area Number |
20K22316
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Okinawa Institute of Science and Technology Graduate University |
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Principal Investigator |
Speyer Liron 沖縄科学技術大学院大学, 表現論と代数的組合せ論ユニット, 准教授(Assistant Professor) (00873762)
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| Project Period (FY) |
2020-09-11 – 2023-03-31
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| Keywords | Cyclotomic KLR algebras / Hecke algebras / Schurian-finiteness / Specht modules |
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| Outline of Final Research Achievements |
We determined the Schurian-finiteness, or equivalently the tau-tilting finiteness, of blocks of type A Iwahori-Hecke algebras, in a preprint (arXiv:2112.11148) submitted for publication. Our main result is that blocks are Schurian-finite if and only if they have finite representation type (known to be the case if and only if they have weight 0 or 1). This project made use of a great breadth of tools, both existing and newly developed for our work.
We've also developed 2 algorithms for computing graded decomposition numbers for cyclotomic KLR algebras R\Lambda_n in type C, and computed all such graded decomposition matrices in level 1, for n<13. In this same project, we also computed the submodule structure of Specht modules in characteristic 0 for n<11, and obtained the first example of characteristic 0 graded decomposition numbers that are not given by the corresponding canonical basis coefficients. The paper is being written up, and we aim to have a preprint submitted by the autumn.
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| Free Research Field |
Representation theory
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| Academic Significance and Societal Importance of the Research Achievements |
Schurian-finiteness is a property which many researchers in finite-dimensional algebras seek to determine for algebras.
The KLR algebras arose from categorification of quantum groups and are studied a lot recently as part of a broader program of categorification. Many open questions remain.
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