2021 Fiscal Year Research-status Report
Cyclotomic KLR algebras in type C: cellularity and blocks
| Project/Area Number |
20K22316
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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 | Cellular algebras / Quiver Hecke algebras / Representation theory / KLR algebras / Specht modules |
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| Outline of Annual Research Achievements |
I have made good progress towards understanding the blocks and simple modules for cyclotomic KLR algebras in finite type C.
In joint with Susumu Ariki, I have studied the Schurian-finiteness of type A Hecke algebras. Using a vast combination of tools including many classical results on these Hecke algebras as well as newer tools derived from them being isomorphic to type A cyclotomic KLR algebras, we have shown that all weight 2 and weight 3 blocks of Hecke algebras are Schurian-infinite (equivalently, tau-tilting infinite) in any characteristic, as are all principal blocks of weight at least 2. We have similar results for other large families of blocks of weight 4 and above.
Besides research done directly on the proposed project, I have been running a fortnightly representation theory seminar over Zoom, bringing in speakers
and an audience from several countries. My paper studying the structure of certain Specht modules for Hecke algebras of type B has been accepted for publication.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
This fiscal year, the covid-19 situation has made it difficult again to meet with international collaborators. Now that travel is finally easing up, we hope progress can speed up during research visits. While our homomorphism methods have not so far yielded the block classification we hoped for, we are making some good progress towards understanding the situation in finite type C. In this case, we still hope to obtain a block classification, a realisation of simple modules as heads of a certain set of Specht modules, and possibly even graded decomposition numbers in many instances.
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| Strategy for Future Research Activity |
The research project will continue, focussing for now on the case of finite type C. In this case, we hope to take our study much deeper, investigating not only the block structure, but also determining the simple modules, decomposition numbers, and certain families of homomorphisms between Specht modules. We expect to have publishable results here in the coming fiscal year.
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| Causes of Carryover |
I will buy a laptop on which to work, and the majority of the rest of the money will be spent on conference travel and collaboration visits.
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