2022 Fiscal Year Annual Research 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 |
During the course of this project, Andrew Mathas and Anton Evseev released a proof of cellularity for the type C cyclotomic KLR algebras. We have redirected our efforts into related structural questions for type C KLR algebras. We have almost completed a paper, joint with Chris Chung and Andrew Mathas, in which we determine graded decomposition matrices for small rank level 1 cyclotomic KLR algebras in type C, as well as the structures of many Specht modules. We have another ongoing project studying arbitrary cyclotomic quotients in finite type C in further detail.
Our joint work with Susumu Ariki last year showed that all weight 2 and weight 3 blocks of type A Hecke algebras are Schurian-infinite, along with several families of blocks of larger weight. This year, we extended this with Sinead Lyle to prove that all blocks of weight at least 2 are Schurian-infinite.
Finally, I have supervised PhD students working on projects relating to type C KLR algebras, too. My student Berta Hudak, together with my postdoc Chris Chung, has recently completed a preprint in which the representation type of all level 1 cyclotomic KLR algebras in type C are determined, generalising results of Ariki and Park. My other student Martin Forsberg Conde is working on homomorphisms between Specht modules in type C, in particular aiming to prove a Carter-Payne type theorem there.
Besides research done for the proposed project, I have been running a fortnightly representation theory seminar over Zoom, bringing in speakers and an audience from several countries.
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