2023 Fiscal Year Research-status Report
KLR algebras and wreath zigzag algebras
| Project/Area Number |
23K03043
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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) |
2023-04-01 – 2026-03-31
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| Keywords | KLR algebras / Quiver Hecke algebras / Hecke algebras / skew Specht modules / simple modules / Schurian-finiteness / Strictly wild |
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| Outline of Annual Research Achievements |
Our results with Susumu Ariki and Sinead Lyle determining that representation infinite blocks of type A Hecke algebras are Schurian-infinite appeared in the Journal of the LMS. We proved that outside of quantum characteristic 2, a block of a type A Hecke algebra is representation infinite (which is known to always be wild in this case) if and only it is Schurian-infinite. We have since begun studying the analogue of this problem for type B Hecke algebras.
I've also made progress studying another related problem, to determine when wild blocks of type A Hecke algebras are strictly wild.
I also completed joint work with Robert Muth, Thomas Nicewicz and Louise Sutton - the preprint is now available online as arXiv:2405.15759. We showed that for an arbitrary convex preorder, the simple modules for type A KLR algebras, which are known to be indexed by root partitions, appear as the heads of skew Specht modules given by explicit skew diagrams that we construct. This fully relates the theories of cuspidal systems and skew Specht modules for the first time - previously such a connection was only made for real roots.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Our work with Muth, Nicewicz and Sutton, now on the arXiv (arXiv:2405.15759), develops the combinatorics of skew diagrams and RoCK blocks, which will be used in a crucial way to relate KLR algebras and wreath zigzag algebras in our next paper.
Using this combinatorics, we were able to give the most explicit description of simple modules known for type A KLR algebras. For an arbitrary convex preorder, the simple modules for type A KLR algebras are known to be indexed by root partitions. For each root partition, we construct an explicit skew diagram, and the skew Specht module indexed by this diagram has simple head isomorphic to the simple module indexed by that root partition.
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| Strategy for Future Research Activity |
The combinatorics we already developed for RoCK blocks and skew diagrams will allow us to take truncations of RoCK blocks of cyclotomic KLR algebras, corresponding to cutting out multicores from each multipartition in the block. In this setting, when we cut out a fixed multicore of defect 0, we showed that our truncation is Morita equivalent to a skew cyclotomic KLR algebra, which we introduced in our work arXiv:2405.15759. Next, we will show that this skew cyclotomic KLR algebra is isomorphic to a cyclotomic wreath zigzag algebra, providing a `local object’ for the higher level cyclotomic KLR algebras, analogous to the level 1 situation.
Separately, I am also completing a paper that determines the graded decomposition numbers for type C KLR algebras, and determines structures of Specht modules. I am also working on determining which representation-wild blocks of type A Hecke algebras are strictly wild.
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| Causes of Carryover |
Other funding sources were used to support travel this year.
Next fiscal year's funds will help cover research visits from my collaborators to visit OIST, as well as my travel to conferences, including the Conference in Algebraic Representation Theory 2024, and the Mathematical Society of Japan Autumn Meeting 2024.
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