2020 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 – 2022-03-31
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| Keywords | Cellular algebras / Quiver Hecke algebras / Representation theory / KLR algebras |
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| Outline of Annual Research Achievements |
Our proposed project is to investigate the cellularity and block structure of cyclotomic KLR algebras in type C. So far, we've made progress in several ways. We are working on row- and column-removal theorems for homomorphism spaces for these modules. We expect that these will simplify the necessary computations in our strategy for proving that what we expect to be the type C blocks are indeed indecomposable. Along with Chris Bowman (University of York) and Maud De Visscher (City University, University of London), we're in the process of developing the necessary type C abacus combinatorics that will also play a large role in our proof, as well as determining some important homomorphisms we will use.
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. I have submitted another research paper for publication, which studies the structure of certain Specht modules for Hecke algebras of type B, and have had another paper accepted for publication, in which we determined all multiplicity-free subgroups of type B Coxeter groups.
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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
In order to prove the block result we want, we must develop the underlying theory in several directions, including analyses of homomorphism spaces and the necessary combinatorial framework of the abacus. This theory will also help us to prove the cellularity of these algebras. Given the necessary work, it is expected to take some time to complete. On top of this, the current covid-19 pandemic has
added difficulty in scheduling discussions with international collaborators. We hope that the next fiscal year will allow research travel, in which this collaboration can be conducted more intensively.
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| Strategy for Future Research Activity |
We will continue the research project roughly as planned. We hope to have our row- and column-removal results ready for submission within the next fiscal year, and to use these results and our other work on the abacus and homomorphisms to complete the block structure. The work on cellularity will also depend on these homomorphism results.
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| Causes of Carryover |
Due to the covid-19 pandemic, travel for research collaboration and attending conferences and seminars has not been possible. I hope to be able to conduct these research trips once travel becomes possible again.
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