| Project/Area Number |
20K22316
|
|---|
| Research Category |
Grant-in-Aid for Research Activity Start-up
|
| Allocation Type | Multi-year Fund |
|---|
| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
|
|---|
| Research Institution | Okinawa Institute of Science and Technology Graduate University |
|---|
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)
|
|---|
| Project Period (FY) |
2020-09-11 – 2023-03-31
|
| Project Status |
Completed (Fiscal Year 2022)
|
| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
|
|---|
| Keywords | Cyclotomic KLR algebras / Hecke algebras / Schurian-finiteness / Specht modules / Cellular algebras / Quiver Hecke algebras / Representation theory / KLR algebras |
|---|
| Outline of Research at the Start |
In joint work with Ariki and Park, I have introduced families of Specht modules over KLR algebras in (finite and affine) type C, and highlighted their importance in subsequent work. We expect that these algebras are also cellular, and that our Specht modules coincide with the cell modules. The major goals of this project are to prove that these algebras are indeed cellular, and subsequently use our techniques and results to further study their block structure. These results will open the door to further study of these algebras, including their decomposition numbers.
|
| 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.
|
| 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.
|
