Calculating the $p$-canonical basis of Hecke algebras
| Autor: | Gibson, Joel, Jensen, Lars Thorge, Williamson, Geordie |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke category inside a semisimple category to build a "model" for indecomposable objects and bases of their morphism spaces. Inside this semisimple category, objects are sequences of Coxeter group elements, and morphisms are (sparse) matrices over a fraction field, making it quite amenable to computations. This strategy works for the full Hecke category over any base field, but in the antispherical case we must instead work over $\mathbb{Z}_{(p)}$ and use an idempotent lifting argument to deduce the result for a field of characteristic $p > 0$. We also describe a less sophisticated algorithm which is much more suited to the case of finite groups. We provide complete implementations of both algorithms in the MAGMA computer algebra system. Comment: 21 pages |
| Databáze: | arXiv |
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