Papers › Calculating the p-canonical basis of Hecke algebras
Calculating the p-canonical basis of Hecke algebras
Joel Gibson, Lars Thorge Jensen, Geordie Williamson
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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 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.
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