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We describe an algorithm for computing the $p$-canonical basis of the Heckealgebra, or one of its antispherical modules. The algorithm does not operate inthe Hecke category directly, but rather uses a faithful embedding of the Heckecategory inside a semisimple category to build a "model" for indecomposableobjects 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. Thisstrategy works for the full Hecke category over any base field, but in theantispherical case we must instead work over $\mathbb{Z}_{(p)}$ and use anidempotent lifting argument to deduce the result for a field of characteristic$p > 0$. We also describe a less sophisticated algorithm which is much moresuited to the case of finite groups. We provide complete implementations ofboth algorithms in the MAGMA computer algebra system.

Calculating the $p$-canonical basis of Hecke algebras