Calculating the $p$-canonical basis of Hecke algebras

TL;DR

This paper introduces an algorithm to compute the $p$-canonical basis of Hecke algebras using a semisimple category embedding, enabling calculations over various fields and addressing challenges in positive characteristic cases.

Contribution

It presents a novel algorithm that embeds the Hecke category into a semisimple category to facilitate computation of the $p$-canonical basis, including over fields of positive characteristic.

Findings

Algorithm successfully computes the $p$-canonical basis.

Implementation provided in MAGMA system.

Applicable to full Hecke category and antispherical modules.

Abstract

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.