论文标题
计算Hecke代数的$ p $ - 适用基础
Calculating the $p$-canonical basis of Hecke algebras
论文作者
论文摘要
我们描述了一种用于计算Hecke代数的$ p $的算法，或它的一种反球形模块。该算法不直接在Hecke类别中运行，而是在半模拟类别中使用Hecke类别的忠实嵌入来构建一个“模型”，以实现其形态空间的不可塑性对象和基础。在此半神经类别中，对象是Coxeter组元素的序列，并且形态在分数字段上是（稀疏）矩阵，使其适合计算。该策略适用于任何基本领域的全部Hecke类别，但是在反球形案例中，我们必须通过$ \ Mathbb {z} _ {（p）} $使用$ \ Mathbb {z} _ {（p）} $，并使用IDEMTOTENT LIPTTING参数来推论特征性$ P> 0 $的领域的结果。我们还描述了一种不太复杂的算法，它更适合有限群体。我们在岩浆计算机代数系统中提供了两种算法的完整实现。
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.
