Representations of Hecke algebras

本文利用Artin代数的方法研究了经典型Hecke代数的模表示。由于Artin代数的一般理论不足以得到Hecke代数的具体结果，所以我们的想法是使用我以前的结果，它允许我们计算经典型Hecke代数的分解数（特别是B型），联合阿廷代数的一般理论。经过两年的研究，我们对Hecke代数的表示类型的判定获得了满意的结果。主要结果如下：定理。设W是Weyl群，Pw（x）是W的Poincare多项式，F是代数闭域，q ∈ F使得q <$0，1.伴随的Hecke代数是F-代数，记为Hw（q）。则Hw（q）是有限的，驯服的，野性的，如果（x-q）^2不整除Pw（x），否则q = -1 <$1，（x + 1）^2 <$Pw（x）.

The goal of this research is to study the modular representations of Hecke algebras of classical type, by using methods of Artin algebras. As the general theory of Artin algebras does not suffice to obtain concrete results for the Hecke algebras, the idea was to use my previous result, which allows us to compute the decomposition numbers of Hecke algebras of classical type (type B in particular), jointly with the general theory of Artin algebras.After two years of research, we have obtained satisfactory results for the determination of representation types of the Hecke algebras. Main result is as follows.Theorem. Let W be a Weyl group, Pw(x) be the Poincare polynomial of W, F an algebraically closed field, q ∈ F so that q ≠ 0,1. The associated Hecke algebra, which is an F-algebra, is denoted by Hw(q). Then, Hw(q) is finite, tame, wild if (x-q)^2 does not divide Pw(x), q = -1 ≠ 1 and (x + 1)^2 ‖ Pw(x) otherwise, respectively.We also have the result for two parameter Hecke algebras in type B case.