A Tower Construction for the Radical in Brauer′s Centralizer Algebras
A Tower Construction for the Radical in Brauer′s Centralizer Algebras
复制标题
布劳尔扶正代数激进式的塔式结构
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
D. B. Wales
中科院分区：
文献类型：
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作者：
P. Hanlon;D. B. Wales
本文研究了乘法常数x为有理数时Brauer中心化子代数的结构。几位作者研究了这些代数的结构为一般值的x。特别地，汉斯·温茨尔证明了布劳尔代数可以从琼斯的基本构造中得到，并且他利用这一事实证明了当x不是有理数时，布劳尔代数是半单的。塔构造是研究半单代数塔的一种方法。因此，它不适用于我们的情况下，代数在塔最终有根。本文的第一步是修改塔构造，使其同时进行代数塔的根式和半单商的塔构造。其余的文件是专门描述这些建设明确的布劳尔代数的情况下。一个令人惊讶的推论，这种方法是一个连接之间的两个看似不同的标准简单的某些子环的布劳尔代数。可以明确地确定Brauer代数的某些子环，这些子环是对应于半单情况下的不可约表示的矩阵环。在以前的工作中，这些作者给出了一个组合条件，这些单独的矩阵环简单时，x是一个合理的整数。塔构造给出了第二个代数条件的简单性。很难理解为什么这两个条件是等价的。然而，本文中使用的方法清楚地表明了这一点。
Abstract In this paper we study the structure of the Brauer centralizer algebras in the case that the multiplication constant x is a rational integer. Several authors have studied the structure of these algebras for generic values of x . In particular, Hans Wenzl showed that the Brauer algebras can be obtained from Jones′ Basic Construction and he used that fact to prove that the Brauer algebras are semisimple when x is not a rational integer. The Tower Construction is a method to study towers of semisimple algebras. Hence it does not apply in our case where the algebras in the tower eventually have radicals. Our first step in this paper is to modify the Tower Construction so that it does a simultaneous Tower Construction of the radicals and the semisimple quotients of a tower of algebras. The rest of the paper is devoted to describing these constructions explicitly in the Brauer algebra case. One surprising corollary of this method is a connection between two seemingly distinct criteria for the simplicity of certain subrings of the Brauer algebras. It is possible to identify explicitly certain subrings of the Brauer algebras which are the matrix rings corresponding to irreducible representations in the semisimple case. In previous work, these authors gave a combinatorial condition for simplicity of these individual matrix rings when x is a rational integer. The Tower Construction gives a second algebraic condition for simplicity. It is difficult to see why these two conditions are equivalent. However, the methods used in this paper make that clear.