On the decomposition of Brauer's centralizer algebras
On the decomposition of Brauer's centralizer algebras
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DOI：
10.1016/0021-8693(89)90076-8
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发表时间：
1989-03
影响因子：
0.9
通讯作者：
P. Hanlon;D. B. Wales
中科院分区：
文献类型：
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作者：
P. Hanlon;D. B. Wales
本文分析了Richard Brauer[2]引入的两个有限维实代数的表示。引入了与作用于End(Tr(V))的李群O(n，R)和Sp(2n，[w)的中心化子代数有关的代数，其中V是基础的n维或2n维实向量空间。这些代数是对称群的群代数的类似物，对称群的作用与S1(n，IX)相同。Brauer描述了两个代数‘9Iy)和B$！’)和到End(TFC V)的同态)。各个像是末端作用(Tf(i‘))的完全中心化子代数。Brown[3，4]讨论了代数‘9.i：’)。他在文[4]中证明了半单的充要条件是n&f-1。Weyl[143已经证明，如果n>；2f Brown的方法显示‘13$’，则它是半单的，当且仅当naf-1。当根数为非零时，没有关于根数的进一步信息。这是我们工作的起点。由于中心化子代数中的象是半单的，所以我们知道每个代数的根一定在核中。因此，在每种情况下，上述同态都是由其根在代数的商上定义的。这个商代数同构于矩阵环的直和。这些矩阵环中的每一个必须在核中，否则必须平凡地与核相交。我们的目的是首先找出9I$“)和SF)的根，然后刻画根对商的韦德伯恩分解。本文将代数2i$‘)和S$’‘的问题归结为某些对称矩阵T+(X)的特征值和特征空间的确定。布朗[3，4]使用了类似的想法。对于m和k的某些值，我们可以使用
In this paper we analyze representations of two finite-dimensional real algebras introduced by Richard Brauer [2]. The algebras were introduced in connection with the centralizer algebras of the Lie groups O (n, R) and Sp (2n,[w) acting on End (Tr (V)), where V is the underlying n-or 2n-dimensional real vector space. These algebras are analogues of the group algebra of the symmetric group which plays the same role for Sl (n, IX). Brauer describes two algebras ‘9Iy) and B $!‘) and homomorphisms into End (Tfc V)). The respective images are the complete centralizer algebras for the action on End (Tf (I’)). Brown [3, 4] discusses the algebra ‘9. I:“). He shows in [4] that it is semisimple if and only if n> f-1. Weyl [143 had shown it was semisimple if n> 2f Brown’s methods show ‘13$‘) is semisimple if and only if n af-1. No further information about the radical when it is nonzero was known. This was the starting point of our work. As the image in the centralizer algebra is semisimple, we know that the radical of each algebra must be in the kernel. So in each case, the above homomorphism is defined on the quotient of the algebra by its radical. This quotient algebra is isomorphic to a direct sum of matrix rings. Each of these matrix rings must be in the kernel or else must intersect the kernel trivially. Our aim is first to find the radicals of 9I $“) and SF) and then to describe the Wedderburn decomposition of the quotients by the radicals. In this paper we reduce questions about the algebras 2I $!‘) and S $!‘) to the determination of the eigenvalues and eigenspaces of certain symmetric matrices T+(X). Brown [3, 4] used similar ideas. For some values of m and k we can determine these eigenvalues and eigenspaces using the