Unipotent representations of the finite general linear groups
Unipotent representations of the finite general linear groups
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有限一般线性群的单能表示
DOI：
10.1016/0021-8693(82)90033-3
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
G. James
中科院分区：
文献类型：
--
作者：
G. James
有限一般线性群 CL,+,(q) 的不可约单能表示是 Bore1 子群上的置换表示的组成因子。在特征为零的域上，存在与 1-t 1 的分区一样多的不等价不可约单能表示； Steinberg [3] 使用字符证明了这一点。在本文中，我们将在特征不整除 q 的任意域上为 I+1 的每个分区构造一个不可约单能表示。在未来的论文中，我们打算给出一种完全不同的构造，它可以产生其中一些表示，并导出特定分区的更详细信息。由于单能表示的内核包含 GL [+,(q) 的中心，因此单能表示可以被视为射影一般线性群的表示。这一观察使我们能够将我们的一些结果置于有限 Chevalley 群表示论的更一般框架中；尽管这些方法在这种情况下几乎没有给出不可约的表示，但我们将尽可能使用Chevalley群的符号，因为结果似乎具有足够的意义，值得进一步研究。
The irreducible unipotent representations of the finite general linear groups CL,+,(q) are the composition factors of the permutation representation on a Bore1 subgroup. Over a field of characteristic zero, there are exactly as many inequivalent irreducible unipotent representations as there are partitions of 1-t 1; this was proved by Steinberg [3] using characters. In this paper we shall construct an irreducible unipotent representation for each partition of I+ 1, over an arbitrary field whose characteristic does not divide q. In a future paper we intend to give an entirely different construction which produces some of these representations and derive more detailed information for particular partitons.Since the kernel of a unipotent representation contains the centre of GL [+,(q), a unipotent representation may be regarded as a representation of the projective general linear group. This observation enables us to place some of our results in the more general framework of the representation theory of finite Chevalley groups; although the methods give few irreducible representations in this situation, we shall use the notation of Chevalley groups as far as we can, since the results appear to be of sufficient significance to merit further investigation.