Permutation representations and a fragment of the decomposition matrix of symplectic and special linear groups over a finite field
Permutation representations and a fragment of the decomposition matrix of symplectic and special linear groups over a finite field
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有限域上辛线性群和特殊线性群的置换表示和分解矩阵的片段
DOI：
10.1007/bf00974487
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发表时间：
1990
影响因子：
0.5
通讯作者：
I. D. Suprunenko
中科院分区：
文献类型：
--
作者：
A. Zalesskii;I. D. Suprunenko
AE Zalesskii和ID Suprunenko UDC 512.547。23 i.引言iI之间的联系普通和模块表示的有限群是总结了相当程度上的分解矩阵，介绍了在1937年由布劳尔和Nesperson [i]。目前，这是一个主要议题正在调查的理论模块表示的有限群（比照。例如[2，Chap. XII; 3]。大量的工作致力于最重要群的分解矩阵的显式构造，主要是简单群和与它们相关的群[4]。去年，我们对Chevalley群的分解矩阵和可由分解矩阵表示的Cartan矩阵进行了深入的研究。本文的目的是构造群SLn（q）和Spn（q）的模p分解矩阵的一个片段的显式形式，其中p> 2，n是任意的。更确切地说，我们将提供进入置换表示He的不可约复表示的模p分解数的公式，与这些群在它们的自然实现空间上的作用有关。特别地，在SLn（q）的情况下，片段由分解矩阵的q行组成，在SPn（q）的情况下，片段由（q+ 7）/2行组成，p> 2。
AE Zalesskii and ID Suprunenko UDC 512.547. 23 i. Introduction iI The connection between ordinary and modular representations of a finite group is summarized to a considerable extent in the decomposition matrix, introduced in 1937 by Brauer and Nesbitt [i]. At present this is one of the main topics under investigation in the theory of modular representations of finite groups (cf. for example [2, Chap. XII; 3]. Great efforts are devoted to the explicit construction of decomposition matrices of the most important groups, mainly for simple groups and groups related to them [4]. Last year, an intensive study of the decomposition matrices and the Cartan matrices which can be expressed by them was undertaken for Chevalley groups.Let p be a prime, q= pd. The aim of the present paper is the construction in explicit form of a fragment of the decomposition matrix modulo p for the groups SLn (q) and Spn (q), p> 2 for arbitrary n. More precisely, we shall provide formulas for the decomposition numbers modulo p of the irreducible complex representations which enter the permutation representation He, associated with the action of these groups on the spaces of their natural realizations. In particular, the fragment consists of q rows of the decomposition matrix in the case of SLn (q) and of (q+ 7)/2 rows in the case of SPn (q), p> 2.