Representations of the general symmetric group as linear groups in finite and infinite fields
Representations of the general symmetric group as linear groups in finite and infinite fields
复制标题
将一般对称群表示为有限域和无限域中的线性群
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
L. Dickson
中科院分区：
文献类型：
--
作者：
L. Dickson
1.在一系列的文章在柏林Berichte，开始于1896年，弗罗贝纽斯制定了一个详细的理论组字符，并将其应用于代表一个给定的有限群Gasa非模块化线性群。后来，伯恩赛德f从连续群的观点来探讨这个问题。作者表明J的方法采用的伯恩赛德可以取代一个只涉及纯粹的理性过程，从而导致有效的结果为一般领域。然而，最后一种处理方法明确排除了域的模能整除G阶数的情况。排除这种情况不仅仅是为了方便，也不仅仅是由于特殊的处理方法的限制;事实上，§群行列式的性质与当模不整除G的阶时所持有的性质本质上不同。当G是q阶时，一般理论没有给出关于模为q的域中的表示的信息，而小模的情况对于应用是最重要的。本文研究了域F中系数为m元的线性齐次群，它与q元上的对称群简单同构。的治疗是基本的，完全独立的上述文件，特别是，调查的所有模量无一例外。主要结果是确定变量数目的最小值。证明了m = q-lormëy~2，根据F没有或有一个能整除q的模p（§ § 8 - 21）。那里
1. In a series of articles in the Berliner Berichte, beginning in 1896, Frobenius has developed an elaborate theory of group-characters and applied it to the representation of a given finite group Gasa non-modular linear group. Later, Burnside f approached the subject from the standpoint of continuous groups. The writer has shown J that the method employed by Burnside may be replaced by one involving only purely rational processes and hence leading to results valid for a general field. The last treatment, however, expressly excludes the case in which the field has a modulus which divides the order of G. The exclusion of this case is not merely a matter of convenience, nor merely a limitation due to the particular method of treatment ; indeed, § the properties of the group-determinant differ essentially from those holding when the modulus does not divide the order of G.\ Thus when G is of order q !, the general theory gives no information as to the representations in a field having a modulus = q, whereas the case of a small modulus is the most important one for the applications. The present paper investigates the linear homogeneous groups on m variables, with coefficients in a field F, which are simply isomorphic with the symmetric group on q letters. The treatment is elementary and entirely independent of the papers cited above ; in particular, the investigation is made for all moduli without exception. The principal result is the determination of the minimum value of the number of variables. It is shown that m = q — lormëy2, according as F has not or has a modulus p which divides q (§§ 8-21). There