群的置换表示与群的图表示

The Representations Between a Group As a Permutation Group and as a Linear Group

考虑群Ｇ在一个集合Ω上传递转换作用和把Ω做成Ｇ的Ｃ［Ω］模空间，得到Ｇ作为线性群在Ｃ［Ω］作用，那么二者有怎样的内在联系呢？本文主要结果是：群Ｇ在一个基数为ｐ的集合Ω上的作用是传递的，那么这个置换特征标π有如下分解式：⑴若Ｇ在Ω上的作用是二传递的，则π＝１Ｇ＋χｉ，ｄｅｇχ＝ｐ－１；⑵若Ｇ在Ω上的作用是单本原的，则π＝１Ｇ＋∑ｒｉ＝１χｉ，ｄｅｇχｉ＝ｍ，ｉ＝１，２，…，ｒ，ｐ－１＝ｍ·ｒ．此处数ｒ称为Ｇ的秩．

In this paper some connection is dealt with between the numerical data describing G as a permutation group on Ω and that describing G as a linear group. The main result of this paper is as follows:Let group G act on Ω transitively with permutation character π. If|Ω|=p, then thd permutation group G on C is multiplicity-free.(1) If G acts on Ω doubly transitively, then π=1G+χ,χ∈Irr(G),degχ=p-1;(2) If G acts on Ω simply primitively, then π=1G+∑r-1iχi,χi∈Irr(G),degχi=f,for i=1,2,…,r-1, where r is rank of G and r·f=p-1.