Counting modular irreducible characters
Counting modular irreducible characters
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DOI：
10.1016/0021-8693(84)90189-3
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发表时间：
1984-10
影响因子：
0.9
通讯作者：
G. Robinson
中科院分区：
文献类型：
--
作者：
G. Robinson
本文研究了给定一个有限群G，一个G的素数p和G的p子群D，在缺陷群为D的p块中G的模不可约字符有多少个？我们将把这个数描述为GF (p)中两个矩阵的秩之差。我们还将讨论如何使用我们所描述的方法来计算特定p块中模不可约字符的个数，其缺陷群（包含在）D中。从现在开始，G， p和D是固定的。设{yi: 1< i< m}是g的p-正则元素的共轭类的完整代表集。对于每一个i，设Qi是C的一个固定的Sy10w - p-子群，（vi），并尽可能选择yi和Qi使Qi< D。标记使Qi< D，对于1< i< r，但对于i， D中不包含Qi的共轭。如果对于某些PE Sylp (G)，我们有：P n Pyi= Qi和n，（Qi） E Sylp (n,(Qi))，我们说yi是有区别的。这个定义与从yi类中选择的特定共轭无关，也与C，（y,）的特定Sylow p子群无关。我们还注意到上述条件意味着Npri (Qt) E SYlP (NG (Qi))，因此P n Pyi是一个温和的黄交，并且
In this paper, we will be concerned with the following question: given a finite group G, a prime divisor p of 1 G 1, and a p-subgroup D of G, how many modular irreducible characters of G lie in p-blocks whose defect group is D? We will give a description of this number as the difference between the ranks of two matrices with entries in GF (p). We will also discuss how the methods we describe can be used to compute the number of modular irreducible characters in a specific p-block whose defect group is (contained in) D. From now on, then, G, p and D are fixed. Let {yi: 1< i< m} be a full set of representatives for the conjugacy classes of p-regular elements of G. For each i, let Qi be a fixed Sy10w p-subgroup of C,(vi), and choose yi and Qi wherever possible so that Qi< D. Label so that Qi< D for 1< i< r, but no conjugate of Qi is contained in D for i> r.DEFINITION. We say that yi is distinguished if, for some PE Sylp (G), we have: P n Pyi= Qi and N,,(Qi) E Sylp (N,(Qi)).Remark. This definition is independent of the particular conjugate chosen from the class of yi, and of the particular Sylow p-subgroup of C,(y,) chosen. We also remark that the above conditions imply that Npri (Qt) E SYlP (NG (Qi)), so that P n Pyi is a tame Sylow intersection, and