相关于次截面的类函数的性质

Properties of Class Functions Associated to Subsections

设G为有限群,p为|G|的一个素因子,x为G的一个p-元,b为CG(x)的一个p-块,则称(x,b)为G的一个次截面(subsection),只要bG成为G的p-块;次截面在模表示理论中具有重要的作用.又设χ为G的不可约特征标,则χ(x,b)称为G的相关于次截面(x,b)的一个类函数.给出了相关于次截面的类函数的一些性质以及关于次截面在一个例子中的获取过程.设χ,ψ为G的不可约特征标,b,e为CG(x)的p-块,S为G的次截面的G-类代表元的完全集.若b≠e,则[χ(x,b),ψ(x,e)]=0;若b=e,则两个类函数的内积和Cartan矩阵存在着一定的联系.而且,χ=∑(x,b)∈Sχ(x,b).

Let G be a finite group,p a prime divisor of EGI ,x a p-element of G and b be a p-block of Cv（x）. Then （x,b） is called a subsection of G provided bc becomes a p-block of G. Subsections play an important role in modular representation theory. And let ）~ be an irreducible character of G; then x（x.b） is a class function associated to （x,b）. Some properties of this kind of class functions are giv- en in this note and an example is also given to show how to obtain subsections. More concretely,let ）~,~ be irreducible characters of G,b,e be p-blocks of Cc （x）, and S be the complete set of representatives of G-classes of subsections of G. If b≠e, then [x（x,b）,φ（x,e）]=0;if b= e, then the inner product of those two class functions has some connections with Cartan matrices. Moreover, x=∑（x,b）∈sx（x,b）.