A generalization of Brauer characters
A generalization of Brauer characters
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DOI：
10.1090/s0002-9947-1965-0181672-x
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发表时间：
1965-02
影响因子：
1.3
通讯作者：
W. Reynolds
中科院分区：
文献类型：
--
作者：
W. Reynolds
1.结果陈述。有限群G关于素数p的Brauer特征标或模特征标仅定义在G的p-正则元上[1]，[3]通过利用G的某些子群的Brauer特征标在一定程度上克服了这一局限性，并据此定义了G的广义分解矩阵D。本文研究了G上与子群的这些Brauer特征标密切相关的一些复值类函数。这些函数，我们称之为0-函数，由Brauer在[3，(7D)]中定义。它们在许多方面都像Brauer字符；例如，它们具有正交性关系(？4)，并且它们可以分布在G(？9)的p-块之间。事实上，我们的中心激励思想一直是，0-函数可以被视为某种类型的群字符。明确地说，如果x是G的任意p元，且vi是G中x的中心化子C(X)的任何不可约Brauer特征标，则在G上定义了唯一的函数0，使得：(A)0是G上的类函数；(B)4(Xy)=+f(Y)如果y在C(X)中是p正则的；(C)如果x1是G的一个不与x共轭的p元，且y在C(X1)中是p正则的，则O(X1y)=0.我们称这些函数为G的素数p的0-函数。(实际上，Brauer特征标的定义取决于在适当的代数数域[6，p.589]中选择p的素数理想因子p；因此，+-函数也依赖于p。)对于x=1，b就是G的Brauer特征标il，如果我们通过在所有p-奇异元上赋予它0来扩展后者(参见？2中对Brauer特征标的重新定义)。我们从C(X)的主要不可分解特征标P出发，对偶定义了G的1D-函数；对于x=1，这些是G的主要不可分解特征标；G恰好有k个不同的+-函数，其中k是G的共轭类的个数。如果XI，“，Xk是G的(普通)不可约特征标，则存在唯一的复数dij，使得
1. Statement of results. The Brauer characters, or modular characters, of a finite group G with respect to a prime p are defined only on the p-regular elements of G. Brauer [1], [3] has overcome this limitation to some extent by using the Brauer characters of certain subgroups of G, in terms of which he has defined the generalized decomposition matrix D of G. In this paper we study some complex-valued class-functions on G which are closely related to these Brauer characters of subgroups. These functions, which we call 0-functions, were defined by Brauer in [3, (7D)]. They behave in many ways like Brauer characters; for example, they have orthogonality relations (?4) and they can be distributed among the p-blocks of G (?9). In fact, our central motivating idea has been that the 0-functions may be regarded as group-characters of a sort. Explicitly, if x is any p-element of G and VI any irreducible Brauer character of the centralizer C(x) of x in G, there is a unique function 0 defined on G such that: (a) 0 is a class-function on G; (b) 4(xy) = +f(y) if y is p-regular in C(x); (c) O(x1y) = 0 if x1 is a p-element of G which is not conjugate to x and if y is p-regular in C(x1). We call these functions 0 the 0-functions of G for the prime p. (Actually the definition of Brauer characters depends on the choice of a prime ideal divisor p of p in a suitable algebraic number field [6, p. 589]; accordingly, the +-functions also depend on p.) For x = 1, b is simply the Brauer character il of G, if we extend the latter by giving it the value 0 on all p-singular elements (see the redefinition of Brauer characters in ?2). Dually we define the 1D-functions of G, starting from principal indecomposable characters P of C(x); for x = 1, these are the principal indecomposable characters of G. G has exactly k distinct +-functions, where k is the number of conjugate classes of G. If XI, ", Xk are the (ordinary) irreducible characters of G, there are unique complex numbers dij such that