Pointed groups and construction of characters
Pointed groups and construction of characters
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DOI：
10.1007/bf01261873
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发表时间：
1981-06
影响因子：
0.8
通讯作者：
L. Puig
中科院分区：
文献类型：
--
作者：
L. Puig
在[13]中，J. Alperin和M. Brou6在有限群G的任意块中引入了所谓的局部结构，在b[3]中，M. Brou6和作者展示了如何使用它来构造G的字符。在这里，我们将引入一个新的概念，即点群，它使我们能够将局部结构和字符的构造推广到内部G代数（见下面的定义3.1）；我们认为这种方法对整个主题提供了更连贯的处理。从局部结构的角度来看，我们的1.2定理与[-1]中的3.10定理和[3]中的1.14定理是同源的（实际上是暗示了它们），完成了Green对缺陷组的定义。定理3.4将Green的“源”概念推广到内部g -代数。从性质的角度出发，定理4.3和推论4.4对布劳尔第二主定理进行了推广，并对广义分解数进行了解释。定理5.2提出了一种构造虚拟字符的方法，并为[33]中的定理2.6提供了一个简单的证明。顺便说一下，我们的命题1.6对[73]的主要结果做了一个简短的证明，而我们的定义2.5给出了一个稍微更一般的布劳尔字符的定义（但我们没有发展它）。本文分为五个部分，每个部分都给出了定义和所需的注释。在本文中，p是素数，(9)是具有特征为0的商域和特征为p的残差域~ f的完整离散赋值环。我们考虑的所有c -代数都与单位元相关联，它们有限地生成为c模，或者是c自由的，或者是被J((9)的Jacobson根湮灭的；类似地，所有the@-modules都是有限生成的，要么不含C，要么被J (C)湮灭。c -代数间的同态f: A~ B不要求是酉的，当Ker (f)={0}且Im (f)= f (1A) Bf （1A）时，我们说f是一个嵌入。对于任意p -代数A，我们用A*表示A的可逆元素群，用N (A)表示A*的共轭类集合。幂等幂的提升定理[10，第13章]的一个简单推论是，如果I是A的双边理想，对于任意~(A) c~ I或~+ I~(A/I)，映射~—~+ I是c~(A)集合的双射，使得~ r到~（A/I）上；特别地，设I= J (A)，我们得到了一个从N (A)到简单集合的双射
In [13 J. Alperin and M. Brou6 introduce a so called local structure in any block of a finite group G, and in [3] M. Brou6 and the author show how to use it to construct characters of G. Here we will introduce a new concept, the pointed groups, which enables us to generalize both, the local structure and the construction of characters, to interior G-algebras (see Definition 3.1 below); we believe this approach provides a more coherent treatment of the whole subject. From the local structure point of view, our Theorem 1.2 is homologous to Theorem 3.10 in [-1] and to Theorem 1.14 in [3](actually it implies them), and completes Green's definition of defect groups. Theorem 3.4 extends Green's concept of" source" to interior G-algebras. From the character point of view, Theorem 4.3 and Corollary 4.4 generalize the Brauer's Second Main Theorem and provide an interpretation of the generalized decomposition numbers. Theorem 5.2 suggests a method to construct virtual characters and provides an easy proof of Theorem 2.6 in [33. By the way, our Proposition 1.6 sketches a short proof of the main result of [73 and our Definition 2.5 leads to a slightly more general definition of Brauer characters (but we do not develop it). The paper is divided in five sections and we give in each one the definitions and notations needed. Troughout the paper, p is a prime number and (9 a complete discrete valuation ring with quotient field of characteristic zero and residual field~ f of characteristic p. All the C-algebras we consider here are associative with unit element, finitely generated as C-modules and either C-free or annihilated by J ((9)(the Jacobson radical of (9); similarly, all the@-modules are finitely generated and either C-free or annihilated by J (C). A homomorphism f: A~ B between C-algebras is not required to be unitary, and we say that f is an embedding if Ker (f)={0} and Im (f)= f (1A) Bf (1A). For any (P-algebra A, we denote by A* the group of invertible elements of A and by N (A) the set of A*-conjugacy classes of primitive idempotents of A. An easy consequence of lifting theorems for idempotents [10, ch. 13 is that if I is a two-sided ideal of A, for any~(A) either c~ I or~+ I~(A/I), and the map~--~+ I is a bijection from the set of c~(A) such that~ r onto~(A/I); in particular, setting I= J (A), we get a bijection from N (A) onto the set of simple