Nilpotent blocks and their source algebras
Nilpotent blocks and their source algebras
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DOI：
10.1007/bf01393688
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发表时间：
1988-02
影响因子：
3.1
通讯作者：
L. Puig
中科院分区：
文献类型：
--
作者：
L. Puig
1.1.设p是素数，~是特征p的代数闭域，（~是具有剩余域的完全离散赋值环。在[3]中，Brou和我们从一个新的观点开始研究有限群G的块B，其假设不再是关于B的亏群P的结构，而是关于P在（9 Gb，即关于B的所谓”局部结构”，它在[3]中被Brauer范畴的等价类所隐含地表示（见[1]）正式定义）。精确地说，假设对P的任意子群Q和CG（Q）中任意与B有关的块f，（Q，f）在G中的稳定子与Q的中心化子的商N~（Q，f）/CG（Q）是p-群，我们确定了B的广义分解数的全矩阵（直到符号），而不管P的结构如何.在[9]中，我们修改了局部结构的概念，以便将它的应用范围扩大到任何内部G-代数（实际上，对于任何G-代数），特别是对于任何（9 G-模的（9-自同态：B的”局部结构”在[9]中隐含地表示为所谓局部范畴的等价类（形式定义见[11]），一般来说比布劳尔的要好与（gG-模的情形一样，”源”的概念在新的上下文中自然产生，并且内部G-代数（9 Gb-
1.1. Letp be a prime number,~ an algebraically closed field ofcharacteristicp and (~ a complete discrete valuation ring with residue field~. In [3] Brou6 and ourself commenced the study of a block b of a finite group G from a new point of view, the hypothesis being no longer on the structure of a defect group P ofb but on the kind of embedding of P in (9 Gb, namely on the so called" local structure" of b which was implicitely represented in [3] by the equivalent class of the Brauer category (see [1] for a formal definition). Precisely, assuming that for any subgroup Q of P and any block f of CG (Q) associated with b, the quotient N~(Q, f)/CG (Q) of the stabilizer of (Q, f) in G by the centralizer of Q is a p-group, we determined (up to signs) the full matrix of generalized decomposition numbers of b, regardless the structure of P.1.2. In [9] we modified the notion of" local structure" in order to enlarge its area of application to any interior G-algebra (actually, to any G-algebra) and in particular, to the full algebra of (9-endomorphisms of any (9 G-module: the" local structure" of b was implicitely represented in [9] by the equivalent class of the so called local category (see [11] for a formal definition), which is in general finer than the Brauer one. As in the (gG-module case; the concept of" source" arises naturally in the new context, and the source of the interior G-algebra (9 Gb-which