Defect zero p-blocks for finite simple groups
Defect zero p-blocks for finite simple groups
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DOI：
10.1090/s0002-9947-96-01481-x
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发表时间：
1996-12
影响因子：
1.3
通讯作者：
A. Granville;K. Ono
中科院分区：
文献类型：
--
作者：
A. Granville;K. Ono
我们对Brauer图（或分解矩阵）具有缺陷为0的p块的有限简单群进行了分类，完成了许多作者的研究。唯一缺陷0 p块未分类的有限单群是交替群An。这里我们表明，对于每一个素数p≥5，这些都有一个缺陷为0的p块。这是通过对每一个对称群Sn证明相同的结果而得出的，反过来，作为t核分割猜想的结果，对于任意t≥4，每一个非负整数至少具有一个t核分割。当t≥17时，我们将这个问题简化为拉格朗日定理，即每个非负整数都可以写成四个平方和。t < 17的唯一情况是t = 13，这在以前的工作中没有涉及到。我们用一个非常不同的论证来证明这一点，通过用模形式解释t核分区的生成函数，然后使用Deligne定理（参见Weil猜想）控制系数的大小。我们还考虑了Sn的p块数的同余，证明了Garvan的一个猜想，该猜想在5 < p < 23时建立了一定的乘法同余。利用Serre关于模形式系数的可整除性的一个结果，我们证明了对于任意给定素数p和正整数m， Sn中缺陷为0的p块的数目是几乎所有n的m的倍数。我们还证明了任意给定素数p可以整除Sn的几乎所有n的p模不可约表示的数目。©1996美国数学学会。
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p-blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (nee the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 < p < 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p-blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p-modularly irreducible representations of Sn, for almost all n. © 1996 American Mathematical Society.