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Eigenvalues of Cartan matrices and Morita equivalences of blocks in finite groups

有限群中块的嘉当矩阵特征值和森田等价

基本信息

批准号：
21540009
负责人：
WADA Tomoyuki
金额：
$ 2.41万
依托单位：
Tokyo University of Agriculture and Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2009
资助国家：
日本
起止时间：
2009 至 2011
项目状态：
已结题

项目摘要

Let G be a finite group and H be a normal subgroup of G with p'-index. Let B be a p-block of G and b be any p-block of H covered by B. Then in Theorem 1. 1 of the work of Okuyama-Wada[ Okuyama-Wada, Contemp. Math., 524, 2010], we have proved that the largest eigenvalues of B and b are equal. Furthermore, we have proved that if B satisfies a certain condition(#) on the degrees of irreducible Brauer characters in B, then there exists an eigenvalueλof B such that the(π)-part ofλis equal to the order of defect group D of B. Any p-block of p-solvable group satisfies(#), but it does not hold in general. There exist counter examples for p> 3, however we could not find a counter example for p=2. How is a 2-block of the symmetric group? Calculating the degrees of irreducible Brauer characters, any 2-block of the symmetric group seems to satisfy(#). Furthermore, Kiyota, Okuyama and Wada have recently proved a stronger result than(#) that any 2-block of the symmetric group of arbitrary degree has a unique irreducible Brauer character of height 0[ Kiyota-Okuyama-Wada, accepted]. This generalizes the theorem of Fong, James that the degree of every non-trivial irreducible 2-Brauer characters of the symmetric group is even. This theorem has never been known. We could not discover this remarkable fact if we would not consider eigenvalues of the Cartan matrices of finite groups.

设 G 为有限群，H 为 G 的正规子群，具有 p' 索引。设 B 是 G 的 p 块，b 是 B 覆盖的 H 的任何 p 块。那么在 Okuyama-Wada 的定理 1. 1 中[ Okuyama-Wada，Contemp。 Math., 524, 2010]，我们证明了B和b的最大特征值是相等的。此外，我们还证明，如果B在B中不可约布劳尔特征的次数上满足一定条件(#)，则存在B的特征值λ，使得λ的(π)部分等于B的缺陷群D的阶。p可解群的任何p块满足(#)，但一般不成立。存在 p> 3 的反例，但是我们找不到 p=2 的反例。对称群的2-块如何？计算不可约布劳尔特征的度，对称群的任意2块似乎都满足(#)。此外，Kiyota、Okuyama 和 Wada 最近证明了比(#)更强的结果，即任意阶对称群的任何 2 块都具有高度为 0 的唯一不可约布劳尔特征[Kiyota-Okuyama-Wada，已接受]。这推广了 Fong, James 的定理，即对称群的每个非平凡不可约 2-Brauer 特征的度都是偶数。这个定理从未被人所知。如果我们不考虑有限群嘉当矩阵的特征值，我们就无法发现这个显着的事实。