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

设G是有限群，H是G的p‘-指标正规子群。设B是G的p-块，b是B所覆盖的H的任意p-块，则在Okuyama-Wada[Okuyama-Wada]的定理1.1中。数学，524,2010]，我们证明了B和b的最大本征值相等。此外，我们还证明了如果B满足B的不可约特征标次的某个条件(#)，则存在B的一个本征值λ，使得π的(λ)-部分等于B的亏群D的阶。p-可解群的任何p-块都满足(#)，但一般不成立。存在p&gt；3的反例，但是我们找不到p=2的反例。对称群的2-块是怎样的？计算不可约Brauer特征标的次数，对称群的任何2-块似乎满足(#)。此外，Kiyota，Okuyama和Wada最近证明了一个比(#)更强的结果，即任意次对称群的任何2-块都有唯一的高度为0的不可约Brauer特征标[Kiyota-Okuyama-Wada，Accept]。这推广了Fong，James关于对称群的每个非平凡不可约2-Brauer特征标的次数为偶数的定理。这个定理从来不为人所知。如果我们不考虑有限群的Cartan矩阵的特征值，我们就无法发现这一显着的事实。

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.