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

设G是一个有限群，H是G的一个带p'-指标的正规子群。设B是G的p块B是B覆盖的H的任意p块，那么在定理1中。1 . Okuyama-Wada的作品[Okuyama-Wada，当代]。数学。， 524, 2010]，我们证明了B和B的最大特征值相等。进一步证明了如果B满足B的不可约Brauer字符的阶数的一定条件（#），则B存在一个特征值λ，使得λ的(π)-部分等于B的缺陷群D的阶数。任何p-可解群的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块是什么？计算不可约布劳尔字符的度，对称群的任何2块似乎都满足（#）。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块具有唯一的高度为0的不可约Brauer特征[Kiyota-Okuyama-Wada，接受]。推广了Fong， James关于对称群的每一个非平凡不可约2-Brauer特征的度是偶的定理。这个定理从来没有人知道。如果我们不考虑有限群的Cartan矩阵的特征值，我们就不能发现这个显著的事实。

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.