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

设G是有限群，H是G的正规子群，且H的指数为p '.设B是G的一个p-块，B是H中被B覆盖的任意一个p-块.然后在定理1中。1的工作Okuyama-Wada[ Okuyama-Wada，Contemp.数学，524，2010]，我们已经证明了B和B的最大特征值相等。进一步证明了：若B在B中的不可约Brauer特征标的阶满足一定条件（#），则存在B的特征值λ，使得λ的（π）-部分等于B的亏群D的阶.任何p-可解群的p-块都满足（#），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Brauer特征标的次数，对称群的任何2-块似乎满足（#）。此外，清田，奥山和田最近证明了一个更强的结果比（#），任何2块的对称群的任意程度有一个独特的不可约布劳尔字符的高度0[清田，奥山和田，接受]。这推广了Fong，James关于对称群的每个非平凡不可约2-Brauer特征标的次数是偶数的定理。这个定理从未为人所知。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.