Study on eigenvalues and elementary divisors of Cartan matrices in finite groups

设G是有限群，F是特征p&gt；0的代数闭域。设B是具有亏群D的群代数FG的块，C是B的Cartan矩阵，ρ是C的Frobenius-Perron特征值。本文的目的是研究当ρ为整数时G或B的结构。我们从许多例子中发现，ρ是整数当且仅当B和它的Brauer对应b是Morita等价的。但当D不是阿贝尔时，这一说法有一个相反的例子。因此，如果D是交换的，我们猜想ρ是一个整数当且仅当B和b是Morita等价的。1.当D是循环的，或p=2且D是4个群时，猜想成立。2.当D是9阶初等交换3-群，B是主3-块时，猜想是真的。3当G有一个交换2-Sylow子群，B是主2-块时，猜想是真的。由于Morita等价意味着导出等价，我们对ρ是整数的关注反过来又与Broue的交换亏群猜想密切相关。

Let G be a finite group and F be an algebraically closed field of characteristic p>0. Let B be a block of the group algebra FG with defect group D. Let C be the Cartan matrix of B and let ρ be the Frobenius-Perron eigenvalue of C. It is the purpose of this research to investigate the structure of G or B when ρ is an integer. We found from many examples that ρ is an integer if and only if B and its Brauer correspondent b are Morita equivalent. But there is a counter example to this statement when D is not abelian. So, if D is abelian, we conjectured that ρ is an integer if and only if B and b are Morita equivalent. We proved that the conjecture is true in the following.1 When D is cyclic, or p=2 and D is the four group, the conjecture is true.2 When D is elementary abelian 3-group of order 9 and B is the principal 3-block, the conjecture is true.3 When G has an abelian 2-Sylow subgroup and B is the principal 2-block, the conjecture is true.Since Morita equivalence means derived equivalence, our concern that ρ is an integer is in turn deeply related to Broue's abelian defect group conjecture.