A STUDY OF THE CARTAN MATRICES OF FINITE GROUPS

Let G be a finite group， FG the group algebra of G over an algebraically closed field F of characteristic p > 0， and B a block of FG。Let C-D2B-D2 be the Cartan matrix of B and p（B） the Perron-Frobenius eigenvalue （i.e. the largest one） of C-D2B-D2。Let k（B）、l（B） be the number of ordinary and modular （i.e. over F） irreducible Management ters， respectively。We have found k（B）[less than or equal] p（B）l（B） holds for any block B of G in [1]， further we posed a conjecture that k（B）[less than or equal] p（B） for any p-solvable groups。这是一个比布劳尔著名的K（B）更强的断言--一个P-可解决群体的概念。在[2]中，我们已经完成了计算一个特定类别的有限可解决组的Cartan矩阵，其中有许多0个条目。我们在本案中也验证了k（B）[小于或等于] p（B）。In [3] we investigated on the fundamental question when eigenvalus and elementary divisors of CイD2BイエD2 coincide。We have had a striking conjecture that p（B） is rational if and only if B is Morita equivalent to the Brauer correspondent b of N-D2G-D2（D）。In [4] we newly obtained a definition of P-good module and p-good group which is related to eigenvalues of C-D2B-D2. [1]T. T. Wada，Perron-Frobenius对Cartan矩阵的基本价值的一个较低的界限，为有限的群体。Arch. Math 73 （1999年）， 407-413。[2]T. T. Wada，一个特定类别的有限可解决群体的Cartan矩阵。（已接受大日的采访）数学（）[3]M. M. Kiyota， M. Murai和T。Wada，“Cartan matrices of the finite group的eigenvalues”。（in preperation） [4] A. Hanaki， M. Kiyota， M. Murai和T。Wada，P-好的模块和P-好的组。（在筹备中）

Let G be a finite group, FG the group algebra of G over an algebraically closed field F of characteristic p > 0, and B a block of FG. Let CィイD2BィエD2 be the Cartan matrix of B and p(B) the Perron-Frobenius eigenvalue (i.e. the largest one) of CィイD2BィエD2. Let k(B), l(B) be the number of ordinary and modular (i.e. over F) irreducible characters, respectively.We have found k(B) 【less than or equal】 p(B)l(B) holds for any block B of G in [1], further we posed a conjecture that k(B) 【less than or equal】 p(B) for any p-solvable groups. This is a stronger assertion than Brauer's famous k(B)-conjecture for p-solvable groups. In [2] we completed to calculate the Cartan matrix of a certain class of finite solvable groups, which have many 0 entries. We also verified that k(B) 【less than or equal】 p(B) in this case. In [3] we investigated on the fundamental question when eigenvalus and elementary divisors of CィイD2BィエD2 coincide. We have had a striking conjecture that p(B) is rational if and only if B is Morita equivalent to the Brauer correspondent b of NィイD2GィエD2(D). In [4] we newly obtained a definition of P-good module and p-good group which is related to eigenvalues of CィイD2BィエD2.[1] T. Wada, A lower bound of the Perron-Frobenius eigenvalue of the Cartan matrix for finite groups. Arch. Math. 73 (1999), 407-413. [2] T. Wada, The Cartan matrix of a certain class of finite solvable groups. (accepted to Osaka Jour. Math.) [3] M. Kiyota, M. Murai and T. Wada, Rationality of eigenvalues of the Cartan matrices of finite groups. (in preperation) [4] A. Hanaki, M. kiyota, M. Murai and T. Wada, P-good modules and p-good groups. (in preperation)