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 be the Cartan matrix of B and p(B) the Perron-Frobeniuseigenvalue (i.e. the largest one) of C y D2B y D2. Let k(B)，l(B) be the number of ordinary and modular (i.e. over F) irreducible charactersrespectively.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 isa stronger assertion than Brauer's famous k(B)-conjecture for p-solvable groups. In [2] wecompleted to calculate the Cartan matrix of a certain class of finite solvable groups有很多0个实体。我们也曾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 D2Bcoincide. We have a striking conjecture that p(B) is rational if and only if B is Moritaequivalent to the Brauer correspondent b of N D2G D2(D). In [4] we newly obtained a definition ofP-good module and P-good group which is related to eigenvalues of C D2B 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，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）

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)