A lower bound of the Perron-Frobenius eigenvalue of the Cartan matrix of a finite group

Abstract.

In this article we exhibit a relation between the number k(B) of ordinary irreducible characters in a p-block B of a finite group G and the Cartan invariants c _ij of B. Next, we give a lower bound of the Perron-Frobenius eigenvalue $\rho (C_B)$ of the Cartan matrix C _B of B in terms of B, that is $k(B) \le \rho (C_B)l(B)$, where l(B) is the number of irreducible Brauer characters in B. For p-solvable groups, we conjecture $k(B) \le \rho (C_B)$ that is closely related to the Brauer's k(B) conjecture.