Eigenvector matrices of Cartan matrices for finite groups.
Let \(p\) be a prime, let \(G\) be a finite group, and let \((K,R,F)\) be a \(p\)-modular system which is sufficiently large for \(G\). Moreover, let \(B\) be a block of the group algebra \(FG\) with defect group \(D\) and Cartan matrix \(C_B\). The author is interested in relations between \(R_B\), the set of eigenvalues of \(C_B\), and \(E_B\), the set of elementary divisors of \(C_B\). All eigenvalues of \(C_B\) are positive real numbers, and there is a unique largest one, denoted by \(\rho(B)\). The author conjectures that the following conditions are equivalent: (a) \(\rho(B)\in\mathbb{Z}\); (b) \(\rho(B)=|D|\); (c) \(R_B=E_B\); (d) \(R_B\subseteq\mathbb{Z}\). In the present paper, he considers the related question whether condition (d) implies the existence of a matrix \(U_B\in\text{GL}(l,R)\) whose columns consist of eigenvectors of \(C_B\). (A positive answer would give that (d) implies (c).) The author shows that his question has indeed a positive answer whenever \(B\) is Morita equivalent to its Brauer correspondent in \(N_G(D)\). He also checks that the answer is positive when \(D\) is cyclic, dihedral, semidihedral or (generalized) quaternion, or when \(G\) is \(p\)-solvable, or when \(B\) is the principal 3-block and \(|D|=9\).
scientific article; zbMATH DE number 5144428