Resolutions over symmetric algebras with radical cube zero

Let Λ be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying J3(Λ)=0. Let S1,…,Sr be representatives of the isomorphism classes of simple Λ-modules, and let E be the r×r matrix whose (i,j) entry is . If there exists an eigenvalue λ of E satisfying |λ|>2 then the minimal resolution of each non-projective finitely generated Λ-module has exponential growth, with radius of convergence . On the other hand, if all eigenvalues λ of E satisfy |λ|⩽2 then the dimensions of the modules in the minimal projective resolution of each finitely generated Λ-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron–Frobenius theorem.