Resolutions over symmetric algebras with radical cube zero

Let Lambda be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying J(3)(Lambda)=- 0. Let S-1 ,..., S-r be representatives of the isomorphism classes of simple A-modules, and let E be the r x r matrix whose (i, j) entry is dim(k) Ext(Lambda)(1)(S-i, S-j). If there exists an eigenvalue lambda of E satisfying vertical bar lambda vertical bar > 2 then the minimal resolution of each non-projective finitely generated A-module has exponential growth, with radius of convergence 1/2 (lambda - root lambda(2) - 4). On the other hand, if all eigenvalues lambda of E satisfy vertical bar lambda vertical bar <= 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated A-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. (C) 2008 Elsevier Inc. All rights reserved.