On exponents and Auslander-Reiten components of irreducible lattices

Let G be a finite group, and let R be a complete discrete rank one valuatio n ring of characteristic zero with maximal ideal max (R) = piR, and residue class field R/piR of characteristic p > 0. The notion of the exponent of a n RG-lattice L is due to J. F. Carlson and the first author [1]. In this no te we use it to show that any non-projective absolutely irreducible RG-latt ice L with indecomposable factor module (L) over bar = L/piL lies at the en d of its connected component Theta of the stable Auslander-Reiten quiver Ga mma (5)(RG) of the group ring RG. Since such lattices L belong to p-blocks B with non-trivial defect groups sigma (B) we also study some relations bet ween the order of sigma (B) and the exponent exp(L).