It was established by Reinhard Knörr that the half of Brauer's height zero conjecture which assumes an abelian defect group would follow from the nonexistence of certain integral representations of abelian p-groups. We establish nonexistence of such representations of rank less than 14 for the elementary abelian group of order 8 and believe these to be the first results of this kind. The result is made possible by imposing a simple rationality condition.

Communicated by Michel BrouéKeywords: Virtually irreducible lattices Brauer's height zero conjecture Abelian defect group Virtually irreducible lattices which satisfy a simple rationality condition and have positive height are constructed for elementary abelian 2-groups. The non-existence of such lattices would have implied part of Brauer's height zero conjecture. A characterization of rank 6 lattices for the elementary abelian group of size 16 is given.

The almost split sequences terminating in certain Knörr lattices for finite groups are shown to have indecomposable middle terms. The tree classes of the connected components of the stable Auslander–Reiten quiver containing certain Knörr lattices for elementary abelian p‐groups are shown to be A∞.

We show that the irreducible characters lying in 2-blocks of finite solvable groups with abelian defect groups have trivial 2-local Schur indices.

We identify a class of symmetric algebras over a complete discrete valuation ring O of characteristic zero to which the characterisation of Knörr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric O-algebras whose extension to the quotient field of O is separable.