This paper classifies the indecomposable modules over a class of quasi-Frobenius algebras over a field k. These algebras have a k-basis described by a Brauer graph.The product of two of these basis elements is either another basis element or zero. The Brauer trees that arise in the study of blocks with cyclic defect group are a special case.The modules may be described as diagrams of vector spaces subject to certain relationships of commutativity and to certain composites being zero. In this interpretation the commutativity of k need not be assumed. As the motivation of this paper is the representation theory of certain finite groups, the presentation is for commutative k only. However no restriction on the characteristic or algebraic closure of k is made.A module is said to be uniserial if it has a unique composition series. A module M is said to be special if it has two uniserial submodules U and V such that Urn V is the socle of M and is simple, M/U and M/V are uniserial and U + V is the unique maximal submodule of M. In particular, uniserial and (by special convention) simple modules are special. If k is algebraically closed, finite dimensional kalgebras whose indecomposable projective modules are special can have their representation theory determined by the methods of this paper.As implied above, the modular representation theory of a block with cyclic defect group is a special case of the work in this paper. More significantly, let k be commutative, have characteristic 2 and contain a splitting field for the group G =$4,$5, A 7 or PSL(2,q), where q is an odd prime power. The indecomposable projective kG-modules are special, and so the indecomposable modular representations of kG are classified by the main theorem of this paper. Section 2 implicitly describes the structure of the indecomposable projective modules in the principal block in each ease. The non-principal blocks cause little difficulty. The details need some modification when k does not contain a splitting field.The groups mentioned above have dihedral Sylow subgroups. Indeed any simple group with dihedral Sylow subgroup is isomorphic to one of them. Thus the classification of their modular representations could, in principle, be deduced from Ringel's classification [7] (also obtained by Bondarenko [1]) of the modular

This paper gives a detailed classification of the indecomposable representations of one non-trivial “commutative quiver”, gives brief details of certain generalisations, and uses methods likely to be more widely applicable.

The indecomposable representations in characteristic two of the groups PSL(2, q) where q is congruent to 3 or 5 modulo 8 are classified. For q = 3 or 5 the classification is obtained by explicit construction of modules, using the Green correspondence to prove completeness. For larger q , the classification is obtained using equivalences between appropriate categories of modules.

Let F be an algebraic number field, and S a subgroup of the general linear group GL(n, F). We shall call S a U-group if S satisfies the condition (U): Every x ∈ S is a matrix all of whose eigenvalues are algebraic integers. (This is equivalent to either of the following conditions: a) the eigenvalues of each matrix (x are all units as algebraic numbers; b) the characteristic polynomial for x has all its coefficients integers in F. In particular, then, every group of matrices with entries in the integers of F is a U-group.

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