  The representation dimension was defined by M. Auslander in 1970 and is, due
to spectacular recent progress, one of the most interesting homological
invariants in representation theory. The precise value is not known in general,
and is very hard to compute even for small examples. For group algebras, it is
known in the case of cyclic Sylow subgroups, due to Auslander's fundamental
work. For some group algebras (in characteristic 2) of rank at least 3 the
precise value of the representation dimension follows from recent work of
R.Rouquier. There is a gap for group algebras of rank 2; here the deep
geometric methods do not work. In this paper we show that for all n and any
field $k$ the commutative algebras k[x,y]/(x^2, y^{2+n}) have representation
dimension 3. For the proof, we give an explicit inductive construction of a
suitable generator-cogenerator. As a consequence, we obtain that the group
algebras in characteristic 2 of the groups C_2 x C_{2^m} have representation
dimension 3. Note that for m>2 these group algebras have wild representation
type.

Representation Theory