The representation dimension of k[x,y]/(x2,yn)

The representation dimension of an Artin algebra was defined by M. Auslander in 1970. 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. 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. In this paper we show that for all n⩾0 and any field k the commutative algebras k[x,y]/(x2,y2+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 C2×Cm2 have representation dimension 3. Note that for m⩾3 these group algebras have wild representation type.