Mathematics – Representation Theory
Scientific paper
2005-05-23
Mathematics
Representation Theory
15 pages
Scientific paper
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.
Holm Thorsten
Hu Wei
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