The indecomposable representations of the dihedral 2-groups

Abstract

Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X , Y)/(X 2, y2) which are finite dimensional as K-vector spaces; here (X 2, y2) stands for the ideal generated by X 2 and y2 in the free associative algebra K (X, Y) in the variables X and Y. If G is the dihedral group of order 4q (where q is a power of 2) generated by the involutions 91 and 92, and if the characteristic of K is 2, then the group algebra K G is a factor ring of R, and the K G-modules KGM which have no non-zero projective submodule correspond to the K-vector spaces (take the underlying space of ~ M ) together with two endomorphisms a and b (namely multiplication by g ~ 1 and g 2 1 , respectively) such that, in addition to a Z b 2 -0 , also (ab) q = (ba) q = 0 is satisfied. We use the methods of Gelfand and Ponomarev developped in their joint paper on the representations of the Lorentz group, where they classify pairs of endomorphisms a, b such that ab = ba = O. The presentation given here follows closely the functorial interpretation of the Gelfand-Ponomarev result by Gabriel, which he exposed in a seminar at Bonn, and the author would like to thank him for many helpful conversations.