| Summary: | Representation theory has seen some new developments recently with the application of Auslander-Reiten theory. Given a finite dimensional algebra A, one can construct a directed graph called the Auslander-Reiten (A-R) quiver whose vertices are the isomorphism classes of indecomposable A-modules. In addition there exists certain non-split short exact sequences called A-R sequences which are closely linked with the A-R quiver. In Chapter 1 we outline this theory including a technique for constructing A-R sequences by J.A. Green [Gr 3]. In Chapter 2 we look at group representation theory concentrating on blocks of cyclic defect group. In particular we are able to construct the A-R quiver for such a block and the A-R sequence for the indecomposable modules. In Chapter 3 we look at some of C. Riedtmann's work ([Rl], [R2]) on abstract quivers and coverings of A-R quivers. In Chapter 4 we combine all these ideas to obtain results on blocks of cyclic defect group. In particular the composition factors (see [Ja] for example) for each indecomposable module are determined and a result concerning the Grothendieck group is obtained ([Bu2]). A-R sequences for blocks of cyclic defect group have already been studied and we refer to [Re] and [GaR] for details concerning both the composition factors for modules and the A-R sequences. In Chapter 5 we look at the group SL(2,pn) . Recently there has been a lot of interest in this group, including work by K. Erdmann on certain filtrations of projective modules [El] and periodic modules [E2]. We look at the simple periodic modules and construct the connected quiver components containing them, using certain pullback techniques. In this way we obtain infinite families of periodic modules of arbitrary large dimension.
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