Chanter, Anthony (1984) On the construction and application of Auslander-Reiten quivers to certain group algebras. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Representation of algebras
Official Date:	March 1984
Dates:	Date Event March 1984 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Green, Sandy
Sponsors:	Science and Engineering Research Council (Great Britain)
Format of File:	pdf
Extent:	124 leaves : illustrations
Language:	eng

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