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SUMMARY:Auslander-Reiten Components of Brauer Graph Algebras  - Drew Duffi
 eld\, University of Leicester
DTSTART:20151023T140000Z
DTEND:20151023T150000Z
UID:TALK61647@talks.cam.ac.uk
CONTACT:Nicolas Dupré
DESCRIPTION:One approach to the representation theory of algebras is to st
 udy\nthe module category of an algebra. This can be achieved\, at least in
  part\,\nby describing the indecomposable modules of an algebra and the ir
 reducible\nmorphisms between them. In some sense\, these can be viewed as 
 the building\nblocks for all modules and morphisms in the module category.
  The\nAuslander-Reiten quiver of an algebra is a means of presenting this\
 ninformation. Of particular interest is a class of algebras known as Braue
 r\ngraph algebras. These are symmetric special biserial algebras that have
  a\npresentation in the form of a (decorated) ribbon graph called a Brauer
 \ngraph. An interesting feature of Brauer graph algebras is that one can o
 ften\nread off aspects of the representation theory by performing a series
  of\ncombinatorial games on the Brauer graph\, which removes the need for\
 npotentially difficult and lengthy calculations. The purpose of this talk 
 is\nshow that one can read off information regarding the Auslander-Reiten 
 theory\nof a Brauer graph algebra from its underlying Brauer graph. We beg
 in by\nproviding an algorithm for constructing the stable Auslander-Reiten
 \ncomponent containing a given indecomposable module of a Brauer graph alg
 ebra\nusing only information from its Brauer graph. We then show that the\
 nstructure of the Auslander-Reiten quiver is closely related to the distin
 ct\nGreen walks around the Brauer graph and detail the relationship betwee
 n the\nprecise shape of the stable Auslander-Reiten components for domesti
 c Brauer\ngraph algebras and their underlying graph. Furthermore\, we show
  that the\nspecific component containing a given simple or indecomposable 
 projective\nmodule for any Brauer graph algebra is determined by the edge 
 in the Brauer\ngraph associated to the module.\n
LOCATION:CMS\, MR15
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