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CATEGORIES:Junior Algebra/Logic/Number Theory seminar
SUMMARY:Auslander-Reiten Components of Brauer Graph Algebr
as - Drew Duffield\, University of Leicester
DTSTART;TZID=Europe/London:20151023T150000
DTEND;TZID=Europe/London:20151023T160000
UID:TALK61647AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/61647
DESCRIPTION:One approach to the representation theory of algeb
ras is to study\nthe module category of an algebra
. This can be achieved\, at least in part\,\nby de
scribing the indecomposable modules of an algebra
and the irreducible\nmorphisms between them. In so
me sense\, these can be viewed as the building\nbl
ocks for all modules and morphisms in the module c
ategory. The\nAuslander-Reiten quiver of an algebr
a is a means of presenting this\ninformation. Of p
articular interest is a class of algebras known as
Brauer\ngraph algebras. These are symmetric speci
al biserial algebras that have a\npresentation in
the form of a (decorated) ribbon graph called a Br
auer\ngraph. An interesting feature of Brauer grap
h algebras is that one can often\nread off aspects
of the representation theory by performing a seri
es of\ncombinatorial games on the Brauer graph\, w
hich removes the need for\npotentially difficult a
nd lengthy calculations. The purpose of this talk
is\nshow that one can read off information regardi
ng the Auslander-Reiten theory\nof a Brauer graph
algebra from its underlying Brauer graph. We begin
by\nproviding an algorithm for constructing the s
table Auslander-Reiten\ncomponent containing a giv
en indecomposable module of a Brauer graph algebra
\nusing only information from its Brauer graph. We
then show that the\nstructure of the Auslander-Re
iten quiver is closely related to the distinct\nGr
een walks around the Brauer graph and detail the r
elationship between the\nprecise shape of the stab
le Auslander-Reiten components for domestic Brauer
\ngraph algebras and their underlying graph. Furth
ermore\, we show that the\nspecific component cont
aining a given simple or indecomposable projective
\nmodule for any Brauer graph algebra is determine
d by the edge in the Brauer\ngraph associated to t
he module.\n
LOCATION:CMS\, MR15
CONTACT:Nicolas DuprÃ©
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