The 47th meeting of the ARTIN network will take place at the University Glasgow, April 8th-9th. Talks will be given by finishing Ph.D. students and post-docs in the ARTIN network; however all members are invited to participate. The workshop is being organized by Charalampos (Haris) Stylianakis. Friday's talks take place in Room 325 of the Mathematics and Statistic Building, and Saturday's talks will be in Room 203.
- Oliver King (University of Leeds) - Constructing some central idempotents in the Brauer algebra
- Drew Duffield (University of Leicester) - TBA
- Sira Gratz (University of Oxford) - Torsion pairs in discrete cluster categories
- Rosie Laking (University of Manchester) - TBA
- Thomas Booker-Price (University of Lancaster) - Graded Cluster Algebras
Schedule and abstracts
8th April 2016, 2:30pm to 3:00pm
Sira Gratz (University of Oxford) - Torsion pairs in discrete cluster categories
8th April 2016, 3:00pm to 4:00pm discrete cluster categories -- Show/hide abstractAbstract: Igusa and Todorov introduced discrete cluster categories of Dynkin type A, which generally are of infinite rank. That is, their clusters contain infinitely many pairwise non-isomorphic indecomposable objects. In joint work with Holm and Joergensen we study torsion pairs in these categories and provide a complete combinatorial classification. Cluster tilting subcategories, t-structures, and co t-structures are all particular instances of torsion pairs and from our classification we are able to describe each of these classes. In particular, there are no co t-structures but, contrary to the finite case, there are a number of interesting t-structures.
Drew Duffield (University of Leicester) - Auslander-Reiten Components of Brauer Graph Algebras
8th April 2016, 4:30pm to 5:30pm -- Show/hide abstractAbstract: One approach to the representation theory of algebras is to study the module category of an algebra. This can be achieved, at least in part, by describing the indecomposable modules of an algebra and the irreducible morphisms between them. The Auslander-Reiten quiver of an algebra is a means of presenting this information. Of particular interest is a class of algebras known as Brauer graph algebras. These are symmetric special biserial algebras that have a presentation in the form of a (decorated) ribbon graph called a Brauer graph. An interesting feature of Brauer graph algebras is that one can often read off aspects of the representation theory by performing a series of combinatorial games on the Brauer graph, which removes the need for potentially difficult and lengthy calculations. The purpose of this talk is show that one can read off information regarding the Auslander-Reiten theory of a Brauer graph algebra from its underlying Brauer graph. We begin by providing an algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a Brauer graph algebra using only information from its Brauer graph. We then show that the structure of the Auslander-Reiten quiver is closely related to the distinct Green walks around the Brauer graph and detail the relationship between the precise shape of the stable Auslander-Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or indecomposable projective module for any Brauer graph algebra is determined by the edge in the Brauer graph associated to the module.
Short talk session
8th April 2016, 5:40pm to 6:40pm
8th April 2016, 7:00pm to 9:00pm Balbir's -- Show/hide abstractAbstract: balbirs.co.uk
Oliver King (University of Leeds) - Constructing some central idempotents in the Brauer algebra
9th April 2016, 9:30am to 10:30am -- Show/hide abstractAbstract: Classical Schur-Weyl duality relates the representations of the general linear group and the symmetric group via their action on tensor space. The Brauer algebra was introduced by Brauer in 1937, to play the role of the symmetric group when one replaces the general linear group with the orthogonal or symplectic groups. In this seminar I will briefly discuss the representation theory of the Brauer algebra, and then provide a new method of constructing central idempotents relating to the splitting of short exact sequences. I will then explain how we can derive some information about the Brauer algebra from these idempotents.
Rosie Laking (University of Manchester) - The Krull-Gabriel dimension of a category
9th April 2016, 10:45am to 11:45am -- Show/hide abstractAbstract: In this talk we will consider categories of finitely presented functors from a module category to the category of abelian groups. Such categories turn out to be a natural setting in which we may study the morphisms between finitely presented modules and the Krull-Gabriel dimension can be seen as a measure of the complexity of the morphism structure in the module category. It is calculated via iterated localisation of the functor category and we will give lots of examples in the context of finite-dimensional algebras in order to demonstrate how the Krull-Gabriel dimension effectively reflects the structure of the module category. In particular I will report on joint work with K. Arnesen, D. Pauksztello, and M. Prest as well as joint work with M. Prest and G. Puninski.
Lunch (sandwiches provided)
9th April 2016, 11:45am to 1:00pm
Thomas Booker-Price (University of Lancaster) - Graded Cluster Algebras
9th April 2016, 1:00pm to 2:00pm -- Show/hide abstractAbstract: A graded cluster algebra assigns degrees to the initial cluster variables in such a way that all exchange relations are homogeneous. This in turn makes all other cluster variables homogeneous and gives the cluster algebra the structure of a Z^n-graded algebra. These gradings have been implicit in the literature for some time, but were formalised (in the sense we are interested in) by Grabowski and Launois in 2013. One question we may ask about such a grading is how the cluster variables are distributed in terms of degrees: whether there are finitely many cluster variables per occurring degree, infinitely many per degree, or a mixture. In this talk we will give a partial classification of graded cluster algebras generated by rank 3 quivers in terms of this question. We will also consider the graded (quantum) cluster algebra structure on the homogeneous coordinate ring of m x n matrices and of quantum Grassmannians, and show that these contain cluster variables in all positive integer degrees.
Time for individual discussions (as desired)
9th April 2016, 2:00pm to 3:30pm
There will be two talks in the afternoon on Friday, and a session of short talks, intended for participants to briefly introduce themselves and their research areas. Saturday there will be three talks, ending mid-afternoon Saturday. The start of Friday's talks will be timed to allow people to make it over after the final talk of the ICMS conference, Representation Theory and Symplectic Singularities, taking place in Edinburgh:
Complete details for ARTIN 47 will appear in due time on the ARTIN webpage,
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