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The 46th ARTIN meeting will take place at the University of Glasgow on the 6th and 7th of November 2015. More details will appear here soon.

Speakers:

  • Giovanni Cerulli Irelli (Rome)
  • Xin Fang (Cologne)
  • Martina Lanini (Edinburgh)
  • Markus Reineke (Wuppertal)
  • Johan Martens (Edinburgh)
  • Ben Martins (Aberdeen)

Tentative Schedule:

  • ARTIN 46
    19th September 2017, 8:47pm to 8:47pm
  • Martina Lanini (Edinburgh) - Degenerate flags and Schubert varieties
    6th November 2015, 2:00pm to 3:00pm -- Show/hide abstract
    Abstract: Introduced in 2010 by Evgeny Feigin, degenerate flag varieties are degenerations of flag manifolds, naturally arising from a representation theoretic context. In this talk, I will discuss joint work with G. Cerulli Irelli, and G. Cerulli Irelli and P. Littelmann, in which we show that such degenerations in type A and C not only share a lot of properties with Schubert varieties (as previously proven by Feigin, Finkelberg and Littelmann), but are in fact Schubert varieties in an appropriate flag manifold.
  • Johan Martens (Edinburgh) - Gelfand-Cetlin revisited
    6th November 2015, 3:00pm to 4:00pm -- Show/hide abstract
    Abstract: In 1950 Gel'fand and Cetlin constructed the first example of what we now call canonical bases for the representations of a semi-simple Lie algebra. Much later Guillemin and Sternberg showed how these could be understood as arising out of the geometric quantization of flag varieties or co-adjoint orbits in various polarisations, relying on a (real) integrable system that is closely related to toric degenerations of these flag varieties. We shall outline this story, and then indicate how this symplectic story generalises using contractions of Hamiltonian spaces. This is based joint work with Joachim Hilgert and Chis Manon.
  • Coffee Break
    6th November 2015, 4:00pm to 5:00pm
  • Giovanni Cerulli Irelli (Bonn) - Quiver Grassmannians associated with Dynkin quivers
    6th November 2015, 5:00pm to 6:00pm -- Show/hide abstract
    Abstract: Given a quiver Q and a finite dimensional Q--representation M, a quiver Grassmannian associated with M parametrizes the subrepresentations of M of a fixed dimension vector. Such projective varieties appeared in the work of Schofield, for the study of general properties of Q--representations. More recently, it was discovered that their Euler characteristic plays a prominent role to categorify cluster algebras associated with Q. It is natural to ask about their geometric properties but it turns out that their geometry can be arbitrarily complicated. In this talk I will restrict my attention to the case when Q is an orientation of a simply--laced Dynkin diagram: In a series of papers with Markus Reineke and Evgeny Feigin, we have developed tools in order to study the geometry of such varieties. Such tools are useful to study classes of quiver Grassmannians of Dynkin type. For example, and this was our motivation, degenerate flag varieties of type A can be realized as a "well--behaved" quiver Grassmannian of Dynkin type. I will overview this tools and provide several applications.
  • Dinner: TBA
    6th November 2015, 7:00pm to 8:00pm
  • Ben Martin (Aberdeen) - Complete reducibility for reductive algebraic groups.
    7th November 2015, 10:00am to 11:00am -- Show/hide abstract
    Abstract: Let G be a reductive algebraic group over a field k of positive characteristic. The notion of a completely reducible subgroup of G generalises the notion of a completely reducible representation (which is the special case when G=GL_n(k)). I will describe a geometric approach to the theory of complete reducibility, based on ideas of R.W. Richardson, and I will discuss some recent work involving non-algebraically closed fields.
  • Coffee Break
    7th November 2015, 11:00am to 12:00pm
  • Xin Fang (Köln) - New filtrations and monomial bases of quantum groups
    7th November 2015, 12:00pm to 1:00pm -- Show/hide abstract
    Abstract: For a Lie algebra of type A or C, E. Feigin, G. Fourier and P. Littelmann constructed a monomial basis for any finite dimensional irreducible representation which is compatible with the PBW-filtration arising from the famous Poincaré-Birkhoff-Witt theorem. This monomial basis is parametrized by the lattice points in a normal polytope, called the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope. This construction solves a conjecture of E. Vinberg. In this talk, I will explain how to define quantum PBW filtrations on quantum groups, via realizing them as Hall algebras of quiver representations. In type A, for any finite dimensional irreducible representation, this construction leads to a canonical monomial basis, which is parametrized exactly by the FFLV polytope. If time permits, I will present results on other types. This talk is based on two joint works: one with G. Fourier and M. Reineke; the other with T. Backhaus and G. Fourier.
  • Markus Reineke (Wuppertal) - Quiver moduli and small resolutions of some GIT quotients
    7th November 2015, 1:00pm to 2:00pm -- Show/hide abstract
    Abstract: A resolution of singularities of a singular space is called small if it satisfies a specific strong dimension estimate for its fibres. The utility of this notion stems from the fact that the intersection cohomology of a singular space can be computed from the cohomology of a small resolution. We construct a class of small resolutions using moduli spaces of representations of quiver. Several classes of examples will be discussed, related to determinantal varieties, moduli of point configurations in the projective line, and certain quotients by Levi actions.
  • Welcome
    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 abstract
    Abstract: 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 abstract
    Abstract: 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
  • Dinner
    8th April 2016, 7:00pm to 9:00pm Balbir's -- Show/hide abstract
    Abstract: 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 abstract
    Abstract: 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 abstract
    Abstract: 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 abstract
    Abstract: 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
  • Vanessa Miemietz (East Anglia) - Baby steps in p-dg 2-representation theory
    1st September 2016, 2:00pm to 3:00pm -- Show/hide abstract
    Abstract: TBA
  • Martin Herschend (Uppsala) - Thick subcategories of n-cluster tilting subcategories
    1st September 2016, 3:10pm to 4:10pm -- Show/hide abstract
    Abstract: In Iyama's higher dimensional Auslander-Reiten theory one shifts attention from module categories of algebras to so called n-cluster tilting subcategories for some fixed positive integer n. These are n-abelian (in the sense of Jasso) and so notions like kernel, cokernel and extension are replaced by their higher analogues: n-kernel, n-cokernel and n-extension, which are similar except that the exact sequences involved are longer. Thus it makes sense to consider subcategories of n-abelian categories that are closed under n-kernels, n-cokernels and n-extensions. We shall refer to these as thick subcategories. In my talk I will present a characterization of thick subcategories of n-cluster tilting subcategories with finitely many indecomposables. This will then be used to classify thick subcategories on n-cluster tilting subacategories for Nakayama algebras of global dimension n.
  • Catharina Stroppel (Bonn) - R-matrices and convolution algebras for Grassmannians
    1st September 2016, 4:40pm to 5:40pm -- Show/hide abstract
    Abstract: In this talk I will describe in detail the combinatorics of equivariant cohomologies of Grassmannians, their Schubert classes (depending on a choice of Borel) and torus fixed point bases and connect them with the representation theory of Lie algebras. In particular we will construct R-matrices as a base change from one choice of Schubert class basis to another. In this way we will construct certain Bethe algebras which are important and interesting from the theory of integrable systems and Bethe bases. Finally we will connect this with the current algebra for sl_2 and the group algebra of the affine Weyl group and compare it briefly with constructions of Ginzburg, Maulik and Okounkov in a related but different setting.
  • Dinner at 'El Coto'
    1st September 2016, 7:30pm to 9:30pm 21 Leazes Park Rd, Newcastle NE1 4PF
  • Gustavo Jasso (Bonn) - Mesh categories of type A-infinity and tubes in higher Auslander--Reiten theory
    2nd September 2016, 9:15am to 10:15am -- Show/hide abstract
    Abstract: This is a report on ongoing work with Julian Kuelshammer. We construct higher analogues of mesh categories of type A-infinity and of the tubes from the viewpoint of Iyama's higher Auslander--Reiten theory. Our construction relies on unpublished work by Darpö and Iyama. We sketch a conjectural construction which relates our categories to spherical objects and to cluster tubes.
  • Kevin McGerty (Oxford) - Springer theory and symplectic resolutions
    2nd September 2016, 10:30am to 11:30am -- Show/hide abstract
    Abstract: We will describe how an analogue of Springer's theory of Weyl group representations can be defined for a symplectic resolution of singularities, and explain what aspects of the classical theory survive in this more general set-up. For finite type Nakajima quiver varieties we will show how one recovers the Weyl group action of Lusztig, Nakajima and Maffei. This is joint work with T. Nevins.
  • Robert Marsh (Leeds) - Dimer models and cluster categories of Grassmannians
    2nd September 2016, 12:00pm to 1:00pm -- Show/hide abstract
    Abstract: The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Pluecker coordinates, which are described by certain diagrams in a disc, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure. In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way. It follows that B can be realised as an idempotent subalgebra of A.
  • Alice Rizzardo (Edinburgh) - From Fourier transforms to Fourier-Mukai functors
    10th November 2016, 1:00pm to 1:50pm Lab 2, Postgraduate Statistics Centre, Lancaster University -- Show/hide abstract
    Abstract: Starting from familiar concepts coming from analysis and classical algebraic geometry, I will introduce Fourier-Mukai functors in the context of derived categories. I will explain what makes them useful, and talk about which functors between derived categories can be expressed in Fourier-Mukai form. This is joint work with Michel Van den Bergh.
  • Kevin de Laet (Antwerp) - Representation theory of Sklyanin algebras at points of finite order
    10th November 2016, 2:00pm to 2:50pm Lab 2, Postgraduate Statistics Centre, Lancaster University -- Show/hide abstract
    Abstract: Sklyanin algebras form a 2-dimensional family of noncommutative algebras and they form deformations of the commutative polynomial ring. They depend on an elliptic curve E and a point t of E. In the case that t is a torsion point, Tate and Van den Bergh showed that such a Sklyanin algebra is a finite module over its center. However, except in the 3-dimensional and in the 4-dimensional case, the PI-degree and a description of the center is not known. After recalling the connection between fat point modules and irreducible representations for Sklyanin algebras, I will show that Sklyanin algebras at points of order 2 of odd global dimension n are Clifford algebras over a polynomial ring in n variables, which gives the PI-degree in this special case. For n=5 in this special case, the ramification locus is calculated, as well as the correspondence between fat points and points projective 4-space.
  • Tea and coffee
    10th November 2016, 3:00pm to 3:30pm B Floor Social Area, Postgraduate Statistics Centre, Lancaster University
  • Kenny de Commer (Brussels) - Torsion-freeness for discrete quantum groups
    10th November 2016, 3:30pm to 4:20pm A54 Lecture Theatre, Postgraduate Statistics Centre, Lancaster University -- Show/hide abstract
    Abstract: In order to formulate the Baum-Connes conjecture in the setting of discrete quantum groups, R. Meyer introduced a notion of torsion-freeness for them. In this talk, we define torsion-freeness for rigid tensor C*-categories and fusion rings. As an application, we then show that the discrete duals of the free unitary quantum groups of Wang and Van Daele are torsion-free, answering a question of C. Voigt. This is joint work with Y. Arano.
  • Dinner
    10th November 2016, 7:00pm to 9:00pm Paulo Gianni's, 15 New Street, Lancaster LA1 1EG, UK
  • Iva Halacheva (Lancaster) - The cactus group, crystals, and shift of argument algebras
    11th November 2016, 9:30am to 10:20am Lab 2, Postgraduate Statistics Centre, Lancaster University -- Show/hide abstract
    Abstract: Two objects arising from a finite-dimensional reductive Lie algebra g and its representation theory are the cactus group defined using the Dynkin diagram of g, and crystals encoding the information of g-representations. We define an action of the cactus group on any crystal, which can be realized both combinatorially and geometrically. On one hand, it can be described in terms of Schützenberger involutions. On the other hand, it is the monodromy action for a covering of a certain moduli space, coming from a family of maximal commutative subalgebras of U(g) known as the shift of argument algebras.
  • Tea and coffee
    11th November 2016, 10:30am to 11:00am B Floor Social Area, Postgraduate Statistics Centre, Lancaster University
  • David Jordan (Edinburgh) - The quantum Springer sheaf
    11th November 2016, 11:00am to 11:50am A54 Lecture Theatre, Postgraduate Statistics Centre, Lancaster University -- Show/hide abstract
    Abstract: In classical Springer theory, one is constructing representations of the Weyl group W of a reductive group, exploiting the geometry of the flag variety. Hotta and Kashiwara gave a striking reformulation of classical Springer theory, in terms of equivariant D-modules on the Lie algebra g (i.e. systems of differential equations on g, with strong symmetry properties). In this talk, I'll review Hotta and Kashiwara's construction, and explain some joint work with Monica Vazirani to define and compute analogs of Hotta-Kashiwara's D-modules in the setting of quantum groups.
(Open in Google Calendar)

Registration and accommodation:


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There is funding available for Ph.D. students and early career researchers.


Page last modified on Thursday 22 of October, 2015 14:35:08 UTC