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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
Combinatorial and Geometric Structures in Representation Theory
Monday 6th July - Thursday 16th July 2009
List of abstracts
Gwyn Bellamy (University of Edinburgh)Poster
Symplectic resolutions via representation theory of rational Cherednik algebras.
I will show how the representation theory of rational Cherednik algebras can be use to construct Poisson deformations of certain symplectic varieties. This allows us to prove the existence (or nonexistence) of symplectic resolutions of symplectic group quotients.

Dave Benson (University of Aberdeen)Tuesday 7th July 17:45
Localising subcategories of the stable module category.
I shall present joint work with Srikanth Iyengar and Henning Krause, in which we classify those localising subcategories of the stable module category of a finite group that are closed under tensor products with the simple modules. I shall also at least mention the corresponding more recent classification of the colocalising subcategories.

Ivan Cherednik (UNC Chapel Hill and RIMS)Friday 10th July 09:15
Nonsymmetric q-Whittaker functions
The "global" symmetric q-Whittaker function, introduced about a year ago, is essentially the generating function for the symmetric q-Hermite polynomials, directly related to the Demazure level-one Kac-Moody characters for dominant weights. It is known/expected to be connected with the Givental-Lee construction in the quantum K-theory of affine flag varieties, the IC-theory of these varieties and, presumably, with the quantum Langlands program. It is an ideal object for the categorization: its coefficients are positive q-integers. The nonsymmetric q-Whittaker function is a generating function for all nonsymmetric q-Hermite polynomials, equivalently, all (level-one) Demazure characters. Its definition is based on a recent theory of nil-DAHA and their spinor representations, which includes the construction of q-Toda Dunkl operators.

Maria Chlouveraki (EPFL)Saturday 11th July 12:15
Families of characters for the Ariki-Koike algebras
The definition of Rouquier for the families of characters, defined originally by Lusztig for Weyl groups, has made possible the generalization of this notion to the case of complex reflection groups. In particular, Rouquier has shown that the families of characters of a Weyl group W coincide with the blocks of the Iwahori-Hecke algebra of W over a suitable coeffient ring, the "Rouquier ring". This definition generalizes without problem to all cyclotomic Hecke algebras associated to any complex reflection group. In this talk, we will give an algorithm for the determination of the families of characters for the Ariki-Koike algebras which demonstrates the important role played by the families of type B. We will use the fact that the families of characters have the property of "semi-continuity".

Anton Cox (City University London)Wednesday 8th July 12:15
Blocks and translation functors for the Brauer algebra
(Joint work with Paul Martin and Maud De Visscher)

The representation theory of the Brauer algebra has many features reminiscent of Lie theory. In this talk I will review some of these features, including the block structure in zero and positive characteristics, and equivalences arising from translation functors.

Peter Fiebig (Freiburg University)Tuesday 14th July 16:30
Lusztig's conjecture as a moment graph problem
We give a direct connection between sheaves on complex affine Schubert varieties (with coefficients in a field k) and the representation theory of the Lie algebra (over k) for the dual root system. The essential tool for this is the theory of sheaves on affine moment graphs. The main application of the above result is a new proof of Lusztig's formula for the simple rational characters of a reductive algebraic group in almost all characteristics. Moreover, the theory of sheaves on moment graphs allows us to calculate an upper bound on the exceptional primes, as well as to give an elementary proof of the multiplicity one case of the conjecture for all relevant characteristics.

Meinolf Geck (University of Aberdeen)Saturday 11th July 11:00
Kazhdan-Lusztig cells and modular principal series representations
Let G be a finite group of Lie type over the field with q elements, and B be a Borel subgroup of G. Let k be an algebraically closed field of characteristic l where l is zero or a prime not dividing q. We are concerned with the problem of classifying Irr_k(G,B), the set of irreducible representations of G which admit non-zero vectors fixed by B. By classical results due to Iwahori, Tits, ..., if l is zero, then Irr_k(G,B) is in bijection with the irreducible representations of W, the Weyl group of G. The solution for positive l is considerably more complicated; it requires some deep results on Kazhdan-Lusztig cells in W. We explain the ideas involved in this solution, and how they might extend to all irreducible representations of G over k.

Christof Geiss (UNAM)Tuesday 14th July 17:45
Categorification of the chamber ansatz
(Joint work with B. Leclerc and J. Schröer)
For an adaptable element w of the Weyl group W the cluster algebra structure on the coordinate ring bbc[cnsupw] of the unipotent cell cnsupw is categorified by a subcategory calcsubw of the modules over the corresponding preprojective algebra. Under the cluster character coming from Lusztig's construction of the semicanonical basis, the initial seed consisting of certain generalized minors corresponds to a canonical cluster tilting object cv in calcsubw.

In order to solve in this context the factorization problem, Berenstein, Fomin and Zelevinsky introduced twisted minors. We show that these twisted minors correspond essentially to the inverse of Auslander-Reiten translate of the summands of cv. More generally, one could say that the Auslander-Reiten translate in calcsubw categorifies the twist automorphism etasubwsupast of bbc[cnsupw]. This allows us to compare the above mentioned cluster character with the Caldero-Fu-Keller cluster character.

Victor Ginzburg (University of Chicago)Saturday 11th July 09:15
Quantized Bezrukavnikov's equivalence
We construct an equivalence between an equivariant derived category of Iwahori-constructible sheaves on the affine flag variety and a certain derived category of _asymptotic' Harish-Chandra D-modules on the square of the flag variety for the Langlands dual group. Our equivalence may be seen as a _quantum/equivariant' deformation of the equivalence established by R. Bezrukavnikov some time ago. At the same time, our equivalence is an extension of the equivalence between an equivariant derived Satake category and a derived category of asymptotic Harish-Chandra bimidules constructed more recently by Bezrukavnikov and Finkelberg.

Mark Haiman (University of California, Berkeley)Tuesday 7th July 11:00
LLT polynomials, k-atoms, and related topics
My tentative plan is to address the following topics, some of them only briefly 1. Combinatorial definition of LLT polynomials 2. (joint work with Ian Grojnowski) Interpretation of LLT polynomials in Kazhdan-Lusztig theory, positivity theorem, generalization to all Lie types 3. The combinatorial approach of Sami Assaf to LLT positivity in type A 4. The "k-atoms" of Lascoux, Lapointe and Morse 5. Positivity conjectures relating k-atoms, LLT, and Macdonald polynomials 6. Li-Chung Chen's modules and the k-atom positivity conjectures 7. Conjectured formulas for k-atoms and characters of Chen's modules and vector bundles on the flag variety.

Mark Haiman (University of California, Berkeley)Wednesday 8th July 16:30
LLT polynomials, k-atoms, and related topics

Mark Haiman (University of California, Berkeley)Thursday 9th July 09:15
LLT polynomials, k-atoms, and related topics

David Hernandez (CNRS - ENS Paris)Tuesday 14th July 09:15
Langlands duality for representations of quantum affine algebras.
(Joint work with E Frenkel)
We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual. We prove this duality for the Kirillov-Reshetikhin modules. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.

Nicolas Jacon (Université de Franche-Comté)Monday 13th July 12:15
Representations of Hecke algebras at roots of unity and constructible characters in type B
The theory of basic sets provides a natural and efficient way to classify the simple modules of Hecke algebras in the non semisimple case. In type B, this theory induces in fact several different parametrisations for these simples modules. The aim of this talk is to show how the theory of constructible characters allows the explicit determination of these parametrisations.

Daniel Juteau (CNRS)Wednesday 15th July 12:15
Modular Springer correspondence and basic sets.
The Springer correspondence makes a link between representations of Weyl groups and perverse sheaves on the nilpotent cone. Such a correspondence also exists for modular representations, and the decomposition matrix of a Weyl group can be seen as a submatrix of a decomposition matrix for perverse sheaves. This allows to define some basic sets for Weyl groups via the Springer correspondence. In a work in progress with Karine Sorlin, we use this to determine the modular Springer correspondence explicitly in the different types, starting with the classical types.

Masaki Kashiwara (Kyoto University)Wednesday 15th July 09:15
Affine Hecke algebras of type B and Symmetric crystals
(Joint work with Enomoto and Miemietz)
Lascoux-Leclerc-Thibon-Ariki established the link between the representations of affine Hecke algebras of type A and the crystal basis of affine quantum group of type A_n^{(1)}. I explain a similar conjecture for the affine Hecke algebras of type B and D . However, we have to replace the crystal basis of affine quantum group with a new object: symmetric crystals.

Bernhard Keller (Université de Paris VII)Monday 13th July 17:45
The Hall algebra of a spherical object
(Joint work with Dong Yang and Guodong Zhou)
We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.

Alastair King (University of Bath)Friday 10th July 12:00
Quivers and Calabi-Yaus
I will discuss some aspects of the construction of Calabi-Yau categories from quivers, focusing on similarities between preprojective algebras and superpotential algebras and their role in conformal field theory.

Kobi Kremnizer (University of Chicago)Friday 10th July 10:45
Localization for finite W-algebras
(Joint work with Chris Dodd)
I'll describe how to use deformation quantization in order to get localization results for finite W-algebras and other algebras.

Aaron Lauda (Columbia University)Saturday 11th July 16:30
Categorifying quantum sl(2)

Crane and Frenkel conjectured that that the quantum enveloping algebra of s(2) could be categorified at generic q using its canonical basis. In my talk I will describe a realization of this conjecture using a diagrammatic calculus.

If time permits I will also explain joint work with Mikhail Khovanov on how this construction can be generalized to quantum sl(n), and conjecturally to any root datum.

Cedric Lecouvey (Université du Littoral Calais)Monday 13th July 15:30
Quantized branching coefficients and affine crystals
(Joint work with M. Okado and M. Shimozono)
This talk is devoted to a conjectured correspondence between one-dimensional sums defined from tensor products of Kirilov-Reshetikhin crystals and generalizations of q-Kostka polynomials. We will first review the cases when this correspondence is established and next present recent advances in the proof of the complete conjecture.

Tony Licata (Stanford University/Max Planck Institut Bonn)Thursday 9th July 12:15
Categorical geometric skew-Howe duality
(Joint work with Sabin Cautis and Joel Kamnitzer.)
We explain how categorical sl(2) actions can be used to categorify the R-matrix isomorphism between tensor products of fundamental representations of U_q(sl_n).

Ivan Losev (MIT)Monday 13th July 11:00
Classification of finite dimensional irreducible representations of W-algebras
Finite W-algebras are certain associative algebras arising in Lie theory. They were studied extensively during the last decade starting from Premet's paper, 2002. I will explain some partial results of mine on the classification of their finite dimensional irreducible modules as well as some conjectures.

Sinead Lyle (University of East Anglia)Tuesday 7th July 15:30
Constructing homomorphisms using Jucy-Murphy elements
(Joint work with Andrew Mathas)
We discuss a method of constructing homomorphims between certain pairs of Specht modules for the Hecke algebras of type A which uses the Jucy-Murphy elements.

Robert Marsh (University of Leeds)Wednesday 8th July 17:45
Cluster structures from 2-Calabi-Yau categories with loops
(Joint work with Aslak Bakke Buan and Dagfinn Vatne (NTNU Trondheim).)
We consider the collection of maximal rigid objects in a 2-Calabi-Yau triangulated category. We show that if this collection satisfies a certain "no 2-cycle" condition then its exchange properties are governed by Fomin-Zelevinsky cluster mutation. In particular, this includes cases where the endomorphism algebra of a maximal rigid object has loops (1-cycles) in its quiver. Thus our result generalises that of Buan-Iyama-Reiten-Scott (where it is assumed that no such loops appear).

Kevin McGerty (Imperial College London)Tuesday 7th July 12:15
Microlocal KZ functors and rational Cherednik algebras.
We describe how the Kashiwara-Rouquier quantization of the Hilbert scheme of n points in the plane naturally yields of a family of exact functors on category O for the rational Cherednik algebra in type A generalizing the KZ functor of Ginzburg-Guay-Opdam-Rouquier.

Markus Reineke (BU Wuppertal)Monday 13th July 16:30
Quiver moduli and Donaldson-Thomas type invariants
M. Kontsevich and Y. Soibelman describe the wall-crossing behaviour for Donaldson-Thomas type invariants of Calabi-Yau categories in terms of factorizations of certain Poisson automorphisms. Using Hall algebras of quivers, such factorizations are related to the cohomology of framed versions of moduli spaces of representations of quivers. Functional equations for this cohomology allow to derive integrality of the Donaldson-Thomas type invariants.

Idun Reiten (NTNU, Trondheim)Wednesday 15th July 11:00
Quasihereditary algebras associated with reduced words
(Joint work with Osamu Iyama)
In work with Buan, Iyama and Scott we have investigated a class of finite dimesnsional algebras associated with reduced words in Coxeter groups. Generalising work of Geiss-Leclerc-Schroer we show that these algebras are quasihereditary and investigate some of their properties.

Konstanze Rietsch (King's College London)Thursday 9th July 17:45
On mirror symmetry for flag varieties

Raphael Rouquier (University of Oxford)Tuesday 7th July 09:15
Higher Representations of Kac-Moody algebras
I will explain the construction of monoidal categories associated with Kac-Moody algebras, following joint work with Chuang in type A. I will explain a unicity theorem for simple 2-representations and the geometrical realization of those simple 2-representations.

Raphael Rouquier (University of Oxford)Wednesday 8th July 11:00
Higher Representations of Kac-Moody algebras

Raphael Rouquier (University of Oxford)Thursday 9th July 16:30
Higher Representations of Kac-Moody algebras

Catharina Stroppel (Mathematisches Institut Bonn)Wednesday 15th July 16:30
The combinatorics of the fusion algebra and Gromov-Witten invariants
I will give a combinatorial description of the fusion algebra of the affine sl(n) for fixed level k and realise it as a quotient of the quantum cohomology of Grassmannians. In particular, we relate fusion constants with Gromov-Witten invariants. The focus of this talk will be to work out the similarities, but also the differences, between the structure of the two rings.

Will Turner (University of Aberdeen)Tuesday 14th July 12:15
Homological and homotopical algebra of GL2
I will describe work with Vanessa Miemietz on homological and homotopical aspects of the rational representation theory of GL2.

Michela Varagnolo (Université Cergy-Pontoise)Monday 13th July 09:15
KLR algebras and canonical bases
(Joint work with E. Vasserot)
I will recall a conjecture by Khovanov and Lauda concerning the categorification of one half the quantum group associated to a simply laced Cartan datum. I will also explain how to prove it.

Eric Vasserot (Université de Paris VII)Tuesday 14th July 11:00
Hall algebras and Hilbert schemes
We'll explain some relation between Hall algebras of smooth projective curves, Hilbert schemes, and geometric Langlands program.

Chelsea Walton (University of Michigan/ University of Manchester)Poster
Point parameter rings
Sklyanin algebras play an important role in the study of physical phenomenon. The focus of this poster is to first discuss techniques of Artin-Tate-van den Bergh (ATV) that describe the ring-theoretic and homological behavior of these structures. In particular, we highlight the significance of __twisted homogeneous coordinate rings”. Secondly, we introduce a generalized twisted homogeneous coordinate ring P associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of ATV, namely that P is a factor of the corresponding degenerate Sklyanin algebra.

Weiqiang Wang (University of Virginia)Saturday 11th July 17:45
A new approach to the representation theory of Lie superalgebras
(Joint work with Shun-Jen Cheng and Ngau Lam (Taiwan))
We formulate and establish an equivalence between parabolic categories O of modules of classical Lie superalgebras to their suitable counterparts for classical Lie algebras. As an intermediate step, we establish isomorphic Kazhdan-Lusztig theories between various Lie superalgebras and Lie algebras. This in particular provides a complete solution to the problem of finding all finite-dimensional irreducible characters of the orth-symplectic Lie superalgebras.

Michael Wemyss (Nagoya University)Wednesday 15th July 15:30
Reconstruction algebras and rational surfaces
I will explain how much of the theory associated to ADE surface singularities can, with some modification, be extended to cover all rational surface singularities. The main tool is the reconstruction algebra (a generalization of the preprojective algebra) which is built combinatorially from the dual graph of the minimal resolution.

Geordie Williamson (Oxford University)Thursday 9th July 15:30
Perverse sheaves and modular representation theory
(Joint work with Carl Mautner and Daniel Juteau)

Modular perverse sheaves are perverse sheaves on complex algebraic varieties with coefficients in a field of positive characteristic. I will briefly discuss some relations (due to Soergel, Mirkovic-Vilonen, Fiebig and Juteau) between categories of modular perverse sheaves and representation theory. I will then describe recent work in which we introduce a new class of sheaves on certain stratified varieties which we call parity sheaves. On the affine Grassmannian they correspond under geometric Satake to tilting modules (except possibly in some small characteristics). I will describe some of their properties, in particular how they allow one to understand the failure of the decomposition theorem.

Andrei Zelevinsky (Northeastern University)Tuesday 7th July 16:30
Cluster algebras via quivers with potentials
This series of three lectures is based on two recent joint papers with Harm Derksen and Jerzy Weyman "Quivers with potentials and their representations I, II." We introduce mutations of quivers with potentials and their representations, and show that they provide a categorification of quiver mutations in the theory of cluster algebras. This representation-theoretic interpretation allows us to prove most of the structural conjectures on cluster algebras made in the paper "Cluster algebras IV" (joint with Sergey Fomin).

Andrei Zelevinsky (Northeastern University)Wednesday 8th July 09:15
Cluster algebras via quivers with potentials

Andrei Zelevinsky (Northeastern University)Thursday 9th July 11:00
Cluster algebras via quivers with potentials

Andrei Zelevinsky (Northeastern University)Tuesday 14th July 15:00
Cluster algebra via quivers with potential IV