Tag - Representation theory

Catharina Stroppel: Schur-Weyl dualities in non-semisimple cases

17th August, 2016

'Schur-Weyl duality' is often used to describe a concept in representation theory involving two kinds of symmetry that determine each other. In its original form it goes back to Schur and Weyl (around 1930) and describes an important interplay between the representation theory of the general linear and the symmetric group over the complex numbers. In this talk we will describe some generalizations of this phenomenon with a focus on modern, still open or recently solved questions. In particular we are interested in situations, where the involved algebras are not semisimple. We will indicate the origin of filtrations, homological properties and hidden gradings on the involved algebras and applications to the representation theory of Lie superalgebras.

Steffen Oppermann: d-tilting bundles for Geigle-Lenzing weighted projective spaces

16th August, 2016

This talk is based on joint work with Martin Herschend, Osamu Iyama, and Hiroyuki Minamoto. Classically, the classes of tame (representation infinite, connected) hereditary algebras and Fano Geigle-Lenzing weighted projective lines coincide up to derived equivalence. With the development of Iyama's higher AR-theory, and our work on Geigle-Lenzing projective spaces, it has become natural to ask if there is a higher dimensional analogue of this fact. Here dimension refers to, on the one side the global dimension of the algebra, and on the other side the dimension of the space. Unfortunately, so far a general answer (or general strategy) is elusive. In my talk I will focus on the hypersurface case, and more specifically certain weight sequences within the hypersurface case. For these, I will explain how one may find suitable tilting bundles on the Geigle-Lenzing weighted projective space.

Ryan Kinser: K-polynomials of type A quiver orbit closures and lacing diagrams

16th August, 2016

Orbit closures of type A quiver representations are algebraic varieties that arise naturally in several areas of mathematics: for example, in Lusztig's geometric realization of Ringel's work on quantum groups; as generalizations of determinantal varieties in commutative algebra; and in the theory of degeneracy loci of maps of vector bundles.

For equioriented type A quivers, a formula due to Knutson-Miller-Shimozono expresses the equivariant cohomology class of each orbit closure as a sum, over certain 'lacing diagrams', of products of Schubert polynomials. Lacing diagrams were introduced by Abeasis and del Fra in 1982 to visualize direct sum decompositions of type A quiver representations.

In joint work with Allen Knutson and Jenna Rajchgot, we proved a 2004 conjecture of Buch and Rimnyi that generalizes this formula in two ways: to arbitrarily oriented type A quivers, and to equivariant K-classes (a.k.a. K-polynomials), from which equivariant cohomology can be recovered.

The aim of this talk is to explain the combinatorics of (K-theoretic) lacing diagrams and carefully state the formula. Time permitting, I will give some idea of the Gröbner degeneration technique used in the proof.

Srikanth Iyengar: Local Serre duality for modular representations of finite group schemes

16th August, 2016

This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. My plan is to explain the statement and proof of a recent result (obtained in collaboration with Dave Benson, Henning Krause, and Julia Pevtsova) to the effect that the stable module category of finite-dimensional representations of G has local Serre duality.

Eleonore Faber: Non-commutative resolutions of discriminants

16th August, 2016

Let G be a finite subgroup of GLn(K) for a field K whose characteristic does not divide the order of G. The group G acts linearly on the polynomial ring S in n variables over K. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of D as a quotient of the skew group ring A = S ∗ G by the idempotent e corresponding to the trivial representation. We will explain how this can be seen in some sense as a McKay correspondence for reflection groups.

Bernard Leclerc: Cluster algebras and quantum loop algebras

16th August, 2016

In 2012, Hernandez and Jimbo introduced a new tensor category of representations of a Borel subalgebra of a quantum loop algebra, and classified its simple objects. This category contains the finite-dimensional representations of the quantum loop algebra, together with some new infinite dimensional representations. The motivation of Hernandez and Jimbo came from mathematical physics, in particular from papers of Bazhanov et al. where some examples of these new representations were used to define analogues of Baxter’s Q-operators in conformal field theory. Recently, using this new category, Frenkel and Hernandez were able to prove a long-standing conjecture of Frenkel and Reshetikhin on the spectra of the transfer matrices of some quantum integrable systems associated with quantum loop algebras. In this talk, I will explain that the new category of Hernandez and Jimbo fits very well with cluster algebras. More precisely I will show that cluster structures occur naturally in its Grothendieck ring, and can be helpful in finding new interesting functional relations. This is a joint work with David Hernandez.

Antoine Touzé: Stabilization and cup products for polynomial representations of GLn(k)

15th August, 2016

It is known for a long time that polynomial representations of GLn(k) stabilize when n grows, i.e. Schur algebras S(n, d) are all Morita equivalent when n ≥ d. A model of the category of stable polynomial representations is given by the strict polynomial functors of Friedlander and Suslin. Using the formalism of strict polynomial functors, we prove a rather counter-intuitive results on cup products, namely that the cup product

Ext∗(M, N) ⊗ Ext∗(P(r), Q(r)) → Ext∗(M ⊗ P(r), N ⊗ Q(r))

induces an isomorphism in low degrees when M, N, P, Q are stable polynomial representations. We shall explain some consequences of these results (including a new proof of the Steinberg tensor product theorem, as well as more general structure theorems which generalize it) and connections with the cohomology of the symmetric group.

Rosanna Laking: Indecomposable objects in the homotopy category of a derived-discrete algebra

15th August, 2016

In this talk I will present joint work with K. Arnesen, D. Pauksztello and M. Prest. We classify the indecomposable pure-injective complexes in the homotopy category of projective modules K(ProjΛ) over a derived-discrete algebra Λ. The set of indecomposable pure-injective complexes are the points of a topological space known as the Ziegler spectrum. We give a complete description of the Ziegler topology and, making use of the interactions between this space and categories of functors, we prove that every indecomposable object in K(ProjΛ) is pure-injective.

Osamu Iyama: Finiteness of global dimension of endomorphism algebras

15th August, 2016

In representation theory, it is basic to study modules whose endomorphism algebras have finite global dimension. They appear naturally in many situations, e.g. Auslander correspondence and representation dimension, Dlab-Ringel's approach to quasi-hereditary algebras of Cline-Parshall-Scott, Rouquier’s dimensions of triangulated categories, and cluster tilting in higher-dimensional Auslander-Reiten theory. Recently such modules are called non-commutative resolutions, and studied in commutative ring theory and algebraic geometry after Van den Bergh's work in birational geometry. In this talk, I will show some of typical examples of non-commutative resolutions, including rings with Krull-dimension at most one, certain hypersurface singularities and Stanley-Reisner rings.

Claire Amiot: Cluster categorification and applications to tilting theory

11th August, 2016

This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite-dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.