Oberseminar Darstellungstheorie, Sommer Term 2016

Catharina Stroppel email
Daniel Tubbenhauer email

Where and when?



Paul Wedrich,
Title: Uniqueness and branching of knot homologies,
Abstract: Knot homology theories are categorified versions of classical (and quantum) knot polynomials that are (conjecturally/almost) functorial under knot cobordisms. The goal of this talk is to present two examples of the role of higher representation theory in the study of categorifications of the Reshetikhin-Turaev invariants of knots colored with $\mathfrak{sl}(N)$ representations. As a first example, I will explain how categorical versions of skew Howe duality have been used to prove that several, superficially very different constructions of $\mathfrak{sl}(N)$ knot homologies (via matrix factorizations, category $\mathcal{O}$, coherent sheaves...) produce isomorphic invariants. The second example concerns categorified branching rules, which provide relationships between these knot homologies in the form of spectral sequences, which are also interesting from a topological perspective. If time permits, I will talk about some open problems related to colored HOMFLY-PT homologies, whose underlying higher representation theory is less well understood.

David Jordan,
Title: Difference operators and double affine Hecke algebras and from topological field theory ,
Abstract: Cherednik introduced his double affine Hecke algebras (DAHAs) in the 90's, to solve some conjectures of Macdonald in the realm of algebraic combinatorics. Since then, they have popped up in remarkably diverse branches of representation theory and mathematical physics. Recently, works of many authors have been devoted to understanding their role in various ``field theoretic" constructions, i.e. to constructing invariants in low-dimensional topology.

In this talk, I'll survey some of these recent works, and I'll report on some joint work with David Ben-Zvi and Adrien Brochier, which constructs the representation theory of the DAHA in Type A in an intrinsically topological fashion: namely, as the factorization homology of a marked torus. This leads us more generally to the construction of a $4$-dimensional ``big sister" to the $3$-dimensional Reshetikhin-Turaev topological field theory, in which the DAHA is expected to play a central role.

Michael Brown,
Title: Topological K-theory of dg categories of graded matrix factorizations,
Abstract: This is a report on joint work with Tobias Dyckerhoff. We consider localizing invariants of a certain dg category, the singularity category, associated to a graded Gorenstein algebra A over a field k. If A is the coordinate ring of a weighted projective hypersurface, the singularity category of A is quasi-equivalent to the dg category of graded matrix factorizations associated to the hypersurface. When k is the complex numbers, one such localizing invariant is topological K-theory, as constructed by A. Blanc. The main goal of the talk will be to discuss a calculation of the topological K-theory of the singularity categories of a large class of weighted projective hypersurfaces in terms of a classical topological invariant of a hypersurface: the Milnor fiber and its monodromy. I will also discuss some applications of this result, and, if time permits, some future directions.

Travis Schedler,
Title: Special polynomials from symplectic singularities,
Abstract: I will explain how to obtain Kostka and Tutte polynomials from the nilpotent cone and hypertoric varieties. The main tool (which I will recall) is a $D$-module on the variety expressing invariance under Hamiltonian flow. This includes work with Bellamy, Etingof, and Proudfoot.

Gus Lehrer,
Title: Invariants of the orthosymplectic super group,
Abstract: A combination of geometric and diagram theoretic methods have led to recent refinements of the classical versions of the second fundamental theorem of invariant theory, and a new theorem in the orthosymplectic case. I shall describe these results, as well as the circle of ideas leading to them. This is joint work with Ruibin Zhang, and partly with Pierre Deligne.

Beren Sanders,
Title: Grothendieck-Neeman duality and the Wirthmüller isomorphism,
Abstract: (Joint work with Paul Balmer and Ivo Dell'Ambrogio.) In this talk, I will discuss an intimate relationship between Grothendieck duality in algebraic geometry and the Wirthmüller isomorphism in equivariant stable homotopy theory. To this end, we will make a general study of the existence and properties of adjoints of an arbitrary coproduct-preserving tensor-triangulated functor between rigidly-compactly generated tensor triangulated categories. It turns out that the more adjoints exist, the more strongly related they must be to each other, and the result is a surprising trichotomy: There exist either exactly three adjoints, exactly five or infinitely many. Moreover, this analysis will provide us with purely formal, canonical constructions of Wirthmüller isomorphisms (when they exist) and demonstrate that Grothendieck duality is in fact a necessary condition for the existence of such an isomorphism. If time permits, I will mention some more recent developments which show that the Adams isomorphism can also be constructed in this way as a (suitably generalized) Wirthmüller isomorphism.

Marco Mackaay,
Title: Simple transitive $2$-representations of Soergel bimodules,
Abstract: I will first recall Soergel bimodules, which categorify Hecke algebras, and some bits of the theory of $2$-representations, which categorify the usual representations.

More specifically, I will explain cell $2$-representations and simple transitive $2$-representations, both due to Mazorchuk and Miemietz. By definition, every cell $2$-representation is simple transitive. Mazorchuk and Miemietz showed that the converse also holds in finite type $A$.

However, this is not true in general, as Kildetoft, Mazorchuk, Zimmermann and I showed in a joint paper (arXiv:1605.01373). In that paper, we classified all transitive $2$-representations (up to equivalence) for the so-called small quotient of the Soergel bimodules in almost all finite Coxeter types. I will sketch our results in my talk.

Andrew Mathas,
Title: Jantzen filtrations and graded Specht modules,
Abstract: I will explain how to give an easy proof of the Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type $A$ using the KLR grading. Then I will discuss some consequences and applications of this approach.