Recent topics in representation theory

The goal of this seminar is to give some introductory talks to topics in representation theory which are of current interest to the speaker. Each speaker gives a long talk (90 min) and then a short talk (30 min) the week after the long talk.


Catharina Stroppel email
Daniel Tubbenhauer email


Where and when?



Daniel Tubbenhauer,
Title: Web calculi in representation theory,
Abstract: A striking question is if one can present the category $\boldsymbol{\mathrm{Mod}}(\mathfrak{g})$ of finite-dimensional representations of some Lie algebra \(\mathfrak{g}\), via generators and relations -- or even better: via diagrammatic generators and relations.
The question itself is very hard and only partial solutions are known: it goes back to work of Schur and Brauer that the subcategory of $\boldsymbol{\mathrm{Mod}}(\mathfrak{g})$ tensor generated by the vector representation can be (almost) described if \(\mathfrak{g}\) is of classical type and certain dimensions are "big enough". Moreover, Rumer, Teller and Weyl showed (more or less) already in the 30ties that the Temperley-Lieb algebra can be seen as a diagrammatic realization of the representation category of \(\mathfrak{sl}_2\)-modules tensor generated by the vector representation of \(\mathfrak{sl}_2\) -- providing a topological (and fun!) tool to study the latter.
In this talk I explain "howe" one can prove such a (diagrammatic) realization.
Everything in this talk can be quantized (roughly: put quantum brackets everywhere) and everything is amenable to categorification, but I will keep it easy and explain the \(\mathfrak{sl}_2\)-case in details and give some applications to knot theory of the whole calculus.

Arik Wilbert,
Title: Topology of two-block Springer fibers for the even orthogonal and symplectic group,
Abstract: In this talk we construct an explicit topological model (similar to the topological Springer fibers of type $A$ appearing in work of Khovanov and Russell) for every two-block Springer fiber corresponding to the even orthogonal group (type $D$) and explain how to prove that the respective topological model is indeed homeomorphic to its corresponding Springer fiber (which confirms a conjecture by Ehrig and Stroppel). Furthermore, we discuss how the two-block Springer fibers for the symplectic group (type $C$) fit into this picture. If time permits, some applications in the context of Springer theory (representations of Weyl groups on the (co)homology of (topological) Springer fibers) are provided. Since the talk is supposed to be as self-contained as possible we do not assume any familiarity with Springer fibers.

Thomas Czempik,
Title: Schur-Weyl duality for partition algebras,
Abstract: First we will repeat the classical case of Schur-Weyl duality. Then we construct the partition algebra, where I will give some nice visualization (referring to Ceccherini-Silberstein, Scarabotti and Tolli or the bachelor thesis of Heike Herr). Then we define representations of the symmetrical group and partition algebra on the tensor product $V^{\otimes k}$ of some complex vector space v. The talk we will very basic, so everyone can follow.

Joanna Meinel,
Title: Combinatorics of particle configurations,
Abstract: We consider bosonic and fermionic particle configurations: which relations are satisfied by the operators that move the particles from one position to another along a line or a circle? They come from the action of the negative half of (affine) $\mathfrak{sl}(n)$ on symmetric and alternating products of the natural representation, giving rise to 'new' algebras such as the (affine) nilTemperley-Lieb algebra and the partic algebra, for which we construct normal forms and embeddings.

Tina Kanstrup,
Title: Braid group actions on matrix factorizations,
Abstract: Let $X$ be a smooth scheme with an action of a reductive algebraic group $G$ over an algebraically closed field $k$ of characteristic zero. We construct an action of the extended affine Braid group on the $G$-equivariant absolute derived category of matrix factorizations on the Grothendieck variety times $TX$ with potential given by the Grothendieck-Springer resolution times the moment map composed with the natural pairing.

Deniz Kus,
Title: Demazure flags and Macdonald polynomials,
Abstract: Demazure modules are finite-dimensional representations and occur in a highest weight integrable representation of an affine Lie algebra. In this talk, we introduce a family of representations which are indexed by partitions and show that these representations admit a Demazure flag, i.e. a flag where the successive quotients are Demazure modules. For special partitions, this implies a positivity result: the specialized Macdonald polynomial can be written as a linear combination of Demazure characters (for any fixed level $m$) with non-negative integer coefficients.

Jacinta Torres,
Title: Branching rules and paths,
Abstract: The study of restrictions of representations is basic in representation theory. I will introduce the path model associated to an irreducible finite-dimensional representation of a simple Lie algebra (focusing on examples) and I will explain how it can be used to produce branching rules for Levi subalgebras. Then I will talk about branching rules for non-Levi subalgebras (in particular, $\mathfrak{sp}_{2n}$ as a sub Lie algebra of $\mathfrak{sl}_{2n}$), and I will explain why in the first situation there is compatibility of the crystal structures, while in the second one there is not (I will provide examples of this). Then, if time allows, I will present a mysterious conjecture by Naito-Sagaki.

Catharina Stroppel,
Title: Quiver Schur algebras and representations of the general $p$-adic group,
Abstract: The representation theory of the general linear group $\mathrm{GL}(n,\mathbb{Q}_p)$ over the field $\mathbb{Q}_p$ of $p$-adic numbers is rather difficult and involved and part of the so-called local Langlands program. Motivated by the Schur-Weyl duality over finite fields, I will define an affine Schur algebra and explain its relation to representations of $\mathrm{GL}(n,\mathbb{Q}_p)$ and to the affine Hecke algebra. At the end I will connect it with KLR algebras and quiver Schur algebras, which can be defined either diagrammatically or geometrically and play an important role in categorification.

Olaf Schnürer,
Title: Homological smoothness and enriched resolutions,
Abstract: This talk is meant as an introduction to my talk on Friday. We discuss the notions of smoothness in algebraic geometry and in dg category theory (non-commutative geometry) in easy examples. Then we discuss the existence of additive resolution functors and, possibly, of dg enriched resolution functors.

Aleksei Ilin (Moscow),
Title: Degeneration of Bethe subalgebras in the Yangian,
Abstract: We study the family of Bethe subalgebras in Yangian for \(\mathfrak{gl}_n\), parametrized by regular diagonal matrices. We prove that all limit subalgebras of this family are free and maximal commutative and explicitly describe “simple” limits.

Michael Ehrig,
Title: Foams (Or: Web calculi now in $3D$),
Abstract: In this talk I will explain how one can use certain types of enhanced cobordisms to construct foam categories, i.e. categories of cobordisms with singular seams. This is done by gluing together different types of TQFTs, which I will explain and discuss what kind of freedom one has in this construction. In all of this \(\mathfrak{gl}_2\)-webs will play a prominent role.
After the topological construction I will explain how one can translate this into an algebraic model similar, but more involved, than Khovanov's original arc algebra. We will then discuss how this construction generalizes/unifies the work of Khovanov, Bar-Natan, Blanchet and others to categorify Khovanov homology and obtaining link/tangle invariants. I will give a rough idea how to show that via this generalized construction one shows that all of these theories in the end are more or less equivalent. If time permits we will discuss the relations to category $\mathcal{O}$, categorification of quantum groups and higher representation theory, tangle invariants, and/or possible generalizations to \(\mathfrak{gl}_n\)-webs.