Oberseminar Darstellungstheorie, Winter Term 2015/2016

Catharina Stroppel email
Daniel Tubbenhauer email

Where and when?



Alexander Kleshchev,
Title: Stratifying Khovanov-Lauda-Rouquier algebras,
Abstract: We discuss standard module theory for KLR algebras of finite and affine types, its connections with PBW bases in quantum groups, and affine highest weight categories.

Daniel Bergh,
Title: Applications of destackification,
Abstract: A stacky blow-up is a particularly well-behaved birational modification of algebraic stacks. We give an algorithm which can be used to modify certain smooth stacks such that they become smooth schemes by using sequences of stacky blow-ups. Since this removes the “stackiness” from the stack in a controlled way, we call the process destackification. I will briefly describe what destackification is and mention some possible ways to use it, including:

Vanessa Miemietz,
Title: Quasi-hereditary algebras and bocses,
Abstract: I will report on some recent work with Julian Kuelshammer on bocses related to quasi-hereditary algebras.

Thorsten Heidersdorf,
Title: Pro-reductive groups attached to representations of the general linear supergroup $Gl(m|n)$,
Abstract: The tensor category $Rep(Gl(m|n))$ is not semisimple and the decomposition of tensor products into the indecomposable constituents is only known in very special cases. Every nice tensor category C has a unique proper tensor ideal $\mathcal{N}$, the negligible morphisms, such that the quotient $C/\mathcal{N}$ is a semisimple tensor category. I will apply this construction to the tensor category $C = Rep(Gl(m|n))$. The quotient is the representation category of a pro-reductive supergroup scheme. I will show some results about this supergroup scheme and what this implies about tensor product decompositions. This is joint work with Rainer Weissauer.

Vassily Gorbounov,
Title: Quantum integrable systems in cohomology theories,
Abstract: In these lectures we describe a geometric construction of two Yang Baxter algebras, quantum group type objects, each is a degeneration of the so-called six vertex model from statistical physics. These algebras are defined by the Chern classes of the natural vector bundles over partial flag varieties via the convolution construction common in the geometric representation theory. The major ingredient of the construction of a Yang Baxter algebra is the $R$ matrix, which is a part of a solution of the Yang Baxter equation. It turns out that in our case the $R$ matrix encodes the natural relations between the above Chern classes. As an output we obtain a new description of the Schubert calculus on the cohomology of Grassmannians.

Oded Yacobi,
Title: Truncated shifted Yangians and Nakajima monomial crystals,
Abstract: In geometric representation theory slices to Schubert varieties in the affine Grassmannian are affine varieties which arise naturally via the Satake correspondence. This talk centers on algebras called truncated shifted Yangians, which are quantizations of these slices. In particular, we will describe the highest weight theory of these algebras using Nakajima's monomial crystal. This leads to conjectures about categorical $g^L$-action (Langlands dual Lie algebra) on representation categories of truncated shifted Yangians.

Jonathan Comes,
Title: Ideals in Deligne's $\underline{\mathrm{Re}}\mathrm{p}(O_\delta)$ and representations of orthosymplectic supergroups,
Abstract: Deligne defined a category $\underline{\mathrm{Re}}\mathrm{p}(O_\delta)$ that permits the simultaneous study of the tensor powers of the natural representations of the orthogonal/symplectic groups and the orthosymplectic supergroups. In this talk I will first give a definition of Deligne's category in terms of Brauer diagrams. Next, I will describe a classification of the indecomposable objects in $\underline{\mathrm{Re}}\mathrm{p}(O_\delta)$ via Young diagrams and their corresponding weight diagrams. I will then explain a recent classification of thick ideals in $\underline{\mathrm{Re}}\mathrm{p}(O_\delta)$, and the consequences of this classification in terms of representations of the orthosymplectic supergroups.

Ruslan Maksimau,
Title: Affine category $\mathcal{O}$ and categorical actions,
Abstract: The parabolic category $\mathcal{O}$ for affine $\mathfrak{gl}_N$ at level -N-e admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors E and F. Our goal is to prove that the functors E and F are Koszul dual to the Zuckerman functors. To do this, it is enough to show that the functor F for the category $\mathcal{O}$ at level -N-e can be decomposed in terms of the components of the functor F for the category $\mathcal{O}$ at level -N-e-1. To get such a decomposition, we prove a general fact about categorical representations: a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. The proof of this claim can be reduced to a statement about KLR algebras: there is an isomorphism between the KLR algebra associated with the quiver $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra associated with the quiver $A_{e}^{(1)}$.

Ivo Dell'Ambrogio,
Title: Restriction to subgroups as étale extensions,
Abstract: It is a rather surprising fact that, in several domains of equivariant mathematics, the restriction functor to a finite-index subgroup can be treated as an extension of scalars along a finite étale map. We explain how, in order to make this geometric intuition precise and to unify all such phenomena, it suffices to step from algebraic geometry to the more general setting of tensor triangular geometry. This is joint work with Paul Balmer and Beren Sanders.

Olaf Schnürer,
Title: Six operations on dg enhancements of derived categories of sheaves and applications,
Abstract: We lift Grothendieck's six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Then we explain applications concerning homological smoothness of derived categories of schemes.

Martina Lanini,
Title: Moment graphs combinatorics for semi-infinite flags,
Abstract: Moment graphs techniques have been successfully applied to the study of the geometry of flag varieties and their Schubert varieties. As such, they provide a way of translating into combinatorics (of Bruhat graphs) some problems arising, for example, in the representation theory of finite-dimensional semisimple Lie algebras, which are known to be controlled by these varieties. In the case of quantum groups at a root of unity and Lie algebras in positive characteristic, flag varieties have to be replaced by an affine variant: the semi-infinite flags. In this talk we discuss why a certain graph we discovered is expected to be the right substitute for Bruhat graphs in this setting.

Vera Serganova,
Title: New tensor categories related to orthogonal and symplectic groups and the strange supergroup $P(n)$,
Abstract: We start with discussing the category of tensor representations of $O(\infty)$. We show that this category is Koszul and that its Koszul dual is the category of tensor representations of the strange supergroup $P(\infty)$. One can use this relation to calculate Ext groups between simple objects in both categories. The above categories are monoidal symmetric categories but they are missing the duality functor. It is possible to extend these categories to certain rigid tensor categories satisfying a nice universality property. In the case of $O(\infty)$ such extension depends on a parameter $t$ and is closely related to the Deligne's category Rep $O(t)$. When t is integer, this new category is a highest weight category. We finish by formulating several open questions concerning these new categories.