• Follow-up Workshop to TP "On the Interaction of Representation Theory with Geometry and Combinatorics"
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This talk is based on joint work with Frederik Marks and Jorge Vitória.

Silting complexes, first introduced by Keller and Vossieck in 1988 as a natural generalization of Rickard's tilting complexes, were recently rediscovered through the work of several authors. They provide an appropriate setting for mutation, and they correspond bijectively to certain t-structures and co-t-structures in the derived category. Hereby, a prominent role is played by the silting complexes of length two already investigated by Hoshino, Kato and Miyachi in 2002. Their module-theoretic counterpart are the support τ-tilting modules recently introduced by Adachi, Iyama and Reiten in the category of finitely generated modules over a finite dimensional algebra.

Aim of the talk is to present a more general framework for these investigations by considering the notion of a silting module over an arbitrary ring, which provides a common generalisation of support τ-tilting modules and of tilting modules.

The role of silting theory in the study of ring epimorphisms and localisations will be discussed in the talk by Frederik Marks.

This talk is a summary of the results of my PhD project. First, I will outline the construction of model categorical enhancements of singularity categories within the framework of abelian model structures and cotorsion pairs. Afterwards, I will explain how a suitable model structure on the category of linear factorizations (enhancing the homotopy category of matrix factorizations) can be used to obtain a description of the Khovanov-Rozansky knot invariant in terms of Hochschild homology of Soergel bimodules.

(joint work with Baohua Fu)

Let G be a simple algebraic group of adjoint type, and V an irreducible projective representation.The closure of the image of G in the projectivization of End(V) is an equivariant  compactification X(V) of the group G. We answer the following questions: Is X(V) covered by lines? If not, what is the minimal degree of a family of rational curves covering X(V), and what is the structure of such a family? In particular, one may find V such that X(V) is covered by lines, except when G has type E₈. The combinatorics of the highest short coroot play an important role in these questions.

I will review the definition of Kac-Moody 2-category due to Khovanov, Lauda and Rouquier. I also hope to say something about cyclotomic quotients and/or super analogs.

We study the abelian degeneration of the affine Grassmannians of type A. We describe the corresponding ind-variety for the group GL(n) and explain the connection with the semi-infinite orbits in the affine Grassmannians for the larger groups. We also realize a version of the Schubert varieties in terms of the quiver Grassmannians for the one loop quiver. Finally, we state several open questions of algebraic and geometric origin. Based on the joint work with M. Finkelberg and M. Reineke.

Our aim is to define a PBW-type filtration for quantum groups, following the framework from recent years in the non-quantum setup. For this we recall main properties of this classical filtration which guideline us towards a quantum PBW filtration. In type A_n we obtain an appropriate degree function via dimensions of homomorphism space of the corresponding oriented quiver.

By specializing to q=1, we obtain a new filtration on the universal enveloping algebra and hence on any simple module. We can give a basis for the associated graded module and see that the annihilating ideal is monomial, in contrast to the standard PBW filtration.

We finish with exploring the relations to flag varieties, Schubert varieties, quiver Grassmanian and their degenerations to toric varieties.

This is joint work with X. Fang and M. Reineke.

We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.

Motivated by examples arising in algebraic geometry, we study objects of k-linear triangulated categories with two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical, i.e. a Calabi-Yau object. In many examples, both from representation theory and geometry these spherical subcategories admit explicit descriptions. Furthermore, the collection of all spherical subcategories ordered by inclusion yields a new invariant for triangulated categories. We derive coarser invariants like height, width and cardinality of this poset. This talk is based on joint work with A. Hochenegger & D. Ploog.

The variety of complexes Com(V), introduced by Buchsbaum and Eisenbud, is the "set" of all differentials making a given grade vector space V into a complex. It is known that Com(V) has the "spherical" property: the action of the group GL(V) on the coordinate algebra k[Com(V)] has simple spectrum (k is a field of characterstic 0).

It turns out that k[Com(V)] is the degree 0 part of the cohomology algebra of the nilpotent radical of a certain parabolic subalgebra in the Lie superalgebra gl(M|N) for some M,N. All parabolic subalgebras (standard as well as non-standard) can appear for different choices of V. We prove that the full cohomology algebra also has simple spectrum. This generalizes both the classical theorem of Kostant and the spherical property of Com(V). Joint work with S. Pimenov.

In 1937 Brauer diagrammatically defined an algebra and proved that his eponymous algebra gives the endomorphisms of the tensor powers of the natural representation for the symplectic and orthogonal groups.  We now know that Brauer's results have a common generalization in which the Brauer algebra provides the endomorphisms of the tensor powers of the natural representation for the orthosymplectic Lie supergroup. This is the case when the underlying bilinear form is even.  When it is odd we are instead studying the type P Lie supergroup. Moon described the relevant endomorphism algebras by generators and relations. We show that they also admit a natural diagrammatic description which generalizes Brauer's original construction. This is joint work with Ben Tharp.

In this talk we discuss the category of graded representations of (twisted) current algebras and prove a special case of a conjecture made by Feigin and Loktev on fusion products. This allows us to realize certain truncated Weyl modules as fusion products of irreducible finite-dimensional representations. Finally, in the case of sl₂ we give a PBW type basis for truncated Weyl modules of the associated current algebra.

This is based on a joint work with Christof Geiss and Jan Schröer (arXiv 1502.01565). Let g be a symmetrizable Kac-Moody algebra, and let n be its nilpotent positive part. We realize the Hopf algebra U(n) as a convolution algebra of constructible functions on varieties of representations of some Iwanaga-Gorenstein algebras of dimension 1.

In my talk I will generalize the notion of holonomic D-modules and the Bernstein inequality to a wide class of filtered algebras of interest for representation theory. The talk is based on arXiv 1501.01260.

Silting modules generalise simultaneously tilting modules over any ring and support τ-tilting modules over finite dimensional algebras. In this talk, we study ring epimorphisms and universal localisations, as defined by Cohn and Schofield, that arise from partial silting modules. These ring epimorphisms are described explicitly by an idempotent quotient of the endomorphism ring of a completion of the given partial silting. In the context of hereditary rings, the above construction yields a bijection between universal localisations and certain minimal silting modules. Moreover, it turns out that all universal localisations over any ring arise from partial silting modules. This talk contains ongoing work with Lidia Angeleri Hügel and Jorge Vitoria, and with Jan Stovicek.

I will give an overview over developments in the 2-representation theory of finitary 2-categories. This is joint work with Volodymyr Mazorchuk.

We define a degenerate affine and cyclotomic versions of the walled Brauer algebra, and we prove an higher version of mixed Schur-Weyl duality. We concentrate then on level 2 cyclotomic quotients, and we show via an explicit isomorphism that the classical walled Brauer algebras, for any choice of the parameter delta, can be realized as idempotent truncations of such quotients. (The last part is joint work with C. Stroppel).

I will explain how to relate the center of a cyclotomic quiver Hecke algebras with the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.

Daniel Tubbenhauer: U_q(sl_n) diagram categories via q-Howe duality

The Temperley-Lieb algebra TL_d has its origin in the study of sl₂-modules: Rumer, Teller and Weyl showed (more or less) already in the 30ties that TL_d can be seen as a diagrammatic realization of the representation category of sl₂-modules — providing a topological (and fun!) tool to study the latter.

In this talk I try to explain how one can proof such a realization and to discuss some related diagram categories, e.g. representation categories of sl_n-modules consisting of alternating tensors, representation categories consisting of symmetric tensors and a generalization containing both tensors at once. Our main tool for all of these is q-Howe duality — either skew, symmetric or super.

In principal, everything in this talk is amenable to categorification, but we have to stay in the uncategorified world for the moment.

We'll first explain how to compute the decomposition matrices of Cherednik algebras via categorical representations. Then, we'll apply categorical representations to finite reductive groups of classical types.

One of the most basic operations in representation theory is restricting representations under homomorphisms between Lie algebras or other algebras. For example, tensor products of simple modules and Fock spaces of all levels are restrictions of simple modules under such a map. In categorified representation theory, such homomorphisms don't lift to functors between 2-categories, but in some cases, there is a more subtle manifestation of these maps. I'll discuss what it means to restrict a categorical representation, and how unique such a restriction can be.

For Kleinian singularities, Siedel-Thomas constructed a braid group action on the derived category of the minimal resolution using spherical twists. For 3-folds, it has long been thought that a similar result holds for flops. In the talk I will explain that this is partially true — there is always some form of braid group action arising from any 3-fold flop, but the combinatorics are more delicate. This is joint work with Will Donovan.

I will try to explain a conjecture with Simon Riche. It says that two categorifications of the "anti-spherical" or "polynomial" module for the affine Weyl group are equivalent. The first categorification is via Soergel bimodules. The second is via tilting modules for reductive algebraic groups. The conjecture has two nice consequences: a) the principal block of a reductive algebraic group admits a grading, which is defined over the integers; b) the characters of indecomposable tilting modules are given by the p-canonical basis in the anti-spherical module. Examples starting from SL₃ suggest that the grading cannot be chosen to be positive, which might upset some viewers. Our conjecture should be Koszul dual to a conjecture of Finkelberg-Mirkovic. If time permits I will outline work in progress which uses work of Chuang-Rouquier, Mackaay-Thiel, Cautis-Lauda and Brundan to give a proof of our conjecture for GL_n.

Open Richardson varieties are the intersections of opposite Schubert cells  in full flag varieties. Leclerc defined a cluster algebra inside the coordinate ring of each open Richardson variety for a symmetric Kac-Moody group, and Muller and Speyer studied these cluster algebras in the case of Richardson varieties in Grassmannnians. We will show how to realize the quantized coordinate ring of each open Richardson variety as a normal localization of a prime factor of a quantum Schubert cell algebra. Using a combination of ring theoretic and representation theoretic methods, we produce large families of toric frames for all quantum Richardson varieties by constructing sequences of normal elements in chains of subalgebras. This gives a method to control the size of Leclerc's cluster algebras from below and ultimately to relate them to the coordinate rings of the Richardson varieties (for the general case of all symmetrizable
Kac-Moody algebras and their quantizations). This is a joint work with Tom Lenagan (University of Edinburgh).