Köln Algebra and Representation Theory Seminar

Abstract: The notion of stability conditions for representations of algebras was introduced by King in the '90s as an application of Mumford's Geometric Invariant Theory. In this talk we will show an explicit and surprising relation between King's stability conditions and tau-tiling theory, an homological theory introduced by Adachi, Iyama and Reiten at the beginning of the last decade. Time permitting, we will show how the wall-and-chamber structure of an algebra can be used to solve a tau-tilting version of the first Brauer-Thrall conjecture. Some of the results presented in this talk are the result of collaborations with Thomas Brüstle, David Smith and Sibylle Schroll. 

Abstract: In this talk, we discuss a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is studied through a generalization of the notion of T-Koszul algebras, as introduced by Madsen and Green–Reiten–Solberg. We present a higher version of classical Koszul duality and sketch some applications for n-hereditary algebras. In particular, we see that an important class of our generalized Koszul algebras can be characterized in terms of n-representation infinite algebras. As a consequence, we show that an algebra is n-representation infinite if and only if its trivial extension is (n+1)-Koszul with respect to its degree 0 part. Furthermore, we see that when an n-representation infinite algebra is n-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated (n+1)-preprojective algebra are equivalent. A generalized notion of almost Koszulity in the sense of Brenner–Butler–King yields similar results in the n-representation finite case.This is based on joint work with Mads H. Sandøy.

Abstract: I will talk about Schubert varieties in the homogeneous spaces G/P, where G is an exceptional reductive algebraic group and P is a minuscule parabolic subgroup. By computational methods as well as hands-on analysis, we explicitly describe the defining ideals of the intersections of these Schubert varieties with the big open cell. We show that some of them as well as their  resolutions are related to well-known ideals respectively resolutions in other contexts. This is joint work with Sara Angela Filippini and Jerzy Weyman. 

Abstract: Simple-minded systems (SMSs) were introduced by Koenig-Liu as an abstraction of nonprojective simple modules in stable module categories: the idea was to use SMSs as a way to get around the lack of projective generators to help develop a Morita theory for stable module categories.

Recent developments have shown that SMSs in negative Calabi-Yau categories admit mutation theories and combinatorics that are highly suggestive of cluster-tilting theory. In this talk, we explain one such development: that negative Calabi-Yau orbit categories of bounded derived categories of acyclic quivers serve as categorical models of positive Fuss-Catalan combinatorics and one can think of SMSs as negative cluster-tilting objects. 

Along the way, we will make use of the rather surprising observation that in a triangulated category of finite homological dimension, functorial finiteness of the heart of a t-structure is related to the property of the heart having enough injectives and enough projectives. This is surprising because it says that some feature of how a heart behaves within an ambient triangulated category can be detected intrinsically in the heart.

This talk is based on joint work with Raquel Coelho Simoes and David Ploog.

Abstract: Toric degenerations with projections and standard monomials. Abstract: I will report on joint work in progress with Takuya Murata. We study toric degeneration, i.e. flat families over the affine line whose special fibre is a projective toric variety. Using valuations and Gröbner theory it turns out that we may assume that our family comes endowed with an embedding. In the case of curves a toric degeneration admits a projection from the generic fibre to the special fibre. So one may ask which kind of (embedded) toric degenerations admit such a projection. We obtain a positive answer for a large class of examples coming from cluster algebras. More generally, the notion of standard monomials in Gröbner theory provides a useful tool in tackling this question. 

Abstract: Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this talk, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. We also construct a specific basis of an irreducible highest weight representation which is naturally parametrized by the set of lattice points in a marked chain-order polytope. This basis is an analog of an essential basis introduced by Feigin- Fourier-Littelmann and by Fang-Fourier-Littelmann. 

Abstract: If k is a field, A a finite dimensional k-algebra, then the simple A-modules form a simple minded collection in the derived category D^b(mod A).  Their extension closure is mod A; in particular, it is abelian.  This situation is emulated by a general simple minded collection S in a suitable triangulated category C.  In particular, the extension closure <S> is abelian, and there is a tilting theory for such abelian subcategories of C.  These statements follow from <S> being the heart of a bounded t-structure. 

It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree.  Relaxing this to vanishing in degrees {-w+1, ..., -1} where w is a positive integer leads to the rich, parallel notion of w-simple minded systems, which have recently been the subject of vigorous interest within negative cluster tilting theory.

If S is a w-simple minded system for some w>=2, then <S> is typically not the heart of a t-structure.  However, it is possible to prove by different means that <S> is still abelian and that there is a tilting theory for such abelian subcategories.  We will explain the theory behind this, which is based on Quillen's notion of exact categories. 

Abstract: In this talk we will study singular foliations of a smooth manifold compatible with a Riemannian metric. These foliations are generalizations of smooth group actions by compact Lie groups. In particular we will consider foliations whose leaves are homeomorphic to tori, and study the local invariants of these foliations, covering the basics of the theory of singular foliations. The main result is that for such foliations of large dimension their local invariants characterize the foliation. These techniques extend the ones used for classifying smooth torus actions in the settings of Riemannian geometry (bounded  curvature)  or symplectic geometry (hamiltonian actions). 

Abstract: Motivated by the study of scattering particles in mathematical physics, N. Arkani-Hamed, Y. Bai, S. He and G. Yan gave a new realization of the classical associahedron as an intersection of a high dimensional positive orthant with well-chosen affine spaces. Their result is interpreted in representation theory of algebras by V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yildirim, allowing them to generalize ABHY's construction to generalized associahedra coming from cluster algebras of finite type, with acyclic initial seed.

Inspired from their approach, A. Padrol, V. Pilaud, P.-G. Plamondon and I give a different interpretation of ABHY's construction. By making use of cluster categories, we further extend ABHY's construction to cluster algebras of finite type with non-necessarily acyclic initial seed. Our approach also applies to a family of "generalized accordiohedra" arising from the tau-tilting theory of gentle algebras. 

Abstract: First, we will discuss a polygon model of the Auslander-Reiten quiver of a type A quiver. Next, we will introduce a new Catalan object which we call a maximal almost rigid representation. Finally, we will define a partial order on the set of maximal almost rigid representations and show that this partial order is a Tamari or Cambrian lattice. This work is joint with Emily Barnard, Emily Meehan, and Ralf Schiffler. 

Abstract: The Kostka-Foulkes polynomials are q-analogues of the weight multiplicities of representations of reductive groups. In type A, Lascoux and Schützenberger gave a combinatorial meaning to the coefficients of Kostka-Foulkes polynomials by defining a statistic, called the charge, on the set of semistandard tableaux. In this talk we will discuss a new geometric approach to the charge statistic. Motivated by the Satake equivalence, we relate the construction of a charge statistic to the geometry of the affine Grassmannian. The main idea consists in looking at how hyperbolic localization changes for a family of cocharacters and then mimicking this procedure at the level of crystal graphs. As a consequence, we obtain a new formula for the charge in terms of crystal operators.