Workshop on Representations and Cohomology

Schedule of Talks

Titles and Abstracts

Eric Friedlander (University of Southern California)
Finite group schemes, cohomology, and cohomological supports
In this first lecture, we compare and contrast various finite group schemes over a field, especially those associated to finite groups and algebraic groups. We review some of what is known about general properties of cohomology algebras of these structures, then discuss cohomological support varieties. We remark upon the limited information provided by these support varieties, and the challenge to seek refinements.
π-points, Jordan types, and refined local invariants
We recall shifted cyclic subgroups for elementary abelian groups, 1-parameter subgroups for infinitesimal group schemes, and π-points for arbitrary finite group schemes. We present a theorem proved jointly with J. Pevtsova which compares the representation-theoretic scheme Π(G) to the cohomological variety. As an advertisement of the perspective of π-points, we define and investigate modules of constant Jordan type.
Constructions for infinitesimal group schemes
The representation theory of these finite group schemes is challenging, but we exploit 1-parameter subgroups to obtain structures not yet available for representations of finite groups. In particular, we discuss the universal p-nilpotent operator and constructions of vector bundles.

Srikanth Iyengar ( University of Nebraska, Lincoln)
Stratifying modular representations of finite groups
The notion of support is a fundamental concept which provides a geometric approach for studying various algebraic structures. The prototype for this has been Quillen's description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Carlson introduced support varieties for modular representations. A central problem, in any context where one has a notion of support, is to characterize when representations have the same support. I will discuss recent work with Dave Benson and Henning Krause which provides a complete solution for representations of finite groups.

Changchang Xi (Bejing Normal University)
Homological dimensions and algebra extensions
In this series of lectures, we shall consider homological dimensions (like the global dimension, the finitistic dimension and other related dimensions), discuss which are invariant under what, and investigate some homological conjectures like the finitistic dimension conjecture and the strong no loop conjecture by introducing an inductive procedure via algebra extensions.

Claire Amiot (NTNU Trondheim)
A generalization of cluster categories
In 2005 Buan, Marsh, Reineke, Reiten and Todorov have introduced the cluster category associated with an acyclic quiver. Their aim was to categorify acyclic cluster algebras. In this talk I will define these categories and show how it is possible to generalize this construction replacing an acyclic quiver by a finite dimensional algebra of global dimension 2, or by a quiver with potential.

Steffen Oppermann (NTNU Trondheim)
Representation dimension of quasi-tilted algebras
The representation dimension (introduced by Auslander) provides a homological criterion for when a finite dimensional algebra is representation finite. One can hope that for representation infinite algebras it measures how complicated the homological algebra of the module category is. Hence it is natural to ask what the representation dimension of quasi-tilted algebras, the homologically simplest algebras, is. In my talk I will first give definitions of and background on representation dimension and quasi-tilted algebras. It will follow that we may restrict ourselves to algebras derived equivalent to coherent sheaves on a projective line. Therefore we will recall the basic structure of this category. Finally, I will sketch how one can use certain torsion sheaves to show that the representation dimension in this setup is always three.

Matthias Künzer (RWTH Aachen)
Learning Neeman's K-theory
Abstract

Dan Zacharia (Syracuse University)
Auslander-Reiten theory for modules of finite complexity over selfinjective algebras
Abstract

Osamu Iyama (Nagoya)
n-representation-finite algebras of type A
We call a finite dimensional algebra A of global dimension at most n n-representation-finite if there exists an n-cluster tilting object M in the module category mod-A. In this case, M is unique up to mulitiplicities of direct summands. Gabriel's Theorem asserts that 1-representation-finite algebras are path algebras of Dynkin quivers. In my talk, a class of n-representation-finite algebras of `type A' will be constructed. We also classify n-representation-finite algebras with dominant dimension at least n. Our main tools are n-AR translation functor and n-APR tilting modules. An interesting combinatorics will appear in our consideration as slices in n-cluster tilting subcategories of derived categories.

Bernhard Keller (Paris 7)
The periodicity conjecture via 2-Calabi-Yau categories
The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Ravanini-Valleriani-Tateo and Kuniba-Nakanishi. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the dual Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (A_n, A_1), by Fomin-Zelevinsky in the case where one of the diagrams is A_1 and by Volkov and independently Szenes when both diagrams are of type A. The conjecture is proved in ongoing work by Hernandez-Leclerc in the case where one of the diagrams is of type A. We will sketch a proof of the general case which is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated 2-Calabi-Yau categories. An important role is played by Coxeter transformations and by Amiot's recent work on cluster categories for algebras of global dimension 2.

Lidia Angeleri Hügel (Verona)
Recollements and tilting objects
This is a report on joint work with Steffen Koenig and Qunhua Liu. We consider recollements of the derived category D(Mod R) of a ring R which are induced by tilting complexes. By a result of Nicolas and Saorin every recollement of D(Mod R) is associated to a differential graded homological epimorphism from R to S. We will focus on the case where this is a homological ring epimorphism or even a universal localization.
Slides

Giovanni Cerulli Irelli (Padova)
Euler-Poincare characteristic of quiver Grassmannians
In the last few years, after the works by Caldero and Keller, the Euler-Poincare characteristic of quiver grassmannians have become of great importance in the theory of cluster algebras. We will talk about some methods to determine such numbers in the special case of a quiver of type $A_{n,1}$.

Xiao-Wu Chen (Paderborn)
An extension of Grothendieck duality via relative homological algebra
We show that for a Gorenstein ring (more generally a left-Gorenstein ring), the homotopy category of Gorenstein-projective modules is equivalent to the homotopy category of Gorenstein-injective modules, and thus the homotopy category of projective modules is equivalent to the homotopy category of injective modules. This is inspired by Iyengar- Krause's extension of Grothendieck duality. Our result follows from a rather easy observation in relative homological algebra.

Yuriy Drozd (Kiev)
Tilting, deformations and representations of linear groups over Euclidean algebras
We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.

Evgeny Feigin (Moscow)
Homological approach to the affine Littlewood-Richardson functions
We compute the generating functions of Littlewood-Richardson coefficients for an affine Kac-Moody algebras using homological arguments. The result is given in terms of the affine Weyl group and string functions.

Martin Herschend (Nagoya)
Solution to the Clebsch-Gordan problem for string algebras
String algebras are a class of tame algebras whose indecomposable finite dimensional modules are classified by so- called strings and bands. They originate from the classification of Harish-Chandra modules over the Lorentz group by Gelfand and Ponomarev and have since appeared in many other areas including modular group representation theory. By definition, any string algebra is presented as the path algebra of a quiver subject to certain monomial relations. Hence its module category is equipped with a tensor product defined point-wise and arrow-wise. This leads to the following problem: given two indecomposable modules over a string algebra, decompose their tensor product into a direct sum of indecomposables. In my talk I will present the solution to this problem, called the Clebsch-Gordan problem, for all string algebras. Moreover, I will describe the corresponding representation ring.

Elena Kireeva (Moscow)
Double Centralizing Theorem for wreath products of the alternating group
The double centralizing theorem between the action of the symmetric group and the action of the general linear group was obtained by I. Schur. Recently A. Regev obtained the double centralizing theorems for the alternating group and for the wreath product of a finite group and the symmetric group. We obtain a double centralizing theorem for the wreath product of a finite group and the alternating group.

Justyna Kosakowska (Torun)
On Lie algebras associated with representation directed algebras
Abstract

Sefi Ladkani (MPI Bonn)
On derived equivalences of triangle, rectangles and lines
I will present new results on derived equivalences of certain finite-dimensional algebras. These derived equivalences can be viewed as categorical interpretations of linear algebra statements concerning equivalences of bilinear forms expressed as explicit matrix shapes. These results imply unexpected derived equivalences among Auslander algebras, endomorphism algebras of certain initial modules in the sense of Geiss-Leclerc-Schroer, incidence algebras of posets and other algebras generalizing the ADE-chain. Among the quivers of these algebras one can find shapes of triangles, rectangles and lines.
Slides

Jue Le (Paderborn)
Auslander-Reiten theory on the homotopy category of projective modules
An Auslander-Reiten formula in the homotopy category of complexes of projective modules is presented. This formula guarantees the explicit description of Auslander-Reiten triangles in the homotopy category. Furthermore, almost split sequences in a module category can be deduced from these Auslander-Reiten triangles. It is interesting to compare this with the case in the homotopy category of injective modules.
Slides

Yuming Liu (Cologne)
Stable Hochschild homology and Auslander-Reiten conjecture II
This is a joint work with Guodong Zhou and Alexander Zimmermann. Let A and B be two finite dimensional algebras which are stably equivalent of Morita type. The Auslander-Reiten conjecture states that A and B have the same number of isomorphism classes of non-projective simple modules. In this talk, we will give several equivalent conditions of this conjecture, involving Hochschild homology groups, centers and projective centers. The key point of our approach is to define the concept of the 0-degree stable Hochschild homology group. In the first half of this talk, Guodong Zhou will give a general introduction and in the second half of this talk, I will mainly concentrate on the definition and properties of stable Hochschild homology.

Piotr Malicki (Torun)
Degenerations in the additive categories of almost cyclic coherent Auslander-Reiten components
Abstract
Slides
Examples

Ehud Meir (Technion Haifa)
Moore's conjecture and nilpotency of certain cohomology elements
Let G be a group and H a finite index subgroup. We consider the following question of Moore: Suppose that every nontrivial cyclic subgroup of G intersects H nontrivially. Is it true that a G- module which is projective over H is also projective over G? In many cases the answer is known to be positive, including for example finite groups and groups of finite cohomological dimension. A theorem of Aljadeff says that the conjecture also holds in case the group G has a finite index normal subgroup K subgroup of G/K intersects H/K trivially. This can be used also to prove that the conjecture is true for profinite groups. In this talk we will show that in all the cases mentioned above, the conjecture holds due to the nilpotency of a certain element (the Bockstein) in the cohomology ring of G (this was actually the way the conjecture was proved for finite groups). We will construct examples for pairs (G,H) of a group G and a finite index subgroup H such that the conjecture is true for G and H, eventhough the Bockstein element is not nilpotent. We will also generalize a result of Aljadeff, Cornick, Ginosar and Kropholler, and show that the conjecture is true for all groups G inside Kropholler's hierarchy LHF, and not just in case the module under consideration is finitely generated.

Vanessa Miemietz (Oxford)
Braidings on the derived categories of polynomial representations for GL_2
Braidings on the derived categories of certain small algebras arising in Lie theory have been studied by Khovanov/Seidel and Rouquier/Zimmermann. We lift those braiding to the derived categories of Schur algebras for $GL_2$.

Ryo Narasaki (RWTH Aachen)
Fusion systems and blocks of finite groups}
Every finite group G gives rise to a saturated fusion system F_P(G) over a Sylow p-subgroup P of G. Here we assume that finite groups G and H have a common Sylow p-subgroup P, and G and H give rise to the same fusion system over P. From the Broue's abelian defect group conjecture of view, if P is abelian, then we might expect the principal blocks B and B' of G and H, respectively, are derived equivalent and are perfect isometric. It doesn't hold for the cases of non-abelian Sylow subgroups. But we might expect that there exists some generalized perfect isometry between B and B'.
Slides

Selene Sánchez-Flores (Montpellier)
The Lie module structure on the Hochschild cohomology groups of monomial algebras of radical square zero
Abstract

Markus Schmidmeier (Florida Atlantic)
The entries in the LR-tableau
Let T be the Littlewood-Richardson tableau corresponding to an embedding M of a subgroup in a finite abelian p -group. For each individual entry in T , we give an interpretation in terms of the functor Hom(M,-) defined on the category of embeddings, and in terms of the direct sum decomposition of a certain subfactor of M .

Vladimir Shchigolev (Moscow)
Local criterion for Weyl modules over groups of type A
Abstract

Adam-Christiaan van Roosmalen (Bonn)
Classification of hereditary categories which are (fractionally) Calabi-Yau
In order to better understand hereditary categories and Calabi-Yau categories, we will discuss the classification (up to derived equivalence) of abelian 1-Calabi-Yau categories and hereditary categories which are fractionally Calabi-Yau. We have shown that all hereditary (fractionally) Calabi-Yau categories are derived equivalent to one of the following: a) the category of nilpotent representations of an oriented cycle, b) the category of coherent sheaves over an elliptic curve, c) the category of coherent sheaves over a weighted projective line of tubular type, d) the category of finite dimensional representations of a Dynkin quiver. The emphasis of this talk will lie on examples and properties of these categories rather than on the proof of this classification.
Slides

Christian Weber (RWTH Aachen)
Some results about cohomology of integral Specht modules
Abstract
Slides

Fei Xu (Nantes)
Hochschild and ordinary cohomology rings of small categories
Let C be a small (often finite) category and k a field. We consider the (associative) category algebra kC and the classifying space BC. We show there exists a natural split surjection from the Hochschild cohomology ring HH*(kC) to the ordinary cohomology ring H*(BC, k), generalizing the well-known results for groups and posets. As an application, we assert that there exists a finite-dimension associative algebra whoseochschild cohomology ring, modulo nilpotent elements, is not finitely generated.

Guodong Zhou (Cologne)
Stable Hochschild homology and Auslander-Reiten conjecture I
This is a joint work with Yuming Liu and Alexander Zimmermann. Let A and B be two finite dimensional algebras which are stably equivalent of Morita type. The Auslander-Reiten conjecture states that A and B have the same number of isomorphism classes of non-projective simple modules. We will give several equivalent conditions of this conjecture, involving Hochschild homology groups, centers and projective centers. The key point of our approach is to define the concept of the 0-degree stable Hochschild homology group. In the first half of this talk, a general introduction to this work will be given. I will state the Auslander-Reiten conjecture and explain the equivalent conditions obtained. More details will be found in the second half of this talk given by Yuming Liu.