Bliem, Thomas (University of Cologne/San Francisco State University):
Polyhedral models of representations
The goal of the project is the development of combinatorial methods and their application in representation theory. The combinatorial side concerns the following problem: How many points with integral coordinates are contained in a given convex polytope? How does this number change if the facets of the polytope are displaced? Important figures in the representation theory of Lie algebras, the so-called weight multiplicities, can be encoded as families of polytopes obtained from one another by displacement of the facets. Thus, in the project, the existing methods for describing this combinatorial situation shall be further developed and exploited with respect to representation theory.

Bürgisser, Peter (University of Paderborn):
Geometry and representation theory in computational complexity
Can one compute the permanent of an n by n matrix with a number of arithmetic operations bounded by a polynomial in n? How many arithmetic operations are sufficient for calculating the product of two matrices? These two questions are undoubtedly the most important open problems of algebraic complexity theory. The first question is closely related with the famous P versus NP problem. Surprisingly, both problems allow a natural formulation in terms of geometric invariant theory and representation theory. For the matrix multiplication problem, this has been realized long ago by Strassen. For the algebraic P versus NP problem, this connection was discovered and considerably pushed further by Mulmuley and Sohoni in the last years. In both cases, questions of computational complexity are first reduced to geometric questions, and then to specific questions about the splitting of irreducible representations when restricting to the subgroup of symmetries of the problem under investigation. One ends up with questions about special instances of Kronecker products and plethysms. Unfortunately, those classical problems concerning the representations of symmetric groups and general linear groups are not well understood. The goal of this project is to prove better lower complexity bounds for the permanent and matrix multiplication based on the Mulmuley Sohoni approach, in collaboration with specialists of representation theory.

Chen, Bo (University of Stuttgart):
The Gabriel-Roiter measure for finite dimensional algebras
The objective of this project is to study the Gabriel-Roiter measure for finite dimensional algebras.The Gabriel-Roiter measure has been used by Roiter in his proof of the fundamental Brauer-Thrall conjecture I,dealing with algebras of finite type. Ringel has suggested to develop the Gabriel-Roiter measure into a foundational tool for representation theory of algebras of any representation type, thus providing alternative methods to those of Auslander-Reiten theory. So far, however, results have been obtained mainly on the Gabriel-Roiter measures in the case of algebras of finite representation type and tame hereditary type. The Gabriel-Roiter measure for wild (hereditary) algebras as well as the interaction between the Gabriel-Roiter measure and stable-equivalence,tilting theory, and Hall algebras will be studied.

Christandl, Matthias (ETH Zürich):
Representation Theory of the Unitary and Symmetric Group with a view towards the Quantum Marginal Problem
The quantum marginal problem asks for the compatibility of a set of partial descriptions of a quantum system. This problem is fundamental to many areas of quantum physics such as condensed matter physics and quantum information theory. Its fermionic version known as the N-representability problem is relevant to quantum chemistry where an efficient solution would allow for the calculation of properties of molecules such as their binding energies. Recently it has been shown that certain instances of the quantum marginal problem are equivalent to questions arising in the representation theory of Lie groups and finite groups. The aim of this project is two-fold. Firstly, we wish to provide a unified treatment of the instances that have been discussed in the literature and tackle the representation-theoretic conjectures that have arisen in this context. Secondly, we wish to extend the connection between the quantum marginal problem and representation theory from specific instances to arbitrary instances, before applying our findings in quantum information theory and quantum physics more generally.

Ebeling,Wolfgang (Leibniz University Hannover):
Homological mirror symmetry for singularities
The primary objective of the project is to study homological mirror symmetry for singularities in order to gain, by means of representation theory, a better understanding of some mysterious phenomena discovered in singularity theory. A two-year postdoc position (David Ploog, University of Toronto) will be financed by the DFG and funds for travel and research visits are approved.

Farnsteiner, Rolf (University of Kiel):
Combinatorial and geometric aspects of the representation theory of
finite group schemes
This project seeks to apply Lie theoretic and geometric methods to the study of module categories affording tensor products. The primary focal points are the representation types and the stable Auslander-Reiten quivers of the relevant Frobenius categories. Given a finite group scheme G over a field k of characteristic p>0, the Friedlander-Suslin Theorem provides a procedure, by which we can associate a cohomological support variety to each G-module M. Support varieties and rank varieties turn out to define invariants of the connected components of AR-quivers, and they also are an indispensable tool in the classification of representation-finite and tame blocks of finite group schemes. The recently defined \Pi-supports of modules lead to refinements, whose ramifications are only emerging. The interaction between combinatorial data given by AR-components and \pi-points as well as the investigation of new classes of modules with certain sets of Jordan types should shed new light on questions that cannot be tackled by varieties only. For certain groups of tame representation type, a classification of their indecomposables is intended, thus furnishing a solid testing ground for general conjectures involving \pi-points.

Fiebig,Peter (University of Erlangen):
Critical level representations of affine Kac-Moody algebras and the geometric Langlands program
The main objective of the project is to link the category O of an affine Kac-Moody algebra at the critical level to the topology of the Langlands dual affine Grassmannian. As an application we want to give a proof of the Feigin-Frenkel formula for the simple critical highest weight characters.

Hartmann, Julia (RWTH Aachen):
Symmetry groups of differential or difference equations and their representations on the solution spaces
The aim of the project is to study symmetry groups of differential or difference equations by means of their representations on the solution spaces. To a differential or difference equation one associates an algebraic group which describes the symmetries and encodes a lot of information about the solutions of the equation. By Tannakian theory, this group is determined by the structure of its representations on objects of the tensor category generated by the solution space. The aim of this project is to apply methods from the structure and representation theory of algebraic groups to study differential (or difference) equations. The first application is the computation of the symmetry group (Galois group) of a given equation. A second goal is the construction of objects with interesting symmetry groups.-->

Heinzner, Peter; Huckleberry, Alan; Püttmann, Annett and Winkelmann, Jörg (University of Bochum):
Complex geometry of actions and related representations
The interplay between actions of a given group on geometric objects having rich structure and linear representations on associated function spaces is studied. The groups being considered are complex Lie groups and their real forms. The geometric objects on which they act are of a complex analytic nature, e.g., complex manifolds. The representations are on spaces of functions which respect the complex structure, often just vector spaces of special holomorphic functions. A typical goal is to attempt to understand attributes of the function algebra in terms of the group structure and its action. In many aspects of the project the information of interest comes from restricting a given action of a complex Lie group to a real form. In that setting the Hamiltonian method is applied in order to identify special regions of representation theoretic relevance. This is often an orbit of the real form. In that case a nonlinear transform is introduced in order to realize the representation space as a function space where the introduction of invariant linear geometry,unitarization, is one particular goal of the project.

Hiß, Gerhard(University of Aachen):
Investigations on Alperin's weight conjecture through Hecke algebras
In its original version, the conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by "local data", i.e. by data constructed from proper subgroups, the other set only by "global data" of the group itself. Thus the philosophy underlying the conjecture agrees with that behind so many other famous conjectures of group theory and representation theory: global data should be determined by local data. This project follows a suggestion of Alperin to study his conjecture through the modular Hecke algebra. (If G is a finite group and p a prime, the modular Hecke algebra (with respect to p) is the endomorphism ring of the G-permutation module on the cosets of a Sylow p-subgroup of G.) This could serve as a mediator between local and global theory. To date there are only few attempts to investigate the modular Hecke algebra systematically, in particular its role in Alperin's conjecture. The experimental results of Naehrig's 2008 PhD thesis suggest that a deeper investigation of the representation theory of this object might well be worthwhile. This is what we pursue in our project.

Hiß, Gerhard(University of Aachen); Malle, Gunter(TU Kaiserslautern):
Investigations on the conjectures of McKay and Alperin-McKay
This research project is located in the representation theory of finite groups. Our aim is the investigation of two prominent and long standing open conjectures, the McKay conjecture and its refinement by Alperin, the Alperin-McKay conjecture, respectively. These conjectures were formulated in the mid 70th of the last century. In its original formulation, the McKay conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by "local data", i.e. by data constructed from proper subgroups, the other set only by "global data" of the group itself. That global data should be determined by local data, is the philosophy behind the McKay conjecture and other famous conjectures of representation theory. Two recent developments have inspired this project. Firstly, in 2007, Isaacs, Malle and Navarro published a powerful reduction theorem for the McKay conjecture. This leaves one to verify some rather complicated conditions for the finite simple groups. Secondly, the results of Späth's 2007 PhD thesis provide a means to verify these complicated conditions in the local situation, at least in special cases. The main objectives of this project are to construct suitable bijections between the two sets of objects of the McKay conjecture in local configurations of classical groups and to prove the McKay conjecture for finite groups of Lie type in non-defining characteristic. A further objective is to prove the Alperin-McKay conjecture for unipotent blocks of finite groups of Lie type in non-defining characteristic.

Hilgert, Joachim (University of Paderborn):
Branching laws for 1-parameter families of representations of Lie groups and their asymptotic behavior
The determination of branching laws, i.e., the decomposition of representations of a group G into irreducible ones upon restriction to a subgroup H is a fundamental problem of representation theory. In physical applications it describes the breaking of symmetries for quantum mechanical systems. Many branching laws are known in principle in the form of complicated combinatorial or topological formulas expressing multiplicities as alternating sums. Exceptions to this rule are results of Littelmann and Knutson, which so far are available only for commutative H. If a representation can be realized by holomorphic sections (as in the case of compact Lie groups via the Borel-Weil Theorem) it admits a reproducing kernel which carries the entire information of the representation. Thus de- compositions of this kernel into suitable pieces invariant under the action of H yield decompositions of the restricted representations. In specific examples such decompositions have been obtained by Taylor expansions transversal to an H-orbit, an idea going back to S. Martens in the 1970s. We propose a systematic study of reproducing kernels associated with repre- sentations of compact Lie groups G with the goal to describe cancelation free branching laws and their asymptotic behavior.

Hille, Lutz (University of Münster):
Actions of Algebraic Groups, Fans and Tilting Modules
To any finite dimensional algebra of finite global dimension one can associate a fan of tilting modules of projective dimension at most one. This fan is connected for representation finite and tame hereditary algebras. Moreover, the support of the fan classifies the actions of the corresponding group on the representation space with a dense orbit. In particular, the action of a parabolic subgroup in a General Linear Group on a subideal in the Lie algebra of the unipotent radical can be described in this way.The principal aim of this project is to understand the fan for hereditary algebras, in particular its connected components. Moreover, for actions of parabolic groups the fan might be connected as well, which in turn would classify all such actions with a dense orbit. Finally, we would like to generalize the results to actions for parabolic subgroups in the classical linear algebraic groups.

Holm, Thorsten (Leibniz University Hannover):
Cluster categories, cluster-tilted algebras and derived equivalences
Cluster algebras form an exciting new area of mathematics, linking
representation theory, algebraic Lie theory, combinatorics and algebraic
geometry. Cluster categories form a categorification of cluster algebras
and allow to apply deep techniques from representation theory.
In the proposed project we plan to study the structure of
cluster categories and of the corresponding
cluster-tilted algebras (endomorphism algebras of certain
objects in the cluster categories). One of the main goals is to
understand when cluster-tilted algebras, e.g. of Dynkin or extended
Dynkin types, have equivalent derived categories. This will also
require to obtain new results on derived invariants of these algebras.
Moreover, we are looking for new and surprising occurrences of cluster
behaviour in different locations. For instance, we recently studied an
interesting triangulated category which shares cluster phenomena, but
would be of type A infinity. We aim at finding more such surprising
examples which should lead to completely new insights.

Knop, Friedrich (University of Erlangen-Nürnberg):
Multiplicity Free Actions
In this project, we study spaces, called spherical varieties, which have in a precise sense the highest possible degree of symmetry. They are generalizations of the round 2-sphere. In recent years, extensive research was devoted towards the goal of finding a complete list of spherical varieties. One aim of this project is to finalize this classification. Another goal is to study the geometric properties of spherical varieties. These are controlled by their set of spherical functions, which are generalizations of spherical harmonics on the 2-sphere. We study, in particular, the multiplicative properties of spherical functions. Finally, certain generalizations of spherical varieties are considered where functions are also allowed to anticommute. These structures are encountered, for example, in the geometry of compact Hermitian symmetric spaces.

Koenig, Steffen (University of Stuttgart):
Homological structures at the interface of abstract representation theory and algebraic Lie theory
This project investigates and uses homological structures of finite dimensional algebras, in particular homological dimensions such as representation dimension or dominant dimension. Results are to be used in other branches of representation theory, in particular in algebraic Lie theory.

Krause, Henning (University of Bielefeld):
The telescope conjecture for derived categories arising in representation theory
The objective of this project is to study the telescope conjecture for triangulated categories arising in representation theory. We follow the original approach of Neeman and establish a stratification of derived categories over appropriate differential graded algebras. Another promising approach is that of Krause and Stovicek for derived categories of hereditary rings. Positive results will be applied in particular to representations of finite dimensional algebras, for instance in the context of cohomological support varieties or for classifying thick and localizing subcategories.

Ladkani, Sefi (University of Bonn):
Derived categories of sheaves over finite partially ordered sets and their homological properties
Triangulated and derived categories have been successfully used to relate objects of different mathematical origins (e.g. Kontsevich's Homological mirror symmetry conjecture) as well as objects of the same nature (e.g. Rickard's Morita theory, Broue's conjecture). In this project we will investigate derived categories arising from combinatorial objects, such as certain quivers with relations and finite partially ordered sets (posets), leading to categories which can be described in algebraic ("modules") as well as topological ("sheaves") terms. We will study both abstract posets, as well as specific ones arising from combinatorial contexts such as cluster algebras and cluster categories. The main questions are: (a) When two such objects lead to equivalent derived categories? (b) Is there an algorithm which decides on this question? (c) Are there basic combinatorial operations ("mutations") taking an object to a derived equivalent one, with the property that any two derived equivalent objects are related by a sequence of such operations? To address them, we will: 1. Develop new constructions of derived equivalences; 2. Investigate the role of combinatorial and numerical invariants, especially the Euler bilinear form, in determining derived equivalence. For these investigations, the process of interpreting certain linear algebra statements as categorical ones (known as "categorification") will play an important role. Outside mathematics, one can find applications to chemistry concerning structure discriminators of molecules.

Littelmann, Peter (University of Cologne):
Geometry and the compression of combinatorial formulas for Macdonald polynomals
The aim of the project is to give a better understanding of the connection between work of Gaussent, Littelmann, Schwer, Ram and Yip and the work of Haglund, Haiman and Loehr. Both methods lead to rather different combinatorial formulas for Macdonald polynomials.It is expected that the connection between these formulas has a geometric background. The project has a geometric and combinatorial part.The aim of the geometric part is to generalize the known methods to a different class of Bott-Samelson varieties. The combinatorial part has as a goal to translate the geometry over a finite field into a statistic on (generalized) Young diagrams.

Malle, Gunter (TU Kaiserslautern):
Structure and representations of cyclotomic Hecke algebras
The project concerns the structure and representation theory of cyclotomic Hecke algebras. These deformations of group algebras of complex reflection groups arise as analogues of the Iwahori-Hecke algebras which in turn play a central role in the representation theory of finite groups of Lie type. It has become apparent in recent years that many properties of Iwahori-Hecke algebras have counterparts in the theory of cyclotomic Hecke algebras. Often, though, new approaches have to be found for their proof. The project aims at investigating analogues of several recent constructions for Weyl groups. The first is the new construction by M. Geck of Lusztig's algebra $J$ in the case of Coxeter groups, which seems to lend itself to a generalization to cyclotomic Hecke algebras. Together with a suitably modified concept of $W$-graph for the explicit construction of irreducible representations this should lead to a better understanding of these algebras. A further topic concerns representations of rational Cherednik algebras attached to complex reflection groups.

Nebe, Gabriele(University of Aachen):
p-Adic Group Rings of Finite Groups
The purpose of this project is to develop new and improve existing methods to describe group rings (respectively their basic algebras) over a discrete valuation ring R via their embedding in a product of matrix algebras over Quot(R). In particular, this gives explicitly a presentation of the basic algebra over the residue field of R as a quotient of a quiver algebra. The existing methods, originally developed by W. Plesken, are limited by constraints on the decomposition numbers as well as (implicitly) on the defect of the block in question. Improving upon these constraints is an important objective of this project.Moreover, we would also like to enhance the algorithmic tools to calculate basic algebras of group rings, which will endow us with further guiding examples. Apart from theory, this project should yield a variety of examples of basic algebras of blocks of groups rings and families of such (e. g. low defect blocks of symmetric groups).

Neeb, Karl-Hermann (University of Erlangen):
Semibounded unitary representations of double extensions of pre-Hilbert--Lie groups
The goal of this project is to develop a geometric approach to the important class of semibounded unitary representations for groups which are so-called double extensions of pre-Hilbert--Lie groups (groups whose Lie algebra carries an invariant scalar product). Typical examples of such groups are oscillator groups, double extensions of Hilbert--Lie groups and affine Kac--Moody groups. Semiboundedness of a unitary representation is a stable version of the ``positive energy'' condition which characterizes many representations arising in mathe\-matical physics, resp., field theories. For a unitary representation of a Lie group it means that the selfadjoint operators from the derived representation are uniformly bounded below on some open subset of the Lie algebra. Our long term goal is to understand the decomposition theory and the irreducible representations for this class.

Perling, Markus (University of Bochum):
Toric structures of exceptional sequences and representations of algebras
In this project we want to construct non-commutative coordinatizations of algebraic varieties by construction of exceptional sequences. On a given variety X, an exceptional sequence is a sequence of coherent sheaves satisfying certain conditions of cohomology vanishing. The endomorphism ring of the direct sum of such a sequence forms a noncommutative coordinate system for X in the sense that the derived category of X, i.e. the category of complexes of coherent sheaves on X, is equivalent to the derived category of modules over the endomorphism algebra. This connection serves as a bridge between geometry and the representation theory of algebras and has attracted much interest in recent years, not only in mathematics but also in the context of superstring theory. Recent results indicate that this correspondence is linked in a deep and unexpected way to the geometry of toric varieties. In this project we want to improve our understanding of this link and we want to use toric geometry to give more general constructions of exceptional sequences and their associated endomorphism rings.

Penkov, Ivan (Jacobs University Bremen):
Structures and representations of infinite- and finite-dimensional Lie algebras
This project concentrates on collaborations with three colleagues: Gregg Zuckermann(Yale University), Vera Serganova (University of California, Berke- lley), Alexander Tikhomirov(Yaroslavl University). With Zuckermann we study the structure of (g; k)-modules of finite type, where g is a finite-dimensional semisimple Lie algebra and k is a reductive (not necessarily symmetric!) subalgebra. The current challenge is to get an understanding (as detailed as possible) of (g; sl(2))-modules. With Serganova we study integrable modules of llocally finite Lie algebras. One of the challenges is to prove that the categories of tensor modules over the classical finite-dimensional Lie algebras so(&infin) and sp(&infin) are equivalent. May be this is infinite Howe duality? With Tikhomirov we address the geometry of homogeneous ind-spaces. The conjecture we are currently working on is to prove that any vector bundle on G=B for G = GL(&infin) has a subbundle of rank 1. This is part of a more general conjecture about finite rank vector bundles on flag ind-varieties.

Plesken, Wilhelm (University of Aachen):
Algebraic group techniques for finite(ly presented) groups
Applications of commutative algebra to representation theory of groups have a long tradition going back to Frobenius. We plan to investigate two new directions: Firstly a Galois descent for the invariant ring of a finite matrix group over an algebraic number field (1st part of the application) and a construction of matrix representations of finitely presented groups into algebraic groups of small ranks. Concerning the first part of the project one can observe that at the end of the 19th century Klein and Maschke and others constructed invariants with rational coefficients of finite irreducible matrix groups with algebraic coefficients. The question arises, when this is possible. Clearly this is the case iff all field automorphisms transform the matrix group into itself. So the real question is: Can one find necessary and sufficient conditions on the group and the character for this to happen. An obvious necessary condition is that the automorphism group acts transitively on the Galois conjugates of the character of the natural representation. To which extent this condition is also sufficient is the main aim of the investigation. The second part of the project stands in the long tradition of investigations, meth- ods and algorithms to construct factor groups of finitely presented groups starting with the work by Todd and Coxeter, nilpotent quotient algorithm, my own soluble quotient algorithm. Until recently there did not exist a general algorithm deciding whether a finitely presented group maps onto groups out of an infinite class of finite simple groups. In a paper on the construction of finite Hurwitz groups by Daniel Robertz and myself a method is outlined how to construct matrix representations of finitely presented groups essentially by assigning matrices with indeterminates to the generators and then turning the group relators into polynomial relations for these indeterminates. In the paper just finished with my student Anna Fabianska for the occasion of this proposal, I manage to avoid the matrix entries in a special but crucial case, namely the construction of representations of 2-generated finitely presented groups into PSL(2; pn). Instead of the matrix entries we work with the traces of the two generators and their product by introducing the concept of generalized Cheby- chev polynomials. Having sorted them into varieties over the integers by means of a primary decomposition over Z, it becomes possible to answer most of the relevant questions. The aim of this project is to extend these methods from PSL(2; ) to algebraic groups of small ranks.

Röhrle, Gerhard (University of Bochum):
Serre's notion of Complete Reducibility and Geometric Invariant Theory
In this proposed research we intend to further investigate J-P. Serre's notion of G-complete reducibility by means of geometric invariant theory, concentrating on rationality and building theoretic questions. Some of the specific principal objectives are as follows. Firstly, we want to introduce a suitable concept of optimality by combining Kempf's instability notion with Hesselink's idea of uniform instability. Apart from being of independent interest, this will then be used to address building theoretic questions. Secondly, we want to generalize Richardson's algebraic characterization of the closed G-orbits in the n-fold Cartesian product of G with itself, under simultaneous conjugation to the action of an arbitrary reductive subgroup of G. This in turn will lead to a generalization of Serre's notion of G-complete reducibility. Further, we will study rationality questions of G-complete reducibility by geometric means, specifically here we will address a general problem posed by Serre concerning the behavior of G-complete reducibility under separable field extensions.

Schweigert, Christoph (University of Hamburg):
Representation theoretic methods for equivariant and orbifold conformal field theories
Rational conformal quantum field theories can be described by separable symmetric Frobenius algebras in modular tensor categories, e.g. in categories of representations of affine Kac-Moody algebras. This makes these quantum field theories amenable to a mathematically rigorous treatment and provides in particular a constructive proof for the existence of a consistent set of correlators. Our project aims at investigating $G$-equivariant generalizations of such theories with representation theoretic methods. Such theories admit in certain cases an orbifold construction which provides new examples of non equivariant theories. We have three principal goals:
1. The construction of $G$-equivariant examples
2. A better conceptual understanding of the orbifold construction
3. A $G$-equivariant generalization of the TFT construction

Soergel, Wolfgang (University of Freiburg):
Koszul duality in representation theory
The project aims at understanding what some standard operations on representations should correspond to under the Koszul-duality equivalence, in particular the operation of tensoring with a finite dimensional representation and Induction.

Stroppel, Catharina(University of Bonn);Littelmann, Peter(University of Cologne):
Combinatorics of affine Grassmannians and affine flag varieties
The aim of the proposed research project is the translation and extension of combinatorial models for studying the complex affine Grassmannian to the complex affine flag variety, and then to both the affine flag variety and Grassmannian in positive characteristic. This should prove to be useful to help understand geometric and representation theoretic problems related to these objects. In the case of the above mentioned complex affine Grassmannian a lot is known thanks to the work of numerous authors which relates the geometry of the affine Grassmannian with the representation theory of algebraic groups, by both combinatorial and geometric approaches. While some of these techniques are already established, others are quite new or need some major modifications to be applicable. The proposed project wants to generalize these established methods and use them in an extended context (the affine flag variety) as well as developing new tools (in particular for the positive characteristic case). The proposed research project is related to the following topics in mathematics: Representation theory of complex algebraic groups, representation theory of algebraic groups/group schemes over a local/finite field, arithmetic geometry, complex algebraic geometry, theory of building over the complex numbers as well as local fields, combinatorics of galleries and alcove walks, and the structure of Iwahori- Hecke algebras.

Wedhorn, Torsten (University of Paderborn):
Stable pieces in a Frobenius linear compactification of a semisimple group
The goal is to determine which stable pieces in a Frobenius linear compactification of a semisimple group are pure and apply these results to families of varieties. One of the most fruitful techniques to understand geometric objects is to attach to them linear invariants. The study of these invariants often leads to a problem within representation theory. One important question for these invariants is whether they have the property to be pure. A positive answer allows to control the degeneration of geometric objects. An important linear invariant is the De Rham cohomology. For certain geometric objects playing a central role in number theory (e.g., abelian varieties, curves, or p-divisible groups in positive characteristic) their De Rham cohomology carries the structure of a so-called truncated crystal. These can be considered in two ways as objects within the area of representation theory: as representation of certain clans in the sense of Crawley-Boevey or as stable pieces in a compactification of the projective linear group. The goal of this project is to study whether the De Rham cohomology is pure focussing on the second interpretation.