DFG- Antrag
Projekte (abg.)

Abgeschlossene Projekte

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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.

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.

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.

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.

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).

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

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.

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