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.

*(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).

*(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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