BIREP – Representations of finite dimensional algebras at Bielefeld

Bad Driburg

The workshop concerns representations of the *general linear group* Γ=Gl_{n}(K) over an inifinite field K, i.e., we want to classify group homomorphism τ : Gl_{n}(K) → Gl_{N}(K) for some N. It is in general a hard problem. (Even the case n=N=1 includes the non-trivial task of describing all field automorphisms of the complex numbers). Therefore, we stick to *polynomial* representations of Γ, i.e., for all g ∈ Gl_{n}(K) all entries of τ(g) are supposed to be polynomials in the entries of g.

The representation theory of Γ has a long and colorful history beginning with I. Schur's dissertation. Schur proved that the category M_{K}(n,r) of r-homogeneous polynomial representations is equivalent to the module category of an algebra S_{K}(n,r) nowadays known as *Schur algebra*. It is constructed as an automorphism algebra of the r-th fold tensor power E^{⊗ r} of an n-dimensional space E using the permutation action of the symmetric group G(r). So, the representation theory of the symmetric group (featuring partitions) enters.

Over the complex numbers, the group Gl_{n}(K) is *reductive*, linear representations are completely reducible, and Schur proved a theorem according to which the isoclasses of irreducible representation in M_{K}(n,r) are in bijection with partitions λ of r in not more than n parts. One can also compute the characters of the irreducible representations in terms of *Schur functions*. The situation is more complicated in characteristic p. Green associates with every such λ two explicitly defined modules D_{λ,K} and V_{λ,K} in M_{K}(n,r). The minimal submodule F_{λ,K} of D_{λ,K} is isomorphic to the maximal factor module of V_{λ,K}. All irreducible representation have the form F_{λ,K}, but they are difficult to compute in general.

Our goal is to understand these results. We follow Green's excellent book which explains the above material very well. It is partly based on Schur's dissertation. We concentrate on the principal part of the book, i.e., chapters 1 - 6.

The school is aimed at graduate students and postdocs. We do not expect the participants to be experts in the field and encourage participants with diverse backgrounds to take part. Most of the talks will be held by the participants (following the tradition of previous spring/summer schools in representation theory): every participant chooses a talk and prepares his/her talk in advance. People from the same university may work together.

Additionally, there will be a lecture series by Professor Dr. Steffen König.

The main reference is the following book:

- J.A. Green:
*Polynomial Representations of GL*. Second corrected and augmented edition. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. Schocker. Lecture Notes in Mathematics, 830. Springer, Berlin, 2007._{n}

The school starts on Monday, August 22, at 12.30 and ends Friday, August 26, after lunch. A preliminary program may be found here.

- Reiner Hermann (Bielefeld)
- Philipp Lampe (Bielefeld)
- Phillip Linke (Bielefeld)

Accomodation and meals are covered by the DFG Sonderforschungsbereich 701 "Spektrale Strukturen und Topologische Methoden in der Mathematik". Travel costs can not be covered.

The summer school takes place at the Hotel Waldcafé Jäger, Waldstraße 1, 33014 Bad Driburg. The hotel
is in walking distance from the train station of Bad Driburg. The seminar room in the hotel is equipped with a whiteboard and video and overhead projectors.

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If you would like to participate in the summer school please email one of the organizers. Please include a brief description of your background and possible talks you are interested in. Please also indicate special meal requests. Deadline for application is 24.07.2011. By capacity restrictions the number of participants is limited to 20.