BIREP – Representations of finite dimensional algebras at Bielefeld

Homological structures on the category of strict polynomial functors

I will sketch the program aiming at computing the Ext groups between simple objects in the category P_{d} of strict polynomial functors over a field of positive characteristic. I will describe combinatorial structures (p-cores and p-quotients of Young diagrams) which help to organize computations. Then I will explain how homolgical formality phenomena allows one, using the language of dg-algebras, to relate our situation to representation theory of the Kac-Moody groups A_{n}. If time permits, I will also introduce certain stratification of the derived category of P_{d}, which may be useful in computing cohomology of the symmetric groups.

Polynomial functors (I): classical notions and uses

We will recall the classical notion of polynomial functor from a symmetric monoidal category whose unit is a zero-object (for example, an additive category for direct sum) to an abelian category and its basic properties. We will also recall the use of polynomial functors in algebraic topology and representation theory and their application to stable homology of groups (as general linear groups).

Polynomial functors (II): some recent developments around the global structure

We will give a notion of polynomial functor from a symmetric monoidal category whose unit is an initial object to an abelian category, some structure properties and motivations to look at these functors (this is a joint work with Christine Vespa). We will also give some finiteness properties for polynomial functors in this setting.

Generalizing Schur algebras

Assume A is a finite-dimensional selfinjective algebra. Then projective resolutions of A-modules which are not themselves projective, are always infinite. However, for homological algebra one would prefer finite resolutions. One needs a module which is a generator, and whose endomorphism ring S has finite global dimension. When A is the group algebra of a symmetric group, one can take for S a Schur algebra. We discuss such modules more generally, when A is the group algebra of some finite group.

Representation theory of classical Schur algebras

We recall the definition of the algebras S(n,r) via permutation modules of symmetric groups, and the connection between representations of symmetric groups, and general linear groups via Schur functors. As well we describe the quasi-hereditary structure, which provides another connection with symmetric groups. Much of the recent work is motivated by the problems of finding the irreducible representations of general linear groups, and of symmetric groups. We discuss related open questions.

Symmetric powers, Brauer algebras and Schur algebras

A fundamental result in representation theory of orthogonal and symplectic groups is Schur-Weyl duality. This relates the enveloping algebras of O(n) or Sp(n) on tensor space, that is certain Schur algebras, with the Brauer algebras, by a double centraliser property. Another Schur-Weyl duality relates the Brauer algebras with the endomorphism rings of permutation modules; these are the Schur algebras of Brauer algebras. The third connection in the triangle of Schur functors to be discussed in this talk relates the two classes of Schur algebras, using symmetric powers. (Joint work with Steffen Koenig.)

The category F(q) in generic representation theory is coherent

What is generic representation theory? For a finite field with q elements we consider the category F(q) of functors from finite dimensional vector spaces to all vector spaces. Evaluating a functor in F(q) at a vector space V gives rise to a representations of GL(V). By the Yoneda-lemma we know how certain projectives in F(q) look like. For each vector space V, the representable functor Hom(V,-) is projective. Such a projective is called a standard projective. Note that the standard projectives generate the category F(q). In the 1980s Lionel Schwartz conjectured that all standard projectives are noetherian objects. If true this would imply that every finitely generated functor in F(q) admits a projective resolution by finitely generated projectives. There are partial results that back up this conjecture but no solution so far. In this talk we will not reach quite as far. The aim is to give an idea why the category F(q) is at least coherent. That means that every finitely presented functor admits a resolution by finitely generated projectives. To achieve this we will use certain combinatorial properties of the dimension function associated with each functor in F(q).

Graded Schur algebras and decomposition numbers

I will give a survey of some recent advances in our understanding of the cyclotomic q-Schur algebras, which include the classical Schur algebras studied by Schur and Green. The main theme will be to try and explain what the recently discovered KLR grading gives us which is new and what it might give us in the future.

Issai Schur and his algebraic school in Berlin: known and unknown historical documents, with emphasis on persecution in the Third Reich

Schur's Nachlass is apparently still in private possession in Switzerland and not easily accessible. In particular, the historian has so far basically no private correspondence and it is difficult to reconstruct a picture of Issai Schur, the man. However, there do exist plenty of documents which give insight into teaching, research, and finally emigration of Schur and many of his students. This is due to the size of the school with student numbers up to around 500 as early as 1930, a total of 22 Ph.D.s under Schur, and the influence of Schur's students both in research (above all Richard Brauer and Helmut Wielandt) and teaching and organizational work (above all Alfred Brauer, Hans Rohrbach, and Walter Ledermann). Some other influential mathematicians who were partly influenced by Schur but worked mostly in other areas, notably Hermann Weyl, Karl Löwner, Max Schiffer, and Helmut Grunsky, contributed to the fame of the Schur school. Also Schur's analytic work continues to have impact on modern research. The talk focuses on the time of persecution in and emigration from Nazi Germany. The coincidental parallelism of the international rise of abstract algebra of the Noether school in Göttingen since 1930 and of the nationalistic rule in Germany since 1933 brought with it certain peculiarities into the process of international recognition and reception of the two schools which are historically not yet fully described or analyzed. Some new documents may shed light on this parallelism. But in the center remains Schur, his impact and his suffering.

Graded decomposition numbers of cyclotomic q-Schur algebras

From classical Schur-Weyl duality to quantized skew Howe dualities

Classical Schur-Weyl duality connects the representation theory of the general linear group with the symmetric group and goes back to Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups. A quantization of the duality plays an important role in basic knot theory and yields for instance the famous Jones polynomial. The talk will start from the basic construction and explain interesting generalizations. The tricky point here is the ambiguous role played by the symmetric group as a centralizer group as well as a Weyl group. This leaves us with questions like: what are good generalizations? Are there categorical or geometric interpretations of such dualities? Why is it difficult to quantize such generalization? And why should we care?

Strict polynomial functors in homological algebra computations

I will present some questions of homological/homotopical algebra which are related to the representation theory of Schur algebras. To adress these questions, it is useful to describe representations of Schur algebras as strict polynomial functors (in the sense of Friedlander and Suslin). I will explain how the difference between the notion of polynomial functor and a strict polynomial functor can be used to obtain some information about the integral torsion of some classical homological invariants.

Polynomial functors of prime degree

Baues, Dreckmann, Franjou and Pirashvili showed that the category of degree d polynomial functors from free abelian groups to modules over a commutative ring R is equivalent to the category of modules over some R-algebra A. Drozd computed A in case of degree 2 and 3, and R being the 2-adic integers resp. the 3-adic integers. He conjectured a general behaviour in degree p and R being the p-adic integers for any prime p. We show Drozd's conjecture and give an idea of the methods of its proof. This is joint work with Steffen Koenig.