Mathematisches Forschungsinstitut Oberwolfach Mini-workshop: Schur Algebras and Quantum Groups Contents 1 the Workshop. 1 2 the Subject Area. 2 Introduction to Q-schur Algebras

1 The workshop. This mini{workshop aimed at discussing existent and potential connections between certain algebraic and geometric objects, in particular Schur algebras and quantum groups. The main focus was on the known epimorphisms from type A quantized enveloping algebras to Schur algebras, their restriction to Hall algebras and their potential analogues for other types. A description of these problems and a reading list had been circulated beforehand. At the workshop, there were survey lectures and more specialized presentations, including suggestions to attack the central problems (some of them stimulated by the material 1 distributed earlier). During the discussions, new links were discovered between diierent approaches, proofs were simpliied and a conjecture arose on the type C situation. The schedule was very exible and no time constraints were imposed on speakers; this format turned out to be very popular both with speakers and audience. There was general agreement that mini{workshops provide a very welcome new format for stimulating research meetings. There have been 15 participants from Australia, France, Germany, the UK and the USA. 27 per cent of the participants were female, the majority of the participants was younger than 35. 2 The subject area. Schur algebras in arbitrary characteristic (over innnite elds) were introduced 1980 in J.A.Green's innuential monograph 17] on polynomial representations of the general linear group. Schur himself had considered the characteristic zero situation in his thesis. Green also introduced integral Schur algebras and insisted on characteristic free notions (eg codeterminant basis for Weyl modules). Schur algebras over nite elds started to be used only very recently (by topologists in the context of functor cohomology, or topological Hochschild cohomology, see 16]). Classical Schur algebras of type A, as introduced by Green, cover the polynomial representation theory of the general linear group over an innnite eld. Donkin 10, 11] introduced more general Schur algebras which cover the rational representation theory of reductive groups. These algebras somehow lack the nice combinatorial theory of the type A{situation where the symmetric group (which on that occasion does not wear its usual Weyl group hat) is of much help. There is some information available on Brauer algebras, which replace the symmetric group in types B and C, but not much, and type D seems to be completely unknown. Analogues of many of Green's main results (standard basis with multiplication formula, bases for Weyl modules, Schur functor, etc) seem to be unknown in the …