Polynomial Functors and Schur Algebras

The programme for the meetings of the study group includes about twelve one-hour lectures, to be given during the four meetings. The meetings will take place at the Universities of Leicester, Oxford and Warwick. The programme will be distributed to all interested, who then choose the topic they want to prepare and lecture about. We encourage to apply for a specific topic in a mini-group of about two to four people, although this is not necessary. The relevant literature is the following. [F1] and [K1] are survey articles. The spectacular new results mentioned above are mostly contained in [FS] and [FFSS]. Here is the list of topics from which the programme will be composed. (Parts I, II and III have to be covered in this order. Material from Parts IV and V can be chosen according to the participants' interests and the time available.)

Part I: Polynomial functors in algebra and topology - introduction and survey (use [F1], [K1] and some of the other papers, eg [K3]). And a detailed example of an injective object in this category ([K3], relating to work of Doty, Kovacs and Krop and giving evidence for the still open `artinian conjecture').

Part II: Strictly polynomial functors and Schur algebras ([FS], chapters 2 and 3, compare J.A.Green's classical book).

Part III: Cohomology ([FFSS], stability and comparison theorems in chapters 1 to 4, and some of the computational results from later chapters - compare the independent , but not unrelated approach in [T], the important earlier work [CPSvdK] and, for some technical points, [F2]).

Part IV: Filtrations and stratifications ([PS], proving a vanishing of cohomology result, and [K4], which introduces new Schur algebras and reproves classical results such as Steinberg's Tensor Product Theorem).

Part V: Group schemes and finite generation of cohomology ([FS] and subsequent work by Friedlander and Suslin and also by Bendel).

Relevant Literature

[FS] E.Friedlander and A.Suslin: Cohomology of finite group schemes over a field. Invent. math. 127 (1997), 209--270.

[FFSS] V.Franjou, E.Friedlander, A.Scorichenko, and A.Suslin: General linear and functor cohomology over finite fields. Ann. of Math. (2) 150 (1999), 663--728.

[F1] E.Friedlander: Geometry of infinitesimal group schemes. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math . 1998, Extra Vol. II, 55--65.

[K1] N.Kuhn: The generic representation theory of finite fields: a survey of basic structure. Infinite length modules (Bielefeld, 1998), 193--212, Trends Math., Birkhäuser, Basel, 2000.

[K2] N.Kuhn: Rational cohomology and cohomological stability in generic representation theory. Amer. J. Math. 120 (1998), 1317--1341.

[K3] N.Kuhn: Invariant subspaces of the ring of functions on a vector space over a finite field. J. Algebra 191 (1997), 212--227.

[K4] N.Kuhn: A stratification of generic representation theory and generalized Schur algebras. Preprint.

[F2] V.Franjou: Extensions entre puissances extrieures et entre puissances symtriques. J. Algebra 179 (1996), no. 2, 501--522.

[PS] L.Piriou and L.Schwartz: Lionel Extensions de foncteurs simples. $K$-Theory 15 (1998), no. 3, 269--291.

[T] B.Totaro: Projective resolutions of representations of ${\rm GL}(n)$. J. Reine Angew. Math. 482 (1997), 1--13.

[CPSvdK] E.Cline, B.Parshall, L.Scott, and W. van der Kallen: Rational and generic cohomology. Invent. Math. 39 (1977), no. 2, 143--163.

Author: Anne Henke , tel: +44 (0)116 252 5237, e-mail: A.Henke@mcs.le.ac.uk, web-page: http://www.mcs.le.ac.uk/~ahenke
Last updated: 15th October 2001
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