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Proceedings of the London Mathematical Society Advance Access published online on October 27, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp040
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© 2009 London Mathematical Society

Homotopy, homology, and GL2

Vanessa Miemietz

Mathematical Institute
University of Oxford
24–29 St Giles’
Oxford
OX1 3LB
United Kingdom

Will Turner

Department of Mathematics
University of Aberdeen
Fraser Noble Building
King's College
Aberdeen
AB24 3UE
United Kingdom
w.turner@abdn.ac.uk

Received 23 June 2008. Revision received 10 June 2009.

We define weak 2-categories of finite-dimensional algebras with bimodules, along with collections of operators O(c, x) on these 2-categories. We prove that special examples Op of these operators control all homological aspects of the rational representation theory of the algebraic group GL2, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators O(c, x) honour derived equivalences in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight Z+-gradings on Schur algebras S(2, r), and the existence of braid group actions on the derived categories of blocks of these Schur algebras.

2000 Mathematics Subject Classification 16E45, 16W50, 16S37, 20G05.

The first author acknowledges support from the ERC (MEIF-CT-2007-039983).


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