Representation theory of Schur algebras
Start date: 01 Sep 2007,
End date: 31 Aug 2009
Two of the central problems in the representation theory of reductive algebraic groups are, firstly, to understand simple representations and their characters (in particular, to determine the so-called decomposition numbers), and, secondly, to determine ex tensions between simple representations, thus aiming to describe indecomposable ones.Schur algebras are a class of finite-dimensional algebras whose representation theory encapsulates the representation theory of (infinite) reductive algebraic groups. This project will focus on the ring theoretic structure and cohomological properties of Schur algebras and will cover both characteristic independent and characteristic dependent situations. This algebraic approach is different from the geometric approach, as taken in Kazhdan-Lusztig theory, where small characteristics are not covered.The proposed research will combine two main objectives.- The first objective will provide a practicable description of a certain triangulated subcategory of the derived category of hereditary which - applied to Schur algebras - will allow us to explicitly compute extensions between Weyl modules, thus gaining information about decomposition numbers.- The second objective will then focus on structural information on Schur algebras, namely Morita equivalences between sub-and factor algebras and in particular between blocks.The project will be carried out at the Mathematical Institute in Oxford under the supervision of Dr Erdmann and Dr Henke, who are both experienced researchers in the field.
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