Representations and invariants of reductive groups

Supervisor(s)

Contact Dr Rudolf Tange to discuss this project further informally.

Project description

This projects concerns certain representations of groups like the general linear group or the symplectic group over an algebraically closed field.There are three possible problems:

A) The truncated coordinate ring of the Lie algebra g of a reductive group is the quotient of the coordinate ring k[g] by the ideal generated by the p-th powers of the linear functions on g. The problem is to find a spanning set for the invariants in this ring under the conjugation action. The invariants in the truncated coordinate ring of the Lie algebra are directly related to the centre of the restricted enveloping algebra in characterictic p which is also of interest in representation theory.

B) Describe the highest weight vectors in the exterior algebra of the Lie algebra of a classical group in characteristic zero, as a module over the invariants. The method of "transmutation" will most likely be useful here, see [1].

C) Study the relation between the representation theories of the Brauer algebra and the symplectic group. In [2] a functor was given between a category of modules for the symplectic group and the modules for the Brauer algebra. This functor has only been studied under an assumption on the parameters and this limits its applicability.

Problems A and B are strongly motivated by classical invariant theory. 

References:
[1] R. Tange, Highest weight vectors and transmutation, Transformation Groups 
[2] S. Donkin and R.Tange, The Brauer algebra and the symplectic Schur algebra, Math. Zeit. 265 (2010)

 

Entry requirements

Applicants should have, or expect to obtain, a UK first class honours degree in Mathematics or a related discipline, or equivalent.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Representations and invariants of reductive groups’ as well as Dr Rudolf Tange as your proposed supervisor.

We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates.  All scholarships will be awarded on the basis of merit.

If you require any further information please contact the Graduate School Office, e: maps.pgr.admissions@leeds.ac.uk.