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Representations and invariants of reductive groups

  • Full or part time
  • Application Deadline
    Applications accepted all year round
  • Self-Funded PhD Students Only
    Self-Funded PhD Students Only

Project Description

This project 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)

Funding Notes

This project is open to self-funded students and is eligible for funding from the School of Mathematics Scholarships, EPSRC Doctoral Training Partnerships, and Leeds Doctoral Scholarships.

All successful UK/EU and international applicants will be considered for funding, in an open competition across the School of Mathematics. To be considered for this funding, it is recommended to apply no later than 31 March 2019 for funding to start in October 2019. However, earlier applications are welcome, and will be considered on an ongoing basis.

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