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Title: Polynomial representations of symplectic groups
Author: Iano-Fletcher, Maria
ISNI:       0000 0001 3585 8267
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1990
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In Chapter 1 we review some of the classical theory of reductive algebraic groups over an algebraically closed field. In Chapter 2 we summarise the work of Green on polynomial representations of GLn(k) where k is an infinite field. The irreducible polynomial representations of degree r of GLn(k) are parametrised by partitions λ of r. The irreducible polynomial GLn(k)-module F λ k can be obtained in two ways, as the quotient of a Weyl module V λ k by its unique maximal submodule, or as the unique minimal submodule of a Shur module D λ k. The two modules V λ k and D λ k are duals of one another. We describe Green's basis of D λ k consisting of one element DT for each semistandard λ-tableau T. We also describe the results of Carter and Lusztig on Weyl modules and the Carter-Lusztig basis of V λ k consisting of one element ψτ for each semistandard λ-tableau T. The aim of this thesis is to extend Green's work to polynomial representations of symplectic groups over an infinite field. The basis facts about symplectic groups are described in Chapter 3. In Chapter 4 we introduce the idea of symplectic tableaux due to R. C. King. The dimensions of the Weyl module and the Shur module is equal to the number of symplectic λ-tableaux. We shall use symplectic tableaux to parametrise basis vectors of the Weyl and Shur modules. Chapter 5 is a detailed study of the Weyl module. In Chapter 6 we consider the Shur module.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics