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Highest weight vectors for classical reductive groups

Dent, Adam David John (2018) Highest weight vectors for classical reductive groups. PhD thesis, University of Leeds.

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Abstract

A result by Tange from 2015 gave bases for the spaces of highest weight vectors for the action of GL_r×GL_s on k[Mat_{rs}^m] over a field of characteristic zero, and in arbitrary characteristic for certain weights; here, we generalise this to give bases for the spaces of highest weight vectors in k[Mat_{rs}^m] of any given weight in arbitrary characteristic. The motivation for this is to apply the technique of transmutation to describe the highest weight vectors for the conjugation action of GL_n on k[Mat_n]. Then, we use similar methods but in characteristic zero to describe finite spanning sets for the spaces of highest weight vectors for a certain polynomial action of GL_r on k[Mat_r^l] (derived from the GL_r-action on Mat_r given by g·A=gAg^T), and apply this to the conjugation action of the symplectic group Sp_n on k[sp_n].

Item Type: Thesis (PhD)
Keywords: representation theory, algebra, noncommutative, infinite-dimensional, algebraic geometry, invariant theory, characteristic p, symplectic group, Young tableau, bideterminant, transmutation, Specht module
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.794157
Depositing User: Dr Adam David John Dent
Date Deposited: 08 Jan 2020 12:22
Last Modified: 18 Feb 2020 12:51
URI: http://etheses.whiterose.ac.uk/id/eprint/24322

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