Tange, R orcid.org/0000-0003-0867-1573 (2019) HIGHEST WEIGHT VECTORS AND TRANSMUTATION. Transformation Groups, 24 (2). pp. 563-588. ISSN 1083-4362
Abstract
Let G = GLn be the general linear group over an algebraically closed field k, let g=gln be its Lie algebra and let U be the subgroup of G which consists of the upper uni-triangular matrices. Let k[g] be the algebra of polynomial functions on g and let k[g]G be the algebra of invariants under the conjugation action of G. We consider the problem of giving finite homogeneous spanning sets for the k[g]G -modules of highest weight vectors for the conjugation action on k[g] . We prove a general result in arbitrary characteristic which reduces the problem to giving spanning sets for the vector spaces of highest weight vectors for the action of GLr × GLs on tuples of r × s matrices. This requires the technique called “transmutation” by R. Brylinsky which is based on an instance of Howe duality. In characteristic zero, we give for all dominant weights χ ∈ ℤn finite homogeneous spanning sets for the k[g]G -modules k[g]Uχ of highest weight vectors. This result was already stated by J. F. Donin, but he only gave proofs for his related results on skew representations for the symmetric group. We do the same for tuples of n × n-matrices under the diagonal conjugation action.
Metadata
Item Type: | Article |
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Authors/Creators: | |
Copyright, Publisher and Additional Information: | (c) The Author, 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons org . /licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 19 Oct 2017 10:44 |
Last Modified: | 23 Jun 2023 22:37 |
Status: | Published |
Publisher: | Springer US |
Identification Number: | 10.1007/s00031-018-9474-9 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:122803 |
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