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Polarizations and Nullcone of Representations of Reductive Groups

Kraft, Hanspeter and Wallach, Nolan. (2010) Polarizations and Nullcone of Representations of Reductive Groups. Progress in mathematics, Vol. 278. pp. 153-168.

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Official URL: http://edoc.unibas.ch/dok/A5251078

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Abstract

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f_1,f_2,...,f_n be homogeneous invariant functions. Then the polarizations of f_1,f_2,...,f_n define the nullcone of k 0} h(t) x = 0 for all x in L. This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of $SL_2$ on the n-fold tensor product C^2 otimes C^2 otimes ... otimes C^2. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n <= 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. {bf 224} (2006), 119--140.)
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft)
UniBasel Contributors:Kraft, Hanspeter
Item Type:Article, refereed
Article Subtype:Research Article
Bibsysno:Link to catalogue
Note:Also published in: Symmetry and spaces. - Boston : Birkhäuser, 2010. - S. 153-168 -- Publication type according to Uni Basel Research Database: Journal article
Language:English
Last Modified:21 Jun 2018 14:22
Deposited On:22 Mar 2012 13:59

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