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THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

Part of: Rings and algebras with additional structure Basic linear algebra Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  04 December 2019

G. I. LEHRER Open the ORCID record for G. I. LEHRER [Opens in a new window]  and
R. B. ZHANG Open the ORCID record for R. B. ZHANG [Opens in a new window]
G. I. LEHRER
Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W.2006, Australia email gustav.lehrer@sydney.edu.au
R. B. ZHANG
Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W.2006, Australia email ruibin.zhang@sydney.edu.au
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Abstract

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$.

MSC classification

Primary: 16W22: Actions of groups and semigroups; invariant theory 15A72: Vector and tensor algebra, theory of invariants 17B20: Simple, semisimple, reductive (super)algebras
Type
Article
Information
Nagoya Mathematical Journal , Volume 242 , June 2021 , pp. 52 - 76
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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Footnotes

This work has been supported by the Australian Research Council.

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