Abstract
We show Laplacian algebras are maximal, and give applications to the Classical Invariant Theory of real orthogonal representations of compact groups, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is a generating set. And the introduction of a class of generalized polarizations which, in a certain class of representations (including all representations of finite groups), always generates the algebra of invariants of their diagonal representations.
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Acknowledgements
It is a pleasure to thank Harm Derksen for pointing out [22], which was the main inspiration for the proof of Theorem A, and Matyas Domokos for a simplification of the proof of Corollary E using [2]. We would also like to thank Alexander Lytchak and Matyas Domokos for suggestions that improved the exposition.
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The first-named author has been supported by the NSF grant DMS-2005373, and the second-named author by NSF 1810913.
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Mendes, R.A.E., Radeschi, M. Maximality of Laplacian algebras, with applications to Invariant Theory. Annali di Matematica 202, 1011–1031 (2023). https://doi.org/10.1007/s10231-022-01269-9
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DOI: https://doi.org/10.1007/s10231-022-01269-9