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Separating invariants for multisymmetric polynomials


Authors: Artem Lopatin and Fabian Reimers
Journal: Proc. Amer. Math. Soc. 149 (2021), 497-508
MSC (2020): Primary 13A50, 16R30, 20B30
DOI: https://doi.org/10.1090/proc/15292
Published electronically: December 14, 2020
MathSciNet review: 4198060
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Abstract: This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb {K}$. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on $\lfloor \frac {n}{2} \rfloor + 1$ sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for $n \leq 4$ we explicitly give minimal separating sets (with respect to inclusion) for all $m$ in case $\text {char}(\mathbb {K}) = 0$ or $\text {char}(\mathbb {K}) > n$.


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Additional Information

Artem Lopatin
Affiliation: Department of Mathematics, State University of Campinas, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil
MR Author ID: 740650
Email: artem\textunderscore lopatin@yahoo.com

Fabian Reimers
Affiliation: Department of Mathematics, Technische Universität München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching, Germany
MR Author ID: 1271373
ORCID: 0000-0003-4552-8186
Email: reimers@ma.tum.de

Keywords: Invariant theory, separating invariants, symmetric group, multisymmetric polynomials
Received by editor(s): November 12, 2019
Received by editor(s) in revised form: April 9, 2020
Published electronically: December 14, 2020
Additional Notes: The first author was supported by FAPESP 2019/10821-8. We are grateful for this support.
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2020 American Mathematical Society