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It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree $p^k,2p^k,\dots,(q-1)p^k$, $k=0,1,2,\dots$ has the same value on them.
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F. Vaccarino, The ring of multisymmetric functions, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 3, 717-731
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M. Domokos, Vector invariants of a class of pseudorelflection groups and multisymmetric syzygies, J. Lie Theory 19 (2009), 507-525
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H. Derksen, G. Kemper, Computational Invariant Theory, Second Edition, Encyclopaedia of Mathematical Sciences 130, Invariant Theory of Algebraic Transformation Groups VIII, Springer-Verlag, Berlin, Heidelberg, 2015
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F. Reimers, Separating invariants for two copies of the natural S n S_{n} -action, Comm. Alg. 48 (2020), 1584-1590
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A. Lopatin, F. Reimers, Separating invariants for multisymmetric polynomials, Proc. Amer. Math. Soc. 149 (2021), 497-508
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G. Kemper, A. Lopatin, F. Reimers, Separating invariants over finite fields, J. Pure Appl. Alg. 226 (2022), paper no. 106904
2022
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