An investigation on symmetric polynomials
Abstract
A polynomial p(x_1,…,x_n ) in the polynomial algebra K[X_n ]=K[x_1,…,x_n ] in n commuting variables is called symmetric if p(x_1,…,x_n )=p(x_σ(1) ,…,x_σ(n) ) for each permutation σ in the symmetric group S_n. The set 〖K[X_n ]〗^(S_n ) of symmetric polynomials is the algebra of S_n-invariants. The algebra 〖K[X_n ]〗^(S_n ) is generated by n algebraically independent elements called elementary symmetric polynomials. The idea in the commutative algebra K[X_n ] was generalized for noncommutative relatively free algebras F_n of rank n; and generating sets for the algebra F_n^(S_n ) were investigated. In this study, we review recent works on symmetric polynomials in the free metabelian associative algebras and the free metabelian Lie algebras.
Keywords
Supporting Institution
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Ethical Statement
In the study, the authors declare that there is no violation of research and publication ethics and that the study does not require ethics committee approval.
References
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Details
Primary Language
English
Subjects
Algebra and Number Theory
Journal Section
Review
Early Pub Date
December 26, 2025
Publication Date
December 29, 2025
Submission Date
September 3, 2025
Acceptance Date
September 19, 2025
Published in Issue
Year 2025 Volume: 1 Number: 1