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We consider a set \({\mathfrak R}{\mathfrak S}\left( \lambda ,S_n\right) \) of self- and scew-reciprocal polynomials of degree mn in \(\lambda \) , based on polynomial invariants \(I_{n,r}(\textbf{x}^n)\) of symmetric group \(S_n\) , acting on the Euclidean space \({\mathbb E}^n\) over the field of real numbers \({\mathbb R}\) , where \(\textbf{x}^\textbf{n}=\{x_1,\ldots ,x_n\}\in {\mathbb E}^n\) . We prove that \({\mathfrak R}{\mathfrak S}\left( \lambda ,S_n\right) \) exhibits a commutative monoid under multiplication. Real solutions \(\lambda \left( \textbf{x}^\textbf{n}\right) \) of scew-reciprocal equations have many remarkable properties: a homogeneity of the 1st order, a self-dual symmetry under inversion of variables \(x_i\rightarrow x_i^{-1}\) and function \(\lambda \rightarrow \lambda ^{-1}\) , a monotony of \(\lambda \left( \textbf{x}^\textbf{n}\right) \) with respect to every \(x_i\) and others.

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Commutative Monoid of Self-Dual Symmetric Polynomials

  • 会议论文

  • 发表日期：

    2025年

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  • Leonid G. Fel

摘要

We consider a set \({\mathfrak R}{\mathfrak S}\left( \lambda ,S_n\right) \) of self- and scew-reciprocal polynomials of degree mn in \(\lambda \) , based on polynomial invariants \(I_{n,r}(\textbf{x}^n)\) of symmetric group \(S_n\) , acting on the Euclidean space \({\mathbb E}^n\) over the field of real numbers \({\mathbb R}\) , where \(\textbf{x}^\textbf{n}=\{x_1,\ldots ,x_n\}\in {\mathbb E}^n\) . We prove that \({\mathfrak R}{\mathfrak S}\left( \lambda ,S_n\right) \) exhibits a commutative monoid under multiplication. Real solutions \(\lambda \left( \textbf{x}^\textbf{n}\right) \) of scew-reciprocal equations have many remarkable properties: a homogeneity of the 1st order, a self-dual symmetry under inversion of variables \(x_i\rightarrow x_i^{-1}\) and function \(\lambda \rightarrow \lambda ^{-1}\) , a monotony of \(\lambda \left( \textbf{x}^\textbf{n}\right) \) with respect to every \(x_i\) and others.