Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations. (November 2021)

• Record Type: Journal Article
• Authors: Moustrou, Philippe; Riener, Cordian; Verdure, Hugues
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 106
• Page End: 121
• Publication Date: 2021-11
• Subjects: Symmetric group -- Specht polynomials -- Polynomial equations; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.02.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.