Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations

2019/12/11 by Philippe Moustrou, Cordian Riener, Moustrou, Philippe +3 · 1 citation
Computer Science · Mathematics · #12Y05 #13P15 #20C30 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1912.05266

openalex publication_date 2019/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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