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

TL;DR

This paper explores the relationship between symmetric polynomial ideals, Specht polynomials, and solutions to symmetric systems of equations, revealing new insights into their algebraic structure and solution sets.

Contribution

It establishes a connection between symmetric ideals and Specht ideals, linking leading monomials to Specht polynomials and applying this to analyze solutions and representations.

Findings

Connection between leading monomials and Specht polynomials

Insights into solutions of symmetric polynomial systems

Restrictions on isotypic decomposition of symmetric ideals

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.