Symmetric Ideals, Specht Polynomials and Solutions to Symmetric Systems of Equations

Philippe Moustrou (1) , Cordian Riener (2) , Hugues Verdure (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 UiT - The Arctic University of Norway [Tromsø, Norway]

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.

Domaines: Mathématiques [math]; Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]

Philippe Moustrou, Cordian Riener, Hugues Verdure. Symmetric Ideals, Specht Polynomials and Solutions to Symmetric Systems of Equations. 2019. ⟨hal-02436751⟩