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Symmetric Ideals, Specht Polynomials and Solutions to Symmetric Systems of Equations
Symmetric Ideals, Specht Polynomials and Solutions to Symmetric Systems of Equations
This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 813211 (POEMA); 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.
- UNIVERSITE PARIS DESCARTES France
- Université Paris Diderot France
- University of Bordeaux France
arXiv: Mathematics::Commutative Algebra Mathematics::Representation Theory
[MATH]Mathematics [math], [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
[MATH]Mathematics [math], [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
arXiv: Mathematics::Commutative Algebra Mathematics::Representation Theory
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- UNIVERSITE PARIS DESCARTES France
- Université Paris Diderot France
- University of Bordeaux France
This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 813211 (POEMA); 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.