• shareshare
  • link
  • cite
  • add
auto_awesome_motion View all 5 versions
Other research product . Other ORP type . 2019

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

Moustrou, Philippe; Riener, Cordian; Verdure, Hugues;
Open Access
Published: 11 Dec 2019
Publisher: HAL CCSD
Country: 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.

Subjects by Vocabulary

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]

25 references, page 1 of 3

[1] S. Basu and C. Riener. On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities. International Mathematics Research Notices, to appear. 1

[2] G. Blekherman and C. Riener. Symmetric nonnegative forms and sums of squares. arXiv preprint arXiv:1205.3102, 2012. 10, 11 [OpenAIRE]

[3] T. Brylawski. The lattice of integer partitions. Discrete mathematics, 6(3):201{219, 1973. 4 [OpenAIRE]

[4] L. Bus and A. Karasoulou. Resultant of an equivariant polynomial system with respect to the symmetric group. Journal of Symbolic Computation, 76:142 { 157, 2016. 1 [OpenAIRE]

[5] T. Church, J. S. Ellenberg, B. Farb, et al. Fi-modules and stability for representations of symmetric groups. Duke Mathematical Journal, 164(9):1833{1910, 2015. 10

[6] J.-C. Faugere and J. Svartz. Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of n vortices in the plane. In Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, pages 170{ 178. ACM, 2012. 1 [OpenAIRE]

[7] R. Froberg and B. Shapiro. On vandermonde varieties. Math. Scand, 119(1):7391, 2016. 4

[8] C. Goel, S. Kuhlmann, and B. Reznick. On the choi{lam analogue of hilbert's 1888 theorem for symmetric forms. Linear Algebra and its Applications, 496:114{120, 2016. 10

[9] D. Jibetean and M. Laurent. Semide nite approximations for global unconstrained polynomial optimization. SIAM Journal on Optimization, 16(2):490{514, 2005. Pagination: 25. 11 [OpenAIRE]

[10] R. Krone. Equivariant grobner bases of symmetric toric ideals. In Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pages 311{318. ACM, 2016. 1

Funded by
Polynomial Optimization, Efficiency through Moments and Algebra
  • Funder: European Commission (EC)
  • Project Code: 813211
  • Funding stream: H2020 | MSCA-ITN-ETN
Download fromView all 2 sources
Oskar Bordeaux
Other ORP type . 2019
Providers: Oskar Bordeaux