Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank 1 Lie groups

Abstract

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group ${\mathrm{O}}_2(\mathbb{R})$. The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series.

Communicated by Ellen Kirkman

KEYWORDS:

• Circle invariants
• circle representation
• Hilbert series
• multigrading
• multigraded Hilbert series
• ${\mathrm{O}}_2(\mathbb{R})$-invariants
• ${\mathrm{O}}_2(\mathbb{R})$-representation
• symplectic reduction

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A50
• Secondary: 05A15
• 14L30
• 53D20

Acknowledgments

H.-C.H. and C.S. would like to thank Baylor University, D.H. and C.S. would like to thank the Instituto de Matemática Pura e Aplicada (IMPA), and H.-C.H. and D.H. would like to thank Rhodes College for hospitality during work contained here. This paper developed from A.B., S.K., and L.W.’s senior seminar projects in the Rhodes College Department of Mathematics and Computer Science, and the authors gratefully acknowledge the support of the department and college for these activities. A.B., S.K., and L.W. express appreciation to the Rhodes College Summer Research Fellowship program, H.-C.H. to CNPq through the Plataforma Integrada Carlos Chagas, and C.S. to the E.C. Ellett Professorship in Mathematics and the Rhodes College sabbatical program, for financial support.

Disclosure statement

The authors report there are no competing interests to declare.

Additional information

Funding

A.B., S.K., and L.W. were supported by Rhodes College Summer Research Fellowships. H.-C.H. was supported by CNPq through the Plataforma Integrada Carlos Chagas. C.S. was supported by the E.C. Ellett Professorship in Mathematics.