Hilbert series associated to symplectic quotients by ${\mathrm{SU}}_2$

Abstract

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an ${\mathrm{SU}}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t=1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary ${\mathrm{Sl}}_2$- or ${\mathrm{SU}}_2$-module as well its first three Laurent coefficients.

23 pages, 3 figures, 1 table

References in corpus (3)

• Counting Gauge Invariants: the Plethystic Program
• Symplectic quotients have symplectic singularities
• The Hilbert series and $a$-invariant of circle invariants

Cited by in corpus (3)

• Hilbert series of symplectic quotients by the 2-torus
• The partial derivative of ratios of Schur polynomials and applications to symplectic quotients
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups