The Hilbert series of ${\mathrm{SL}}_2$-invariants

Abstract

Let $V$ be a finite dimensional representations of the group ${\mathrm{SL}}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=\mathbb{C}[V{]}^{{\mathrm{SL}}_2}$ be the algebra of ${\mathrm{SL}}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series ${\mathrm{Hilb}}_R(t)={\sum }_{n\ge 0}\dim (R_n)t^n$ as well as formulas for the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_R(t)$ at $t=1$.

33 pages. From v1: added Section 6, minor revisions. From v2: added computations of ${\gamma }_2$ and ${\gamma }_3$, minor revisions and corrections

References in corpus (4)

• The Hilbert series of a linear symplectic circle quotient
• On compositions with $x^2/(1-x)$
• The Hilbert series and $a$-invariant of circle invariants
• The MAPLE package for $S{L}_2$-invariants and kernel of Weitzenböck derivations

Cited by in corpus (3)

• Hilbert series of symplectic quotients by the 2-torus
• Classifying irreducible fixed points of five scalar fields in perturbation theory
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups