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

TL;DR

This paper computes the Hilbert series of the algebra of invariants under SL_2 action on a finite-dimensional representation, providing explicit formulas and Laurent expansion coefficients.

Contribution

It offers explicit calculations and formulas for the Hilbert series and Laurent expansion coefficients of SL_2-invariant polynomial algebras.

Findings

Explicit Hilbert series formulas derived

Laurent expansion coefficients at t=1 calculated

Provides tools for understanding SL_2-invariant algebra structures

Abstract

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