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

Austin Barringer, Hans-Christian Herbig, Daniel Herden, Saad Khalid, Christopher Seaton, Lawton Walker

Abstract

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group $\operatorname{O}_2$. 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.

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 . 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.
Paper Structure (14 sections, 14 theorems, 112 equations)

This paper contains 14 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Invariants and covariants of $\mathbb{S}^1$
  4. The maximally graded Hilbert series
  5. The $\mathbb{N}^2$-grading of a cotangent-lifted representation and the symplectic quotient
  6. Laurent coefficients of the univariate Hilbert series
  7. Laurent coefficients of multigraded Hilbert series
  8. Invariants and covariants of $\operatorname{O}_2(\mathbb{R})$
  9. The irreducible representations of $\operatorname{O}_2(\mathbb{R})$
  10. The Hilbert series of covariants of $\operatorname{O}_2$
  11. Symplectic quotients by $\operatorname{O}_2$
  12. The Laurent coefficients for symplectic quotients by $\operatorname{O}_2(\mathbb{R})$
  13. Other semidirect products of $\mathbb{S}^1$ by finite groups
  14. Algorithms to compute the Hilbert series

Key Result

Theorem 2.1

Let $G$ be a reductive group over $\mathbb{C}$ and $K$ a maximally compact subgroup. Let $V = V_1\oplus V_2\oplus\cdots\oplus V_n$ and $W$ be finite-dimensional rational representations of $G$ over $\mathbb{C}$. Then the multigraded Hilbert series of the module of covariants $\operatorname{Hom}(V,W) where $\boldsymbol{t} = (t_1,\ldots,t_n)$, $\mu$ is a normalized Haar measure on $K$, $\chi_W$ is t

Theorems & Definitions (29)

  • Theorem 2.1: Molien-Weyl Theorem
  • Remark 3.1
  • Theorem 3.2: Maximally graded Hilbert series of $\mathbb{S}^1$-covariants
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • ...and 19 more