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Unitary Invariants of the Finite Heisenberg Group

Josh Katz

Abstract

Polynomial invariants of a group action often appear only in high degree, and in many representations the invariant ring imposes severe degree constraints before any nontrivial invariants can occur. In contrast, the larger class of unitary invariants -- polynomials in both the variables and their conjugates -- typically exhibits very different behavior, and their separating power is comparatively unexplored. We highlight this contrast in the setting of the finite Heisenberg group $H_N$. Although the polynomial invariant ring $\mathbb{C}[V]^{H_N}$ contains no nontrivial elements below degree $N$, we show that degree-six unitary invariants are already sufficient to separate generic $H_N$-orbits up to a global phase factor. These invariants arise from cubic equations involving the magnitudes of a vector and its discrete Fourier transform. A single polynomial invariant in degree $N$ then resolves the remaining global phase, yielding full generic orbit separation. Our proof utilizes fundamental results from phase retrieval. Along the way we will also explore the utility of unitary invariants in obtaining improved degree bounds for representations of cyclic groups. This paper provides a concrete example in which the minimal separating degree for unitary invariants is dramatically lower than the minimal degree for polynomial invariants.

Unitary Invariants of the Finite Heisenberg Group

Abstract

Polynomial invariants of a group action often appear only in high degree, and in many representations the invariant ring imposes severe degree constraints before any nontrivial invariants can occur. In contrast, the larger class of unitary invariants -- polynomials in both the variables and their conjugates -- typically exhibits very different behavior, and their separating power is comparatively unexplored. We highlight this contrast in the setting of the finite Heisenberg group . Although the polynomial invariant ring contains no nontrivial elements below degree , we show that degree-six unitary invariants are already sufficient to separate generic -orbits up to a global phase factor. These invariants arise from cubic equations involving the magnitudes of a vector and its discrete Fourier transform. A single polynomial invariant in degree then resolves the remaining global phase, yielding full generic orbit separation. Our proof utilizes fundamental results from phase retrieval. Along the way we will also explore the utility of unitary invariants in obtaining improved degree bounds for representations of cyclic groups. This paper provides a concrete example in which the minimal separating degree for unitary invariants is dramatically lower than the minimal degree for polynomial invariants.
Paper Structure (14 sections, 16 theorems, 26 equations)

This paper contains 14 sections, 16 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Motivating examples
  3. Cyclic Groups
  4. The Heisenberg Group
  5. Absence of low-degree polynomial invariants
  6. The Heisenberg Bispectrum
  7. Separation of Heisenberg Orbits
  8. Cyclic bispectrum theory
  9. Recovery of magnitude data
  10. Reduction to classical phase retrieval
  11. Eliminating the global phase
  12. Small example with $N=3$
  13. Summary and Discussion
  14. Open problems

Key Result

Proposition 1.5

Let $G$ be a compact group with $V$ a complex representation. Then $\gamma(V, G)$ is finite if and only if $G$ is finite.

Theorems & Definitions (35)

  • Remark 1.1
  • Definition 1.2
  • Example 1.3: Circle action on bandlimited functions
  • Example 1.4: Circle action with weights $1$ and $2$
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • Lemma 1.7: bandeira2023estimation
  • proof : Proof of Theorem \ref{['thm:unitary-separate']}
  • Example 2.1
  • ...and 25 more