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 and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra
New York City College of Technology, City University of New York（纽约市立学院，纽约市立大学）；Ph.D. Programs in Mathematics and Physics, Graduate Center, City University of New York（数学与物理博士项目，研究生院，纽约市立大学）；Department of Physics, Lakehead University（湖首大学物理系）；Department of Mathematics, University of Toronto（多伦多大学数学系）
