Phase retrieval for nilpotent groups

TL;DR

This paper investigates the phase retrieval property for irreducible representations of nilpotent groups, establishing that all such representations in certain classes, including specific Lie and finite p-groups, do phase retrieval.

Contribution

It proves that all irreducible representations of simply connected nilpotent Lie groups and certain finite p-groups have the phase retrieval property, using inductive proof methods.

Findings

All irreducible representations of simply connected nilpotent Lie groups do phase retrieval.

Irreducible representations of p-groups with exponent p and size ≤ p^{2+p/2} do phase retrieval.

Inductive proof methods are effective across different group settings.

Abstract

We study the phase retrieval property for orbits of general irreducible representations of nilpotent groups, for the classes of simply connected connected Lie groups, and for finite groups. We prove by induction that in the Lie group case, all irreducible representations do phase retrieval. For the finite group case, we mostly focus on $p$-groups. Here our main result states that every irreducible representation of an arbitrary $p$-group with exponent $p$ and size $\le p^{2+p/2}$ does phase retrieval. Despite the fundamental differences between the two settings, our inductive proof methods are remarkably similar.