On the phase retrievability of irreducible representations of finite groups

Let G be a finite group and $\pi :G\to U\left(V\right)$ be an irreducible representation of G on a complex Hilbert space V. In this paper we study the phase retrieval property of π and the existence of maximal spanning vectors for $(\pi ,G,V)$. By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form $v\otimes v$ for the representation $(\pi \otimes {\pi }^{⁎},G,V\otimes V)$, we show that if π is unramified, in the sense that each irreducible component of $\pi \otimes {\pi }^{⁎}$ has multiplicity one, then π admits maximal spanning vectors and hence does phase retrieval. Moreover, if $\mathrm{GA}(1,q)$ is the one-dimensional affine group over the finite field $F_q$ and $\pi :\mathrm{GA}(1,q)\to U\left(C^{q-1}\right)$ is the unique $(q-1)$-dimensional irreducible representation of $\mathrm{GA}(1,q)$ (which is ramified), we give a characterization of maximal spanning vectors for $(\pi ,\mathrm{GA}(1,q),C^{q-1})$ by a detailed study of the adjoint representation of $\mathrm{GA}(1,q)$ on $L^2\left(\mathrm{GA}\right(1,q\left)\right)$. In particular, we show that the set of maximal spanning vectors are open dense in $C^{q-1}$ and the representation $(\pi ,\mathrm{GA}(1,q),C^{q-1})$ does phase retrieval. Furthermore, we show that the special representations and the cuspidal representations of ${\mathrm{GL}}_2\left(F_q\right)$ admit maximal spanning vectors and do phase retrieval.