On the phase retrievability of irreducible representations of finite groups

Let C be a finite group and it : C-* U(V) be an irreducible representation of C on a complex Hilbert space V. In this paper we study the phase retrieval property of it and the existence of maximal spanning vectors for (it, C, V ). By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form v (R) v for the representation (it (R) it & lowast;, C, V (R) V ), we show that if it is unramified, in the sense that each irreducible component of it (R) it & lowast; has multiplicity one, then it admits maximal spanning vectors and hence does phase retrieval. Moreover, if GA(1, q) is the one-dimensional affine group over the finite field Fq and it : GA(1, q)-* U(Cq-1) is the unique (q- 1)-dimensional irreducible representation of GA(1, q) (which is ramified), we give a characterization of maximal spanning vectors for (it, GA(1, q), Cq-1) by a detailed study of the adjoint representation of GA(1, q) on L2(GA(1, q)). In particular, we show that the set of maximal spanning vectors are open dense in Cq-1 and the representation (it, GA(1, q), Cq-1) does phase retrieval. Furthermore, we show that the special representations and the cuspidal representations of GL2(Fq) admit maximal spanning vectors and do phase retrieval. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.