On the phase retrievability of irreducible representations of finite groups

IF 1 3区 数学 Q1 MATHEMATICS
Chuangxun Cheng
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引用次数: 0

Abstract

Let G be a finite group and π:GU(V) 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 (π,G,V). By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form vv for the representation (ππ,G,VV), we show that if π is unramified, in the sense that each irreducible component of ππ has multiplicity one, then π 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 π:GA(1,q)U(Cq1) is the unique (q1)-dimensional irreducible representation of GA(1,q) (which is ramified), we give a characterization of maximal spanning vectors for (π,GA(1,q),Cq1) 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 Cq1 and the representation (π,GA(1,q),Cq1) 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.
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
期刊介绍: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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