{"title":"On the phase retrievability of irreducible representations of finite groups","authors":"Chuangxun Cheng","doi":"10.1016/j.laa.2025.03.011","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and <span><math><mi>π</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo>(</mo><mi>V</mi><mo>)</mo></math></span> be an irreducible representation of <em>G</em> on a complex Hilbert space <em>V</em>. In this paper we study the phase retrieval property of <em>π</em> and the existence of maximal spanning vectors for <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>G</mi><mo>,</mo><mi>V</mi><mo>)</mo></math></span>. By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form <span><math><mi>v</mi><mo>⊗</mo><mi>v</mi></math></span> for the representation <span><math><mo>(</mo><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mi>G</mi><mo>,</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>)</mo></math></span>, we show that if <em>π</em> is unramified, in the sense that each irreducible component of <span><math><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has multiplicity one, then <em>π</em> admits maximal spanning vectors and hence does phase retrieval. Moreover, if <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> is the one-dimensional affine group over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><mi>π</mi><mo>:</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>→</mo><mi>U</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> is the unique <span><math><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional irreducible representation of <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> (which is ramified), we give a characterization of maximal spanning vectors for <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> by a detailed study of the adjoint representation of <span><math><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span>. In particular, we show that the set of maximal spanning vectors are open dense in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and the representation <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>GA</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> does phase retrieval. Furthermore, we show that the special representations and the cuspidal representations of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> admit maximal spanning vectors and do phase retrieval.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 64-95"},"PeriodicalIF":1.0000,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525001120","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a finite group and 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 . By translating the existence of maximal spanning vectors into the existence of cyclic vectors of the form for the representation , we show that if π is unramified, in the sense that each irreducible component of has multiplicity one, then π admits maximal spanning vectors and hence does phase retrieval. Moreover, if is the one-dimensional affine group over the finite field and is the unique -dimensional irreducible representation of (which is ramified), we give a characterization of maximal spanning vectors for by a detailed study of the adjoint representation of on . In particular, we show that the set of maximal spanning vectors are open dense in and the representation does phase retrieval. Furthermore, we show that the special representations and the cuspidal representations of admit maximal spanning vectors and do phase retrieval.
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.