Eigenvalue-less matrices from SL₃(\mathbbF₇) to PSL₃(\mathbbF₇)

2023/03/25 by Juan Lucas Callo, Callo, Juan Lucas, George Z. Chen +13
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #General Mathematics (math.GM)

paper · pdf · doi:10.48550/arxiv.2303.14393

openalex publication_date 2023/03/25 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28

Abstract

We analyse the set of matrices in SL₃(\mathbbF₇) without eigenvalues explicitly, extracting nice bijections between the 18 equally sized conjugacy classes contained within. In doing so, we discover a set of 18 commuting matrices for which every conjugacy class is represented and tells us how to decide when collections of commuting matrices are simultaneously conjugate. The main innovation is the proofs are accessible to undergraduates and do not rely on computer calculations nor very general theory. It is also rare that the details of special cases are written down.

Related

• On Asymptotic Reducibility in SL(3,Z)
• On conjugacy classes of GL(n,q) and SL(n,q)
• Expansion of conjugacy classes in PSL(2,q)
• Minimal Presentation of PSL(2,ℤ) Using Continuant Matrices with Integer Coefficients
• Conjugacy in Permutation Representations of the Symmetric Group