Conjugacy in Permutation Representations of the Symmetric Group

2004/11/24 by Yona Cherniavsky, Cherniavsky, Yona, Mishael Sklarz +1
Mathematics · #05E15 #20C30 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.CO #math.RT #msc:05E15 #msc:20C30

paper · pdf · doi:10.48550/arxiv.math/0411554

openalex publication_date 2004/11/24 · arxiv created 2007/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e. that conjugacy classes of S_n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S_n x S_n. We also consider the permutation representations of S_n which arise from the action of S_n on k-tuples, and classify which of them unite conjugacy classes and which do not.

Related

• The conjugacy action of S_n and modules induced from centralisers
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Symmetric groups and conjugacy classes
• On conjugacy classes of S_n containing all irreducibles
• Permutation representations on invertible matrices