Conjugacy in Permutation Representations of the Symmetric Group

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  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.

Key Words:

• Characters
• Conjugacy classes
• Fixed points
• Permutation representations
• Symmetric group

2000 Mathematics Subject Classification:

• Primary 20C30
• Secondary 05E15

ACKNOWLEDGMENTS

This article expands upon the authors theses, and in particular, the second author's thesis, which was supervised by Yuval Roichman and Ron Adin. Both authors are very grateful to them. Also the authors are grateful to Alex Lubotzky, Eli Bagno, Uzi Vishne, and Boris Kunyavskii for helpful discussions.

Supported in part by a grant from the Israel Science Foundation.

Notes

¹We use (n, k) to denote the greatest common divisor of n and k.

Communicated by A. Yu. Olshanskii.