z-Classes and Rational Conjugacy Classes in Alternating Groups

TL;DR

This paper calculates the number of z-classes in symmetric and alternating groups, revealing how these counts relate to specific restricted partitions, and proves a correspondence between rational characters and rational conjugacy classes in A_n.

Contribution

It provides explicit formulas for z-classes in S_n and A_n based on restricted partitions, and establishes a new link between rational characters and conjugacy classes in A_n.

Findings

Number of z-classes in S_n depends on partitions excluding 1 and 2.

Number of z-classes in A_n depends on partitions with distinct, odd parts excluding 1 and 2.

Number of rational-valued irreducible characters in A_n equals the number of rational conjugacy classes.

Abstract

In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n is greater or equal to 3 and alternating group A_n, when n is greater or equal to 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S_n is determined by those restricted partitions of n-2 in which 1 and 2 do not appear as its part. And, in the case of alternating groups, it is determined by those restricted partitions of n-3 which has all its parts distinct, odd and in which 1 (and 2) does not appear as its part, along with an error term. The error term is given by those partitions of n which have each of its part distinct, odd and perfect square. Further, we prove that the number of rational-valued irreducible complex characters for A_n is same as the number of conjugacy classes which are rational.