Extended Orbits-Fixedpoints Relations

TL;DR

This paper generalizes an existing orbits-fixed-points theorem to broader contexts using new proofs and Stirling numbers, illustrating with Mathieu group M24 and potential applications in tensor products of matrix representations.

Contribution

It introduces new proofs and extends the orbits-fixed-points relation beyond symmetric groups, incorporating Stirling numbers and applications to Mathieu groups.

Findings

Extended theorems to new group actions

Utilized Stirling numbers of the second kind

Discussed applications in tensor product representations

Abstract

I extend further, using new proofs, two generalizations of an earlier orbits-fixed-points theorem, which was restricted to group action of the symmetric group. The extended equality makes use of the Stirling numbers of the second kind. An illustration using Mathieu group M24 is discussed. Possible applications using tensor products of matrix permutation representations is indicated.