A recursion formula for the irreducible characters of the symmetric group

TL;DR

This paper introduces a new recursion formula for calculating irreducible characters of symmetric groups, extending the branching theorem to cover all elements and offering an alternative to the Murnaghan-Nakayama formula.

Contribution

It develops a recursive method to compute all irreducible character values of symmetric groups, surpassing the limitations of the branching theorem.

Findings

Provides a complete recursion formula for symmetric group characters.

Offers an alternative computational approach to the Murnaghan-Nakayama formula.

Enhances understanding of symmetric group representations.

Abstract

The branching theorem expresses irreducible character values for the symmetric group $S_n$ in terms of those for $S_{n-1}$, but it gives the values only at elements of $S_n$ having a fixed point. We extend the theorem by providing a recursion formula that handles the remaining cases. It expresses these character values in terms of values for $S_{n-1}$ together with values for $S_n$ that are already known in the recursive process. This provides an alternative to the Murnaghan-Nakayama formula.