Murnaghan-Nakayama Rule The Explanation and Usage of the Algorithm

Abstract

Character values are not the easiest to calculate, so it is important to find good algorithms that can help ease these calculations. In the 20th century, the two mathematicians Murnaghan and Nakayama developed a rule that calculates character values for partitions on some computations. This rule has later been given the name The Murnaghan-Nakayama rule, after these two authors.

The Murnaghan-Nakayama rule is a combinatorial method for computing character values of irreducible representations of symmetric groups. This makes this rule an important part of representation theory. One of the versions of this rule is stated in the recursive Murnaghan-Nakayama rule. Where, in this version, we can use border strips and diagrams to calculate the character values of representations on a given composition. This algorithm is quite fast in these calculations.

The Murnaghan-Nakayama rule can also be considered a central algorithm in representation theory over symmetric groups. It is a fascinating and powerful algorithm that has a strong connection to both combinatorics and representation theory.