Murnaghan–Nakayama rule
In group theory, a branch of mathematics, the Murnaghan–Nakayama rule, named after Francis Murnaghan and Tadashi Nakayama, is a combinatorial method to compute irreducible character values of a symmetric group. There are several generalizations of this rule beyond the representation theory of symmetric groups, but they are not covered here.
The irreducible characters of a group are of interest to mathematicians because they concisely summarize important information about the group, such as the dimensions of the vector spaces in which the elements of the group can be represented by linear transformations that “mix” all the dimensions. For many groups, calculating irreducible character values is very difficult; the existence of simple formulas is the exception rather than the rule.
The Murnaghan–Nakayama rule is a combinatorial rule for computing symmetric group character values χλ
ρ using a particular kind of Young tableaux.
Here λ and ρ are both integer partitions of some integer n, the order of the symmetric group under consideration. The partition λ specifies the irreducible character, while the partition ρ specifies the conjugacy class on whose group elements the character is evaluated to produce the character value. The partitions are represented as weakly decreasing tuples; for example, two of the partitions of 8 are (5,2,1) and (3,3,1,1).
There are two versions of the Murnaghan-Nakayama rule, one non-recursive and one recursive.