Frobenius formula
In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn. Among the other applications, the formula can be used to derive the hook length formula.
Statement
Let be the character of an irreducible representation of the symmetric group corresponding to a partition of n: and. For each partition of n, let denote the conjugacy class in corresponding to it, and let denote the number of times j appears in . Then the Frobenius formula states that the constant value of onis the coefficient of the monomial in the homogeneous polynomial in variables
where is the -th power sum.
Example: Take. Let and hence,,. If , which corresponds to the class of the identity element, then is the coefficient of in
which is 2. Similarly, if and, then, given by
is −1.
For the identity representation, and. The character will be equal to the coefficient of in,
which is 1 for any as expected.