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. 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 (cf. the example below), and let denote the number of times j appears in (so ). Then the Frobenius formula states that the constant value of on is the coefficient of the monomial in the homogeneous polynomial where is the -th power sum. Example: Take and . If , which corresponds to the class of the identity element, then is the coefficient of in which is 2. Similarly, if (the class of a 3-cycle times an 1-cycle), then , given by is −1. In , Arun Ram gives a q-analog of the Frobenius formula.