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 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 (the class of a 3-cycle times an 1-cycle) 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.