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 "S""n". Among the other applications, the formula can be used to derive the hook length formula.
Statement.
Let formula_1 be the character of an irreducible representation of the symmetric group formula_2 corresponding to a partition formula_3 of "n": formula_4 and formula_5. For each partition formula_6 of "n", let formula_7 denote the conjugacy class in formula_2 corresponding to it (cf. the example below), and let formula_9 denote the number of times "j" appears in formula_6 (so formula_11). Then the Frobenius formula states that the constant value of formula_1 on formula_13
is the coefficient of the monomial formula_15 in the homogeneous polynomial
where formula_17 is the formula_18-th power sum.
Example: Take formula_19 and formula_20. If formula_21, which corresponds to the class of the identity element, then formula_22 is the coefficient of formula_23 in
which is 2. Similarly, if formula_25 (the class of a 3-cycle times an 1-cycle), then formula_26, given by
is −1.
Analogues.
In , Arun Ram gives a "q"-analog of the Frobenius formula.