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
formula_14
is the coefficient of the monomial formula_15 in the homogeneous polynomial in formula_16 variables
 formula_17
where formula_18 is the formula_19-th power sum.
Example: Take formula_20. Let formula_21 and hence formula_22, formula_23, formula_24. If formula_25 (formula_26), which corresponds to the class of the identity element, then formula_27 is the coefficient of formula_28 in
formula_29
which is 2. Similarly, if formula_30 (the class of a 3-cycle times an 1-cycle) and formula_31, then formula_32, given by
formula_33
is −1.
For the identity representation, formula_34 and formula_35. The character formula_27 will be equal to the coefficient of formula_37 in formula_38, 
which is 1 for any formula_6 as expected.
Analogues.
Arun Ram gives a "q"-analog of the Frobenius formula.
References.
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