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 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 Ã¢ÂˆÂ’1.

For the identity representation, and . The character will be equal to the coefficient of in , which is 1 for any as expected.

Analogues

Arun Ram gives a q-analog of the Frobenius formula.

See also

• Representation theory of symmetric groups

References

• Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.