Frobenius formula

A formula expressing a relation between the generalized Vandermonde determinant and the ordinary one (see Vandermonde determinant) in terms of sums of powers. The characters of representations of a symmetric group (cf. Representation of the symmetric groups) appear as coefficients in the Frobenius formula.

Let be independent variables. For any -tuple of non-negative integers satisfying the condition , let

so that is the ordinary Vandermonde determinant. Let ; then after discarding zeros the -tuple can be regarded as a partition of the number . Consider the corresponding irreducible representation of the symmetric group . For any partition of one denotes by the value of the character of on the conjugacy class of determined by , and by the order of the centralizer of any permutation in this class. Let , where . Then

where the sum is taken over all (unordered) partitions of . Here, if the partition contains ones, twos, etc., then

If , then Frobenius' formula can be put in the form

where the sum is taken over all partitions of (adding the appropriate number of zeros). The last formula can be used to compute the characters of the symmetric group. Namely, is the coefficient of in the polynomial .

References

[1]	F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)

Comments

See also Character of a representation of a group.

References

[a1]	H. Boerner, "Representations of groups" , North-Holland (1970) (Translated from German)
[a2]	D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[a3]	I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979)
[a4]	B.G. Wybourne, "Symmetry principles and atomic spectroscopy" , Wiley (Interscience) (1970)