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 Sn. Among the other applications, the formula can be used to derive the hook length formula.

In (Ram 1991), Arun Ram gives a q-analog of the Frobenius 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

where is the -th power sum.

Example: Take and . 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), then , given by

is −1.

See also

References

    • A. Ram, A Frobenius formula for the characters of the Hecke algebras, Inventiones mathematicae, vol 106, no 1, pp 461–488, 1991.
    • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
    • 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. ISBN 0-19-853489-2 MR1354144
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