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.

Statement

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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 −1.

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

Analogues

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Arun Ram gives a q-analog of the Frobenius formula.[1]

See also

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References

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  • Ram, Arun (1991). "A Frobenius formula for the characters of the Hecke algebras". Inventiones mathematicae. 106 (1): 461–488.
  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 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