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.

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

Statement

Let $\chi_\lambda$ be the character of an irreducible representation of the symmetric group $S_{n}$ corresponding to a partition $\lambda$ of n: $n=\lambda _{1}+\cdots +\lambda _{k}$ and $\ell _{j}=\lambda _{j}+k-j$ . For each partition $\mu$ of n, let $C(\mu )$ denote the conjugacy class in $S_{n}$ corresponding to it (cf. the example below), and let $i_j$ denote the number of times j appears in $\mu$ (so $\Sigma \,i_{j}j=n$ ). Then the Frobenius formula states that the constant value of $\chi_\lambda$ on $C(\mu ),$

\chi _{\lambda }(C(\mu )),

is the coefficient of the monomial $x_{1}^{\ell _{1}}\dots x_{k}^{\ell _{k}}$ in the homogeneous polynomial

\prod _{i<j}(x_{i}-x_{j})\prod _{j}P_{j}(x_{1},\dots ,x_{k})^{i_{j}},

where $P_{j}(x_{1},\dots ,x_{k})=x_{1}^{j}+\dots +x_{k}^{j}$ is the $j$ -th power sum.

Example: Take $n=4$ and $\lambda :4=2+2$ . If $\mu :4=1+1+1+1$ , which corresponds to the class of the identity element, then $\chi _{\lambda }(C(\mu ))$ is the coefficient of $x_{1}^{3}x_{2}^{2}$ in

(x_{1}-x_{2})(x_{1}+x_{2})^{4}

which is 2. Similarly, if $\mu :4=3+1$ (the class of a 3-cycle times an 1-cycle), then $\chi _{\lambda }(C(\mu ))$ , given by

(x_{1}-x_{2})(x_{1}+x_{2})(x_{1}^{3}+x_{2}^{3}),

is −1.

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 MR 1354144

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Frobenius formula

Statement

See also

References