Frobenius Character 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 $\chi_\lambda$ 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 $x_1^ \dots x_k^$ in the homogeneous polynomial

: $\prod_ (x_i - x_j) \; \prod_j P_j(x_1, \dots, x_k)^,$

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 $\chi_(C(\mu))$, given by

:$(x_1 - x_2)(x_1 + x_2)(x_1^3 + x_2^3),$

is −1.

Analogues

In , 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. 

Representation theory
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