# 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 $\chi _{\lambda }$ 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 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.