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

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

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

${\displaystyle \prod _{i

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

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

${\displaystyle (x_{1}-x_{2})(x_{1}+x_{2})^{4}}$

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

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

is −1.