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.

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 ${\displaystyle \sum _{j}i_{j}j=n}$). 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 in ${\displaystyle k}$ variables

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

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 ${\displaystyle n=4}$. Let ${\displaystyle \lambda :4=2+2=\lambda _{1}+\lambda _{2}}$ and hence ${\displaystyle k=2}$, ${\displaystyle \ell _{1}=3}$, ${\displaystyle \ell _{2}=2}$. If ${\displaystyle \mu :4=1+1+1+1}$ (${\displaystyle i_{1}=4}$), 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

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

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

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

is −1.

For the identity representation, ${\displaystyle k=1}$ and ${\displaystyle \lambda _{1}=n=\ell _{1}}$. The character ${\displaystyle \chi _{\lambda }(C(\mu ))}$ will be equal to the coefficient of ${\displaystyle x_{1}^{n}}$ in ${\displaystyle \prod _{j}P_{j}(x_{1})^{i_{j}}=\prod _{j}x_{1}^{i_{j}j}=x_{1}^{\sum _{j}i_{j}j}=x_{1}^{n}}$, which is 1 for any ${\displaystyle \mu }$ as expected.

Analogues

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

See also

• Representation theory of symmetric groups