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.
Statement
Let
be the character of an irreducible representation of the symmetric group
corresponding to a partition
of n:
and
. For each partition
of n, let
denote the conjugacy class in
corresponding to it (cf. the example below), and let
denote the number of times j appears in
(so
). Then the Frobenius formula states that the constant value of
on
![{\displaystyle \chi _{\lambda }(C(\mu )),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/004c668d3af3a6c6cd89014bdc311f96aabc5e42)
is the coefficient of the monomial
in the homogeneous polynomial in
variables
![{\displaystyle \prod _{i<j}^{k}(x_{i}-x_{j})\;\prod _{j}P_{j}(x_{1},\dots ,x_{k})^{i_{j)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0ac31825fca9d271256870d506d327ab5e594b)
where
is the
-th power sum.
Example: Take
. Let
and hence
,
,
. If
(
), which corresponds to the class of the identity element, then
is the coefficient of
in
![{\displaystyle (x_{1}-x_{2})P_{1}(x_{1},x_{2})^{4}=(x_{1}-x_{2})(x_{1}+x_{2})^{4))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e649695dca30aa6eb6b0711211298e337eb08db1)
which is 2. Similarly, if
(the class of a 3-cycle times an 1-cycle) and
, then
, 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}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66055359ae94f6eb5820f739914fa98bf591c9ab)
is −1.
For the identity representation,
and
. The character
will be equal to the coefficient of
in
,
which is 1 for any
as expected.
Analogues
Arun Ram gives a q-analog of the Frobenius formula.