Frobenius formula explained

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

\chiλ

be the character of an irreducible representation of the symmetric group

Sn

corresponding to a partition

λ

of n:

n=λ1++λk

and

\ellj=λj+k-j

. For each partition

\mu

of n, let

C(\mu)

denote the conjugacy class in

Sn

corresponding to it (cf. the example below), and let

ij

denote the number of times j appears in

\mu

(so

\sumjijj=n

). Then the Frobenius formula states that the constant value of

\chiλ

on

C(\mu),

\chiλ(C(\mu)),

is the coefficient of the monomial

\ell1
x
1

...

\ellk
x
k
in the homogeneous polynomial in

k

variables
k
\prod
i<j

(xi-xj)\prodjPj(x1,...,

ij
x
k)

,

where

Pj(x1,...,xk)=

j
x
1

+...+

j
x
k
is the

j

-th power sum.

Example: Take

n=4

. Let

λ:4=2+2=λ1+λ2

and hence

k=2

,

\ell1=3

,

\ell2=2

. If

\mu:4=1+1+1+1

(

i1=4

), which corresponds to the class of the identity element, then

\chiλ(C(\mu))

is the coefficient of
3
x
1
2
x
2
in

(x1-x2)P1(x1,x

4=(x
1

-x2)(x1+

4
x
2)

which is 2. Similarly, if

\mu:4=3+1

(the class of a 3-cycle times an 1-cycle) and

i1=i3=1

, then

\chiλ(C(\mu))

, given by

(x1-x2)P1(x1,x2)P3(x1,x2)=(x1-x2)(x1+x2)(x

3
1

+

3),
x
2

is −1.

For the identity representation,

k=1

and

λ1=n=\ell1

. The character

\chiλ(C(\mu))

will be equal to the coefficient of
n
x
1
in

\prodjPj(x

ij
1)

=\prodj

ijj
x
1

=

\sumjijj
x
1
n
=x
1
, which is 1 for any

\mu

as expected.

Analogues

Arun Ram gives a q-analog of the Frobenius formula.

See also

References