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.
Let
\chiλ
Sn
λ
n=λ1+ … +λk
\ellj=λj+k-j
\mu
C(\mu)
Sn
ij
\mu
\sumjijj=n
\chiλ
C(\mu),
\chiλ(C(\mu)),
is the coefficient of the monomial
\ell1 | |
x | |
1 |
...
\ellk | |
x | |
k |
k
k | |
\prod | |
i<j |
(xi-xj) \prodjPj(x1,...,
ij | |
x | |
k) |
,
where
Pj(x1,...,xk)=
j | |
x | |
1 |
+...+
j | |
x | |
k |
j
Example: Take
n=4
λ:4=2+2=λ1+λ2
k=2
\ell1=3
\ell2=2
\mu:4=1+1+1+1
i1=4
\chiλ(C(\mu))
3 | |
x | |
1 |
2 | |
x | |
2 |
(x1-x2)P1(x1,x
4=(x | |
1 |
-x2)(x1+
4 | |
x | |
2) |
which is 2. Similarly, if
\mu:4=3+1
i1=i3=1
\chiλ(C(\mu))
(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
λ1=n=\ell1
\chiλ(C(\mu))
n | |
x | |
1 |
\prodjPj(x
ij | |
1) |
=\prodj
ijj | |
x | |
1 |
=
\sumjijj | |
x | |
1 |
n | |
=x | |
1 |
\mu
Arun Ram gives a q-analog of the Frobenius formula.