# Frobenius formula

A formula expressing a relation between the generalized Vandermonde determinant and the ordinary one (see Vandermonde determinant) in terms of sums of powers. The characters of representations of a symmetric group (cf. Representation of the symmetric groups) appear as coefficients in the Frobenius formula.

Let $x _ {1} \dots x _ {n}$ be independent variables. For any $n$-tuple $\lambda = ( \lambda _ {1} \dots \lambda _ {n} )$ of non-negative integers satisfying the condition $\lambda _ {1} \geq \dots \geq \lambda _ {n}$, let

$$W _ \lambda = \left | \begin{array}{lll} x _ {1} ^ {\lambda _ {1} + n - 1 } &\cdots &x _ {n} ^ {\lambda _ {1} + n - 1 } \\ \vdots &\ddots &\vdots \\ x _ {1} ^ {\lambda _ {n} } &\cdots &x _ {n} ^ {\lambda _ {n} } \\ \end{array} \ \right | ,$$

so that $W _ {0}$ is the ordinary Vandermonde determinant. Let $\sum \lambda _ {i} = m$; then after discarding zeros the $n$-tuple $\lambda$ can be regarded as a partition of the number $m$. Consider the corresponding irreducible representation $T _ \lambda$ of the symmetric group $S _ {m}$. For any partition $\mu = ( \mu _ {1} \dots \mu _ {r} )$ of $m$ one denotes by $a _ {\lambda \mu }$ the value of the character of $T _ \lambda$ on the conjugacy class of $S _ {m}$ determined by $\mu$, and by $c _ \mu$ the order of the centralizer of any permutation in this class. Let $s _ \mu = s _ {\mu _ {1} } \dots s _ {\mu _ {r} }$, where $s _ {k} = x _ {1} ^ {k} + \dots + x _ {n} ^ {k}$. Then

$$\frac{W _ \lambda }{W _ {0} } = \ \sum _ \mu a _ {\lambda \mu } c _ \mu ^ {- 1} s _ \mu ,$$

where the sum is taken over all (unordered) partitions of $m$. Here, if the partition $\mu$ contains $k _ {1}$ ones, $k _ {2}$ twos, etc., then

$$c _ \mu = k _ {1} ! k _ {2} ! \dots 1 ^ {k _ {1} } 2 ^ {k _ {2} } \dots .$$

If $n \geq m$, then Frobenius' formula can be put in the form

$$\sum _ \lambda a _ {\lambda \mu } W _ \lambda = \ s _ \mu W _ {0} ,$$

where the sum is taken over all partitions of $m$( adding the appropriate number of zeros). The last formula can be used to compute the characters of the symmetric group. Namely, $a _ {\lambda \mu }$ is the coefficient of $x _ {1} ^ {\lambda _ {1} + n - 1 } {} \dots x _ {n} ^ {\lambda _ {n} }$ in the polynomial $s _ \mu W _ {0}$.

#### References

 [1] F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)