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)

Comments

See also Character of a representation of a group.

References

[a1]	H. Boerner, "Representations of groups" , North-Holland (1970) (Translated from German)
[a2]	D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[a3]	I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979)
[a4]	B.G. Wybourne, "Symmetry principles and atomic spectroscopy" , Wiley (Interscience) (1970)