Power of a determinant with two physical applications

• Main Author: James D. Louck
• Format: Article
• Language: English
• Published: Hindawi Limited 1999-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: Power of a determinant; Clebsch-Gordan coefficients; Vandermonde determinant; map from magic squares to partitions.
• Online Access: http://dx.doi.org/10.1155/S0161171299227457

An expression for the kth power of an n×n determinant in n2 indeterminates (zij) is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary group SU(2), noting also the relation to the  3 F2 hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given.