International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 745-759
doi:10.1155/S0161171299227457
Power of a determinant with two physical applications
Theoretical Division, Los Alamos National Laboratory, Los Alamos 87545, NM, USA
Received 29 September 1997; Revised 7 May 1998
Copyright © 1999 James D. Louck. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.