On rotation distance between binary coupling trees and applications for 3nj-coefficients

Generalized recoupling coefficients or 3nj-coefficients for a Lie algebra ( with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6j-coefficients). In general there exist many such expressions; we say that such an expression is optimal if the number of Ra cah coefficients in such a product (and, correlated, the number of summatio n indices) is minimal. The problem of finding an optimal expression for a g iven 3nj-coefficient is equivalent to finding a shortest path in a graph G( n). The vertices of this graph G(n) consist of binary coupling trees, repre senting the coupling schemes in the bra/kets of the 3nj-coefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and h as order (2n - 1)!!. As the order increases so rapidly, finding a shortest path is computationally achievable only for n < 11. We present some mathema tical tools to compute or estimate the length of such shortest paths betwee n binary coupling trees. The diameter of G(n) is determined explicitly up t o n < 11, and it is shown to grow like n log(n). Thus for n large enough, t he number of Racah coefficients in the expansion of a 3nj-coefficient is of order n log(n). We also show that this problem in Racah-Wigner theory is e quivalent to a problem in mathematical biology, where one is concerned with the quantitative comparison of classifications or dendrograms. From this c ontext, some algorithms for approximating the shortest path can be deduced. (C) 1999 Elsevier Science B.V. All rights reserved.