Gel’fand lattice polynomials and irreducible representations of U(n)

Abstract

A finite difference equation defines the exponential of a square tableau, extension of the usual Gel’fand pattern. These exponentials or ’’K powers’’ are homogeneous polynomials useful in the theory of group representations. The theory of these polynomials is developed, and some important addition and multiplication theorems are deduced. The application to the group U(n) gives explicitly the Gel’fand states for n=4, and it is conjectured that the given relation is true in general for any dimension. The matrix elements with respect to this basis are calculated for n=3 and the Clebsch–Gordan decomposition of the n product of U(2) is also given.