An explicit basis of lowering operators for irreducible representations of unitary groups

Abstract

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the formalism of lowering operators. Here, one identifies a basis for the algebra of all lowering operators and, for each irreducible representation, gives a prescription for choosing a subcollection of lowering operators that yields a basis upon application to the highest weight vector. Bases obtained through lowering operators are particularly convenient for computing matrix coefficients of observables as the calculations reduce to the commutation relations for the standard matrix units. The best known examples of this approach are the extremal projector construction of the Gelfand–Zetlin basis and the crystal (or canonical) bases of Kashiwara and Lusztig. In this paper, we describe another simple method of obtaining bases for the irreducible representations via lowering operators. These bases do not have the algebraic canonicity of the Gelfand–Zetlin and crystal bases, but the combinatorics involved are much more straightforward, making the bases particularly suited for physical applications.
Keywords: unitary group, special unitary group, irreducible representations, lowering operators, spin-free quantum chemistry, many-body problem
PACS: 02.20.-a, 31.15.xh