The parafermion Fock space and explicit so(2n+1) representations

The defining relations (triple relations) of n pairs of parafermion operators f_j^\pm (j=1,...,n) are known to coincide with a set of defining relations for the Lie algebra so(2n+1) in terms of 2n generators. With the common Hermiticity conditions, this means that the ``parafermions of order p'' correspond to a finite-dimensional unitary irreducible representation W(p) of so(2n+1), with highest weight (p/2, p/2,..., p/2). Although the dimension and character of W(p) is known by classical formulas, there is no explicit basis of W(p) available in which the parafermion operators have a natural action. In this paper we construct an orthogonal basis for W(p), and we present the explicit actions of the parafermion generators on these basis vectors. We use group theoretical techniques, in which the u(n) subalgebra of so(2n+1) plays a crucial role: a set of Gelfand-Zetlin patterns of u(n) will be used to label the basis vectors of W(p), and also in the explicit action (matrix elements) certain u(n) Clebsch-Gordan coefficients are essential.