The parafermion Fock space and explicit representations

Abstract

The defining relations (triple relations) of n pairs of parafermion operators f±j (j = 1, …, n) are known to coincide with a set of defining relations for the Lie algebra 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 , with highest weight . Although the dimension and the character of W(p) is known by classical formulae, 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 present the explicit actions of the parafermion generators on these basis vectors. We use group theoretical techniques, in which the subalgebra of plays a crucial role: a set of Gelfand–Zetlin patterns of will be used to label the basis vectors of W(p), and also in the explicit action (matrix elements) certain Clebsch–Gordan coefficients are essential.