The parafermion Fock space and explicit so(2n + 1) representations
نویسندگان
چکیده
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 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 representationW (p) of so(2n+1), with highest weight ( 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.
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