The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp(1|2n)

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b i . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V (p) of osp(1|2n). Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V (p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V (p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching osp(1|2n) ⊃ sp(2n) ⊃ u(n), we find explicit infinitedimensional unitary irreducible lowest weight representations of sp(2n) and their characters.