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 vertical bar 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 vertical bar 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 vertical bar 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 vertical bar 2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1 vertical bar 2n) representations. Furthermore, by considering the branching osp(1 vertical bar 2n) superset of sp(2n) superset of u(n), we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.