Bases for infinite dimensional simple osp(1|2n)-modules respecting the branching osp(1|2n)⊃gl(n)

Abstract

We study the effects of the branching osp(1|2n)⊃gl(n) on a particular class of simple infinite-dimensional osp(1|2n)-modules L(p) characterized by a positive integer p. In the first part (Sec. III), we use combinatorial methods, such as Young tableaux and Young subgroups, to construct a new basis for L(p) that respects this branching, and we express the basis elements explicitly in two distinct ways: first, as monomials of negative root vectors of gl(n) acting on certain gl(n)-highest weight vectors in L(p) and then as polynomials in the generators of osp(1|2n) acting on a osp(1|2n)-lowest weight vector in L(p). In the second part (Sec. IV), we use extremal projectors and the theory of Mickelsson–Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching osp(1|2n)⊃gl(n). We use the raising operators to give new expressions for the elements of the Gel’fand–Zetlin basis for L(p) as monomials of operators from U(osp(1|2n)) acting on a osp(1|2n)-lowest weight vector in L(p). We observe that the Gel’fand–Zetlin basis for L(p) is related to the basis constructed earlier in this paper by a triangular transition matrix. We end this paper (Sec. V) with a detailed example treating the case n = 3.