Polynomial Representations of the Lie Superalgebra osp(1|2n)

A. K. Bisbo, H. De Bie, J. Van der Jeugt

Published 2019-12-13Version 1

We study a particular class of infinite-dimensional representations of osp(1|2n). These representations W_n(p) are characterized by a positive integer p, and are the lowest component in the p-fold tensor product of the metaplectic representation of osp(1|2n). We construct a new polynomial basis for W_n(p) arising from the embedding of osp(1|2n) in osp(1|2np). The basis vectors of W_n(p) are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in np variables. Using combinatorial properties of these tableau vectors it is deduced that they form indeed a basis. The computation of matrix elements of a set of generators of osp(1|2n) on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.