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

Abstract

We study a particular class of infinite-dimensional representations of 𝔬𝔰𝔭(1|2𝑛). These representations 𝐿ₙ(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 𝐿ₙ(𝑝) arising from the embedding 𝔬𝔰𝔭(1|2𝑛𝑝) ⊃ 𝔬𝔰𝔭(1|2𝑛). The basis vectors of 𝐿ₙ(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 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.