Representations of the Lie Superalgebra $\mathfrak{osp}(1|2n)$ with Polynomial Bases

TL;DR

This paper constructs a new polynomial basis for a class of infinite-dimensional representations of the Lie superalgebra fosp(1|2n), using combinatorics and Young tableaux, enabling explicit matrix element calculations.

Contribution

It introduces a novel polynomial basis for fosp(1|2n) representations derived from an embedding, with basis vectors labeled by Young tableaux and expressed as Clifford algebra valued polynomials.

Findings

Basis vectors form a complete basis for the representations

Explicit polynomial expressions for basis vectors are provided

Methods for computing matrix elements using combinatorics are developed

Abstract

We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We construct a new polynomial basis for $L_n(p)$ arising from the embedding $\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of $L_n(p)$ 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 $\mathfrak{osp}(1|2n)$ on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.