Models for irreducible representations of the symplectic algebra using Dirac-type operators

TL;DR

This paper develops polynomial models for finite and infinite-dimensional irreducible representations of the symplectic Lie algebra using Dirac-type operators, linking symplectic Dirac operators to parafermion algebras and constructing a symplectic Rarita-Schwinger operator.

Contribution

It introduces a novel polynomial model for symplectic algebra representations using Dirac operators and constructs a symplectic Rarita-Schwinger operator based on transvector algebra theory.

Findings

Polynomial models for $rak{sp}(2n)$ representations are established.

Symplectic Dirac operators generate parafermion algebras.

A symplectic Rarita-Schwinger operator is constructed.

Abstract

In this paper we will study both the finite and infinite-dimensional representations of the symplectic Lie algebra $\mathfrak{sp}(2n)$ and develop a polynomial model for these representations. This means that we will associate a certain space of homogeneous polynomials in a matrix variable, intersected with the kernel of $\mathfrak{sp}(2n)$-invariant differential operators related to the symplectic Dirac operator with every irreducible representation of $\mathfrak{sp}(2n)$. We will show that the systems of symplectic Dirac operators can be seen as generators of parafermion algebras. As an application of these new models, we construct a symplectic analogue of the Rarita-Schwinger operator using the theory of transvector algebras.