A Lie algebra of Grassmannian Dirac operators and vector variables

TL;DR

This paper explores the algebraic structure generated by Grassmannian Dirac operators and vector variables, revealing its isomorphism to a classical orthogonal Lie algebra and constructing a polynomial basis linked to representation theory.

Contribution

It identifies the Lie algebra generated by Grassmannian Dirac operators and vector variables as so(2m+1), and constructs a basis for the polynomial space using Young tableaux techniques.

Findings

The generated Lie algebra is isomorphic to so(2m+1).

A basis for the polynomial space is explicitly constructed.

Connections to parafermion theory are discussed.

Abstract

The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of polynomials in these vector variables, corresponding to an irreducible $\mathfrak{so}(2m+1)$ representation. In particular, a basis of $\mathcal{P}$ is constructed, using various Young tableaux techniques. Throughout the paper, we also indicate the relation to the theory of parafermions.