Schwinger Representation for the Symmetric Group: Two explicit constructions for the Carrier Space

TL;DR

This paper presents two explicit constructions for the carrier space of the Schwinger representation of the symmetric group, one using antisymmetric monomials and the other based on Fock space linked to Greenberg algebra.

Contribution

It introduces two novel explicit methods for constructing the carrier space of the Schwinger representation of the symmetric group.

Findings

First construction uses monomials in antisymmetric variables

Second construction employs Fock space from Greenberg algebra

Provides explicit formulas for the carrier space

Abstract

We give two explicit construction for the carrier space for the Schwinger representation of the group $S_n$. While the first relies on a class of functions consisting of monomials in antisymmetric variables, the second is based on the Fock space associated with the Greenberg algebra.