Structures of boson and fermion Fock spaces in the space of symmetric functions

TL;DR

This paper constructs representations of infinite-dimensional groups, specifically the Weil and spinor representations, within the space of symmetric functions, advancing the understanding of their algebraic structures.

Contribution

It introduces a realization of the Weil and spinor representations as linear operators acting on the space of symmetric functions in infinitely many variables.

Findings

Realization of the Weil representation in symmetric functions.

Construction of the spinor representation within the same framework.

Provides a new algebraic perspective on infinite-dimensional group representations.

Abstract

We realize the Weil representation of infinite dimensional symplectic group and spinor representation of infinite-dimensional group $GL$ by linear operators in the space of symmetric functions in infinite number of variables.