Vertex operators arising from Jacobi-Trudi identities

TL;DR

This paper interprets the boson-fermion correspondence through Jacobi-Trudi identities, leading to new constructions of Clifford algebra actions and explicit formulas for vertex operators related to classical Lie algebra characters.

Contribution

It introduces a novel perspective connecting Jacobi-Trudi identities with vertex operators and Clifford algebra actions, extending the theory to generalized symmetric functions.

Findings

Derived explicit formulas for vertex operators of classical Lie algebra characters

Established a generalized Giambelli identity from Jacobi-Trudi identities

Constructed Clifford algebra actions on polynomial algebras related to symmetric functions

Abstract

We give an interpretation of the boson-fermion correspondence as a direct consequence of Jacobi-Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi-Trudi identity the action of Clifford algebra on polynomial algebras that arrives as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions associated to linear recurrence relations.