Schur function identities arising from the basic representation of $A^{(2)}_{2}$

TL;DR

This paper provides a Lie theoretic interpretation of formulas involving Schur and Schur Q-functions by exploring two realizations of the basic representation of the affine Lie algebra $A^{(2)}_2$, linking fermionic and bosonic perspectives.

Contribution

It introduces a novel interpretation connecting Schur function identities to the basic representation of $A^{(2)}_2$ using fermionic and bosonic realizations.

Findings

Relations between vacuum expectation values and Schur functions established

Fermionic and bosonic realizations are shown to be equivalent via boson-fermion correspondence

New algebraic identities for Schur functions derived from Lie algebra representations

Abstract

A Lie theoretic interpretation is given for some formulas of Schur functions and Schur $Q$-functions. Two realizations of the basic representation of the Lie algebra $A^{(2)}_2$ are considered; one is on the fermionic Fock space and the other is on the bosonic polynomial space. Via the boson-fermion correspondence, simple relations of the vacuum expectation values of fermions turn out to be algebraic relations of Schur functions.