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

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.