  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.
