Abstract
We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra
U
q
(
sl
^
2
)
in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of
Z
. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig's
Z[q,q]
-form of
U
q
(
sl
^
2
)
on Schur polynomials.