We study the explicit formula of Lusztig’s integral forms of the level one quantum affine algebra
U
q
(
s
l
^
2
)
U_q(\widehat {sl}_2)
in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of
Z
\mathbb 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
−
1
]
\mathbb Z[q, q^{-1}]
-form of
U
q
(
s
l
^
2
)
U_q(\widehat {sl}_2)
on Schur polynomials. As a result the
Z
[
q
,
q
−
1
]
\mathbb Z[q, q^{-1}]
-lattice of Schur functions tensored with the group algebra contains Lusztig’s integral lattice.