  The goal of this paper is twofold. First, we write down the semi-infinite
Pl\"ucker relations, describing the Drinfeld-Pl\"ucker embedding of the (formal
version of) semi-infinite flag varieties in type A. Second, we study the
homogeneous coordinate ring, i.e. the quotient by the ideal generated by the
semi-infinite Pl\"ucker relations. We establish the isomorphism with the
algebra of dual global Weyl modules and derive a new character formula.
