In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic constru
ction of the canonical basis for modified quantum affine $mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $mathfrak{gl}_n$.
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
