  We study the relation between quantum affine algebras of type A and
Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism
from the Grothendieck ring of a certain subcategory $\mathcal{C}_{\ell}$ of
$U_q(\hat{\mathfrak{sl}_n})$-modules to a quotient of the Grassmannian cluster
algebra in which certain frozen variables are set to 1. We explain how this
induces an isomorphism between the monoid of dominant monomials, used to
parameterize simple modules, and a quotient of the monoid of rectangular
semistandard Young tableaux. Via the isomorphism, we define an element ch(T) in
a Grassmannian cluster algebra for every rectangular tableau T. By results of
Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster
monomial is of the form ch(T) for some T. Using formulas of Arakawa-Suzuki, we
give an explicit expression for ch(T), and also give explicit q-character
formulas for finite-dimensional $U_q(\hat{\mathfrak{sl}_n})$-modules. We give a
tableau-theoretic rule for performing mutations in Grassmannian cluster
algebras. We suggest how our formulas might be used to study reality and
primeness of modules, and compatibility of cluster variables.
