Quantum affine algebras and Grassmannians

Understand

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.