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 C ℓ of U q ( 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 with n rows and with entries in [n + ℓ + 1]. 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 a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( 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.

Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez-Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types A n and B n . We prove that prime snake modules are real. We introduce S-systems consisting of equations satisfied by the q-characters of prime snake modules of types A n and B n . Moreover, we show that every equation in the S-system of type A n (respectively, B n ) corresponds to a mutation in the cluster algebra A (respectively, A ′ ) constructed by Hernandez and Leclerc and every prime snake module of type A n (respectively, B n ) corresponds to some cluster variable in A (respectively, A ′ ). In particular, this proves that the Hernandez-Leclerc conjecture is true for all prime snake modules of types A n and B n .

No abstract

The aim of this paper is two-fold: (1) introduce four systems of equations called Msystems and dual M-systems of types An and Bn respectively; (2) make a connection between Msystems (dual M-systems) and cluster algebras and prove that the Hernandez-Leclerc conjecture is true for minimal affinizations of types An and Bn.

No abstract