Quantum affine algebras and Grassmannians

TL;DR

This paper explores the deep connection between quantum affine algebras of type A and Grassmannian cluster algebras, providing explicit formulas and combinatorial rules to understand their structure and representations.

Contribution

It establishes an explicit isomorphism linking modules of quantum affine algebras to Grassmannian cluster variables, and introduces combinatorial methods for mutations and module analysis.

Findings

Explicit formulas for ch(T) in Grassmannian cluster algebras.

A tableau-based rule for mutations in Grassmannian cluster algebras.

Explicit q-character formulas for finite-dimensional modules.

Abstract

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.