Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$

TL;DR

This paper investigates specific irreducible modules for quantum affine algebras, identifying prime modules, their factorizations, and their role in monoidal categorification, revealing new structures like imaginary modules and their implications for cluster algebras.

Contribution

It classifies prime modules supported on one node, proves a unique factorization theorem, and introduces imaginary modules in the context of quantum affine algebras and categorification.

Findings

Prime modules are classified and their tensor product properties analyzed.

Tensor products of higher order Kirillov-Reshetikhin modules contain imaginary modules.

Explicit formulas for Drinfeld polynomials of imaginary modules are provided.

Abstract

We study the family of irreducible modules for quantum affine $\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m$ with $m\le n$. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category $\mathscr C^-$. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov--Reshetikhin module with its dual always contains an imaginary module in its Jordan--Holder series and give an explicit formula for its Drinfeld polynomial. Together with the results of \cite{HL13a} this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of \cite{LM18}. Finally, we use our methods to give a family of imaginary modules in type $D_4$ which do not arise from an embedding of $A_r$ with $r\le 3$ in $D_4$.