Imaginary modules arising from tensor products of snake modules

TL;DR

This paper extends the construction and understanding of imaginary modules in quantum loop algebra representations, providing explicit descriptions and new classes of such modules through tensor products of snake modules.

Contribution

It introduces a new explicit description of socles of tensor products of snake modules and conjectures the simplicity and imaginary nature of certain module quotients.

Findings

Explicit description of socle of tensor products of snake modules.

Identification of a sequence of highest-$\

,

Abstract

Motivated by the limitations of cluster algebra techniques in detecting imaginary modules, we build on the representation-theoretic framework developed by the first author and Chari to extend the construction of such modules beyond previously known cases, which arise from the tensor product of a higher-order Kirillov--Reshetikhin module and its dual. Our first main result gives an explicit description of the socle of tensor products of two snake modules, assuming the corresponding snakes form a covering pair of ladders. By considering a higher-order generalization of the covering relation, we describe a sequence of inclusions of highest-$\ell$-weight submodules of such tensor products. We conjecture all the quotients of subsequent modules in this chain of inclusions are simple and imaginary, except for the socle itself, which might be real. We prove the first such quotient is indeed simple and, assuming an extra mild condition, we also prove it is imaginary, thus giving rise to new classes of imaginary modules within the category of finite-dimensional representations of quantum loop algebras in type A.