Imaginary modules arising from tensor products of snake modules
Matheus Brito, Adriano Moura
TL;DR
This work extends the domain of imaginary simple modules in quantum loop algebras by analyzing tensor products of prime snake modules in type $A$, beyond the previously understood higher-order KR cases. The authors develop a covering-ladder framework to explicitly compute socles, and they construct a chain of highest-$\ell$-weight submodules $M_0\subseteq M_1\subseteq\cdots\subseteq M_p$, with quotients conjectured to be simple and imaginary; the first quotient is proven simple and imaginary under a mild extra condition. A key methodological advance is the use of Mukhin–Young lattice paths to model $\ell$-weights, combined with a detailed analysis of right/left subintervals and combinatorial ladder operations to control tensor products. The results generalize previous BC23-type constructions, provide evidence for broader families of imaginary modules, and offer a concrete combinatorial counting framework for coverings, ladders, and MY-path configurations that underpin the $\ell$-weight structure. Overall, the paper delivers a systematic representation-theoretic route to detect and construct new imaginary objects in the category of finite-dimensional $U_q(\tilde{\mathfrak{g}})$-modules, with potential impact on understanding cluster-algebra limitations and the structure of tensor products in type $A$.
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.