Cluster monomials correspond to simple objects in O^{sh} (monoidal categorification for shifted quantum affine algebras)

Prove that under the isomorphism K(O^{sh}) ≅ (a completion of) the cluster algebra A_{Γ_∞′} from Theorem 9 (Theorem \ref{clsh}), every cluster monomial of A_{Γ_∞′} corresponds to the class of a simple object in the category O^{sh} of representations of shifted quantum affine algebras.

Background

The paper constructs an isomorphism between the Grothendieck ring of finite-length representations of shifted quantum affine algebras and a (completed) cluster algebra defined by an explicit infinite quiver Γ_∞′.

Building on monoidal categorification results for ordinary quantum affine algebras, the conjecture seeks to extend the categorification paradigm to the shifted setting by identifying cluster monomials with classes of simple modules.

This would supply a powerful factorization framework for simple modules in O{sh} and connect QQ-relations with cluster exchange relations.

References

In general, we conjecture the following, which can be seen as a generalization of Theorem \ref{mth} to the category $\mathcal{O}{sh}$. All cluster monomials in $\mathcal{A}{\Gamma\infty'}$ correspond to classes of simple objects in $\mathcal{O}{sh}$ through the isomorphism in Theorem \ref{clsh}.

— Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models  (2510.06437 - Hernandez, 7 Oct 2025) in Conjecture 9 (Conjecture \ref{clc}), Section 9 (Shifted quantum groups and cluster algebras)

These results, however, did not provide a full based categorification: they did not identify the localized quantum Grothendieck ring with the upper quantum cluster algebra or establish a bijection between its normalized simple classes and the common triangular basis. Such a based categorification inside a fixed Hernandez--Leclerc category, together with a flat integral quantization of C[X(β)], remained open.

— Monoidal Categorification and Quantization of Braid Varieties  (2609.02962 - Bi, 2 Sep 2026) in Section 1, Introduction