Jordan-H\"older property for shifted quantum affine algebras

TL;DR

This paper proves that finite length representations of shifted quantum affine algebras are stable under fusion, forming a subring in the Grothendieck ring, and conjectures its isomorphism to a known cluster algebra.

Contribution

It establishes the stability of finite length representations under fusion and links the Grothendieck ring to a cluster algebra, advancing understanding of shifted quantum affine algebras.

Findings

Finite length representations are stable under fusion product.

The Grothendieck ring forms a non-topological subring.

Conjecture: this subring is isomorphic to a specific cluster algebra.

Abstract

We prove that finite length representations of shifted quantum affine algebras in category $\mathcal{O}^{\mathrm{sh}}$ are stable by fusion product. This implies that in the topological Grothendieck ring $K_0(\mathcal{O}^{\mathrm{sh}})$ the Grothendieck group of finite length representations forms a non-topological subring. We also conjecture this subring is isomorphic to the cluster algebra discovered in arXiv:2401.04616. In the course of our proofs, we establish that any simple representation in category $\mathcal{O}^{\mathrm{sh}}$ descends to a truncation, for certain truncation parameters as conjectured in arXiv:2010.06996 in terms of Langlands dual $q$-characters.