Isomorphisms among quantum Grothendieck rings and propagation of positivity

Abstract

Let ($\mathfrak{g},\mathsf{g})$ be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with $\mathsf{g}$ being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories ${\mathcal{C}}_{\mathfrak{g}}$ and ${\mathcal{C}}_{\mathsf{g}}$ of finite-dimensional representations over the quantum loop algebras of $\mathfrak{g}$ and $\mathsf{g}$ respectively. As a consequence, we solve long-standing problems : the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced $\mathfrak{g}$. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [H., Adv. Math., 2004]) for simple modules in remarkable monoidal subcategories of ${\mathcal{C}}_{\mathfrak{g}}$ for any non-simply-laced $\mathfrak{g}$, and for any simple finite-dimensional modules in ${\mathcal{C}}_{\mathfrak{g}}$ for $\mathfrak{g}$ of type ${\mathrm{B}}_n$. In the course of the proof we obtain and combine several new ingredients. In particular we establish a quantum analog of $T$-systems, and also we generalize the isomorphisms of [H.-Leclerc, J. Reine Angew. Math., 2015] and [H.-O., Adv. Math., 2019] to all $\mathfrak{g}$ in a unified way, that is isomorphisms between subalgebras of the quantum group of $\mathsf{g}$ and subalgebras of the quantum Grothendieck ring of ${\mathcal{C}}_{\mathfrak{g}}$.

v3: minor revision, 68 pages, Lemma 3.12 and Lemma 8.3 corrected, to appear in Crelle's journal

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Cited by in corpus (5)

• The $(q,t)$-Cartan matrix specialized at $q=1$
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• Equivariant multiplicities via representations of quantum affine algebras
• Inflations among quantum Grothendieck rings of type A