Q-data and representation theory of untwisted quantum affine algebras

Abstract

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $\Lambda (V,W)$ and ${\Lambda }^{\infty }(V,W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.

arXiv commentv2: 52 pages, a considerable revision. v3 : 52 pages, minor revision, final version

References in corpus (20)

• Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
• Braid Group Action and Quantum Affine Algebras
• Cluster algebras and quantum affine algebras
• The q-characters of representations of quantum affine algebras and deformations of W-algebras
• Baxter's Relations and Spectra of Quantum Integrable Models
• Quantum Grothendieck rings and derived Hall algebras
• Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules
• Yangians, Integrable Quantum Systems and Dorey's rule
• Simplicity of heads and socles of tensor products
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras
• On Level Zero Representations of Quantized Affine Algebras
• Monoidal categorification and quantum affine algebras
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Categorical relations between Langlands dual quantum affine algebras: Exceptional cases
• Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras
• Characters and Blocks for Finite-Dimensional Representations of Quantum Affine Algebras
• Simplicity of tensor products of Kirillov--Reshetikhin modules: nonexceptional affine and G types
• Categories over quantum affine algebras and monoidal categorification

Cited by in corpus (10)

• Monoidal categorification and quantum affine algebras
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Poles of finite-dimensional representations of Yangians
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Deformed Cartan matrices and generalized preprojective algebras I: Finite type
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Deformed Cartan matrices and generalized preprojective algebras II: General type
• Equivariant multiplicities via representations of quantum affine algebras
• From i-boxes to signed words