2020/07/31 by Ryo Fujita, Se-jin Oh, Se‐jin Oh · 23 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Cellular algebra #Combinatorics #Coxeter group #Dynkin diagram #Geometry #Inverse #Lie algebra #Mathematics #Physics #Pure mathematics #Quantum #Quantum affine algebra #Quantum mechanics #Quiver #Representation theory #Simple (philosophy) #Weyl group #math.CO #math.QA #math.RT
paper · pdf · doi:10.1007/s00220-021-04028-8
published in Communications in Mathematical Physics 384(2), 1351-1407 (Springer Science+Business Media) · v2: 52 pages, a considerable revision. v3 : 52 pages, minor revision, final version
openalex publication_date 2021/03/24 · arxiv created 2021/04/02 · arxiv updated 2021/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
For a complex finite-dimensional simple Lie algebra \mathfrakg, 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 \mathfrakg, 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 \mathfrakg. 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 Λ(V,W) and Λ^∞(V, W) introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules V and W.