2012/05/31 by Yoshiyuki Kimura, Fan Qin · 3 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Basis (linear algebra) #Cluster algebra #Geometry #Kronecker delta #Mathematics #Monomial #Noncommutative geometry #Nonlinear Waves and Solitons #Pure mathematics #Quantum #Quantum mechanics #Quiver #Standard basis #math.QA #math.RT
paper · pdf · doi:10.1016/j.aim.2014.05.014
published as Advances in Mathematics 262 (2014): 261-312 · 42 pages, minor corrections, references updated
arxiv created 2012/06/12 · openalex publication_date 2014/06/05 · arxiv updated 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.