2001/11/08 by Alexander Kuznetsov, Kuznetsov, Alexander · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/0111092
LaTeX, 27 pages
arxiv created 2001/11/08 · openalex publication_date 2001/11/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the Γ-equivariant Hilbert scheme XΓ[n] and the Hilbert scheme XΓ[n] (where X=\C2, Γ⊂ SL(\C2) is a finite subgroup, and XΓ is a minimal resolution of X/Γ) are quiver varieties for the affine Dynkin graph, corresponding to Γ via the McKay correspondence, the same dimension vectors, but different parameters ζ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties XΓ[n] and XΓ[n] are diffeomorphic. Computing their cohomology (in the case Γ=\Z/d\Z) via the fixed points of (\C^*×\C^*)-action we deduce the following combinatorial identity: the number UCY(n,d) of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number CY(n,d) of collections of d Young diagrams with the total number of boxes equal to n.