2006/08/31 by Catharina Stroppel · 54 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Coherent sheaf #Cohomology #Derived category #Endomorphism #Endomorphism ring #Equivalence of categories #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Ring (chemistry) #Subalgebra #math.GT #math.RT #msc:14M15 #msc:14M17 #msc:16S99 #msc:17B10 #msc:20C30 #msc:20G05 #msc:57M27
paper · pdf · doi:10.1112/s0010437x09004035
published in Compositio Mathematica 145(4), 954-992 (Cambridge University Press) · 39 pages, 9 figures, added a few remarks
arxiv created 2008/03/06 · openalex publication_date 2009/06/19 · arxiv updated 2014/01/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract For a fixed parabolic subalgebra 𝔭 of \mathfrak gl(n,\mathbb C) we prove that the centre of the principal block 𝒪 0 𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H * (ℬ 𝔭 ) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪 0 𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category Perv B ( G ( k , n )) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋ n is realised as the endomorphism ring of some object from Perv B ( G ( n , n )) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪 0 𝔭 . From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors , Duke Math. J. 126 (3) (2005), 547–596] by restriction.