2023/10/13 by Jens Niklas Eberhardt, Eberhardt, Jens Niklas, Catharina Stroppel +1 · 1 citation
Mathematics · #14F08 #17B05 #17B10 #22E46 #32S60 #55N91 #57R58 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2310.09206
openalex publication_date 2023/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this first of a series of articles on standard extension algebras we study standard perverse sheaves on varieties with \mathbbGm-actions. Based on Braden's hyperbolic localisation, we describe their extension algebra geometrically via a convolution structure on the intersections of attracting with repelling cells. We introduce a multiplicative structure on open Richardson varieties which provides a practical way to compose these extensions in case of flag varieties. For open Richardson varieties in Grassmannians we construct two explicit cell decompositions, of Gauss- and of Deodhar-type. It is shown that the latter is a stratification with the same combinatorics as Deodhar's decomposition. We introduce a calculus of Fukaya diagrams to encode the geometry of the decompositions. It provides a model for the cohomology of open Richardson varieties and thus for standard extensions. The calculus is motivated by the Mak-Smith Fukaya-Seidel category of a natural Lefschetz fibration and should allow to compute morphism spaces in there. We finally discuss the relation of our work to extensions of (parabolic) Verma modules in category O as well as to the computation of R- and R'-polynomials.