Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

Abstract

For each integer $$k\ge 4$$ , we describe diagrammatically a positively graded Koszul algebra $$\mathbb {D}_k$$ such that the category of finite dimensional $$\mathbb {D}_k$$ -modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type $$\mathrm{D}_k$$ or $$\mathrm{B}_{k-1}$$ , constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) “folding” procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.