Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

For each integer k≥ 4 , we describe diagrammatically a positively graded Koszul algebra D_k such that the category of finite dimensional D_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D _k or 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.