Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

TL;DR

This paper constructs a diagrammatic algebra $\

Contribution

It introduces a new algebraic model for perverse sheaves on isotropic Grassmannians, linking geometric categories to graded Koszul algebras.

Findings

The algebra $\

Explicit formulas for graded decomposition numbers are provided.

The algebra $\

Abstract

For each integer $k\geq 4$ 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 ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.