Highest weight categories arising from Khovanov's diagram algebra III: category O

TL;DR

This paper establishes an algebraic proof that integral blocks of parabolic category O for gl(m) x gl(n) are Morita equivalent to quasi-hereditary covers of generalized Khovanov algebras, linking representation theory and combinatorics.

Contribution

Provides a new algebraic proof of Morita equivalence for blocks of category O, avoiding geometric methods, and constructs 2-Kac-Moody representations combinatorially.

Findings

Morita equivalence between category O blocks and generalized Khovanov algebras

Algebraic approach using Schur-Weyl duality

Concrete combinatorial construction of 2-Kac-Moody representations

Abstract

We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A.