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

@article{Brundan2008HighestWC,
title={Highest weight categories arising from Khovanov's diagram algebra III: category O},
author={Jonathan Brundan and Catharina Stroppel},
journal={arXiv: Representation Theory},
year={2008}
}
• Published 5 December 2008
• Mathematics
• arXiv: Representation Theory
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…
175 Citations

## Figures from this paper

Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
• Mathematics
• 2010
This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined
Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity
• Mathematics
• 2011
This is the first of four articles studying some slight generalisations Hn m of Khovanov’s diagram algebra, as well as quasi-hereditary covers Kn m of these algebras in the sense of Rouquier, and
A diagram algebra for Soergel modules corresponding to smooth Schubert varieties
Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from
Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras
• Mathematics
• 2012
In this paper, we prove Khovanov-Lauda’s cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_{q}(\mathfrak{g})$ be the quantum group associated with a
On the Ext algebras of parabolic Verma modules and A infinity-structures
• Mathematics
• 2011
We study the Ext-algebra of the direct sum of all parabolic Verma modules in the principal block of the Bernstein-Gelfand-Gelfand category O for the hermitian symmetric pair $(\mathfrak{gl}_{n+m}, The modular Weyl-Kac character formula • Mathematics • 2020 We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated in one degree. Each of these homogeneous representations is 2-Verma modules • Mathematics Journal für die reine und angewandte Mathematik • 2021 Abstract We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup • Mathematics • 2009 We prove that blocks of the general linear supergroup are Morita equivalent to a limiting version of Khovanov's diagram algebra. We deduce that blocks of the general linear supergroup are Koszul. Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n) • Mathematics • 2013 We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the ## References SHOWING 1-10 OF 102 REFERENCES Highest weight categories arising from Khovanov's diagram algebra II: Koszulity • Mathematics • 2010 This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products • Mathematics • 2005 The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology Abstract For a fixed parabolic subalgebra 𝔭 of$\mathfrak {gl}(n,\mathbb {C})$we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the Duality between sln(C) and the Degenerate Affine Hecke Algebra • Mathematics • 1998 Abstract We construct a family of exact functors from the Bernstein–Gelfand–Gelfand category O of s l n-modules to the category of finite-dimensional representations of the degenerate affine Hecke Projective modules in the category _{}: self-duality Given a parabolic subalgebra ps of a complex, semisimple Lie algebra , there is a naturally defined category °s of g-modules which includes all the g-modules induced from finite-dimensional Representations of Semisimple Lie Algebras in the BGG Category O Review of semisimple Lie algebras Highest weight modules: Category$\mathcal{O}$: Basics Characters of finite dimensional modules Category$\mathcal{O}\$: Methods Highest weight modules I Highest
Category O and slk link invariants
We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k)
KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY O
This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O over a complex semisimple Lie-algebra, established by Beilinson, Ginzburg and
Centers of degenerate cyclotomic Hecke algebras and parabolic category
We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type B_d(l) consists of symmetric polynomials in its commuting generators. The