Highest weight categories arising from Khovanov's diagram algebra I: cellularity

TL;DR

This paper proves that certain generalisations of Khovanov's diagram algebra are cellular and symmetric, establishing their algebraic structure and setting the stage for connections to representation theory.

Contribution

It demonstrates that H(n,m) is a cellular symmetric algebra and K(n,m) is a cellular quasi-hereditary algebra, foundational for further study.

Findings

H(n,m) is a cellular symmetric algebra

K(n,m) is a cellular quasi-hereditary algebra

Establishes algebraic properties for future representation theory links

Abstract

This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.