Cellularity of endomorphism algebras of tilting objects

TL;DR

This paper proves that endomorphism algebras of tilting objects in highest weight categories are inherently cellular, providing new bases and a unified framework that connects category theory, algebra, and reflection groups.

Contribution

It generalizes a recent construction to show these algebras are cellular, even without duality, and applies the theory to Hecke algebras of complex reflection groups.

Findings

Endomorphism algebras of tilting objects are cellular in highest weight categories.

Construction of standard bases similar to cellular bases, applicable without duality.

Standard bases for Hecke algebras of complex reflection groups derived from category $\

Abstract

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases -- and thus a general theory of "cell modules" -- for Hecke algebras associated to finite complex reflection groups (as introduced by Brou\'e, Malle, and Rouquier) via category $\mathcal{O}$ of the rational Cherednik algebra. For real reflection groups these bases are cellular.