2-Auslander algebras associated with reduced words in Coxeter groups

Abstract

In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{\Lambda_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the 2-Auslander algebras introduced in \cite{I1}. Generalizing work in \cite{GLS1} we show that they are quasihereditary, even strongly quasihereditary in the sense of \cite{R}. We also describe the cluster tilting object giving rise to the Ringel dual, and prove that there is a duality between $\Sub{\Lambda_w}$ and the category $\mathcal{F}(\Delta)$ of good modules over the quasihereditary algebra. When $w = uv$ is a reduced word, we show that the 2-Calabi-Yau triangulated category $\underline{\Sub}\Lambda_v$ is equivalent to a specific subfactor category of $\underline{\Sub}\Lambda_w.$ This is applied to show that a standard cluster tilting object $M$ in $\Sub{\Lambda_w}$ and the cluster tilting object $\Lambda_w\oplus\Omega{M}$ lie in the same component in the cluster tilting graph.