2-Auslander algebras associated with reduced words in Coxeter groups

TL;DR

This paper explores the structure of 2-Auslander algebras arising from reduced words in Coxeter groups, revealing their quasihereditary nature, dualities, and relationships within cluster categories.

Contribution

It generalizes previous work by showing these algebras are quasihereditary and establishes dualities and categorical equivalences related to cluster tilting objects.

Findings

They are quasihereditary and strongly quasihereditary.

Existence of a duality between subcategories and modules over the algebra.

Cluster tilting objects are connected within the same component of the cluster tilting graph.

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.