Preprojective algebras and c-sortable words

TL;DR

This paper explores the structure of certain finite-dimensional algebras derived from preprojective algebras of acyclic quivers, focusing on filtrations associated with c-sortable words and their connections to tilting modules.

Contribution

It introduces a detailed analysis of filtrations of algebras linked to c-sortable words, revealing their relation to tilting modules with finite torsionfree classes.

Findings

Filtrations of $ ext{Lambda}_w$ relate to tilting $kQ$-modules.

Consecutive quotients of the filtration can be described via tilting modules.

Special case of c-sortable words yields structured algebraic relations.

Abstract

Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In this paper we look at filtrations of $\Lambda_w$ associated to any reduced expression $\mathbf{w}$ of $w$. We are especially interested in the case where the word $\mathbf{w}$ is $c$-sortable, where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ$-modules with finite torsionfree class.