Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras

TL;DR

This paper proves the connectedness of mutation graphs of maximal rigid modules over preprojective algebras of Dynkin type, confirming a conjecture for finite and tame types, and establishes isomorphisms with tilting graphs in the finite case.

Contribution

It confirms the connectedness conjecture for mutation graphs over preprojective algebras of Dynkin type in finite and tame cases, and relates these graphs to tilting graphs in the finite case.

Findings

Mutation graph is connected for finite type

Mutation graph is connected for tame type

Mutation graph is isomorphic to tilting graph in finite type

Abstract

Let $Q$ be a finite quiver of Dynkin type and $\Lambda=\Lambda_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\mathcal {T}_\Lambda$ be the mutation graph of maximal rigid $\Lambda$ modules. Geiss, Leclerc and Schr$\ddot{\rm o}$er conjectured that $\mathcal {T}_\Lambda$ is connected, see [C.Geiss, B.Leclerc, J.Schr\"{o}er, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $\Lambda$ is of representation finite type or tame type. Moreover, we also prove that $\mathcal {T}_\Lambda$ is isomorphic to the tilting graph of ${\rm End}_\Lambda T$ for each maximal rigid $\Lambda$-module $T$ if $\Lambda$ is representation-finite.