On the number of tilting modules over a class of Auslander algebras

TL;DR

This paper calculates the exact number of tilting modules over Auslander algebras derived from radical square zero algebras of Dynkin quivers, revealing explicit formulas depending on the quiver type.

Contribution

It provides explicit formulas for counting tilting modules over Auslander algebras associated with radical square zero algebras of Dynkin quivers, a specific class not thoroughly analyzed before.

Findings

Number of tilting modules for A_m type is 2^{m-1}.

Number of tilting modules for D_m and E_m types is 2^{m-3} times 14.

Results depend on the Dynkin quiver type and provide exact counts.

Abstract

Let $\Lambda$ be a radical square zero algebra of a Dynkin quiver and let $\Gamma$ be the Auslander algebra of $\Lambda$. Then the number of tilting right $\Gamma$-modules is $2^{m-1}$ if $\Lambda$ is of $A_{m}$ type for $m\geq 1$. Otherwise, the number of tilting right $\Gamma$-modules is $2^{m-3}\times14$ if $\Lambda$ is either of $D_{m}$ type for $m\geq 4$ or of $E_{m}$ type for $m=6,7,8$.