Minimal right determiners of irreducible morphisms in algebras of type ${\mathbb A}_n$

Abstract

Let $\Lambda$ be a finite dimensional algebra of type ${\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\Det(\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\Lambda$-modules. If $\Lambda$ is a path algebra, then we have $$|\Det(\Lambda)|= 2n-2, &\mbox{if $p=0$; } 2n-p-1, &\mbox{if $p\geq 1$,}$$ where $p=|\{i\mid i$ is a source in $Q$ with $2\leq i\leq n-1\}|$. If $\Lambda$ is a bound quiver algebra, then we have $$ |\Det(\Lambda)|= 2n-2, &\mbox{if $r=1$; } 2n-p-q-1, &\mbox{if $r\geq 2$,} $$ where $q$ is the number of non-zero sink ideals of $\Lambda$ and $r=|\{i\mid i$ is a sink in $Q$ with $1\leq i\leq n\}|$.