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

TL;DR

This paper investigates the minimal right determiners of irreducible morphisms in algebras of type A_n, providing explicit formulas based on the quiver structure and algebra type, enhancing understanding of morphism determiners.

Contribution

It derives explicit formulas for the number of minimal right determiners in type A_n algebras, considering path and bound quiver cases, based on quiver sources and sinks.

Findings

Formulas for minimal right determiners in path algebras.

Formulas for minimal right determiners in bound quiver algebras.

Dependence of determiners on quiver sources and sinks.

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 $1\leq i\leq n\}|$.