Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the indecomposable projective $A$-module at vertex $l$ of quiver $Q$.

Let $M$ be a $\tau$-rigid $A$-module, where $\tau$ is the Auslander-Reiten translate. Suppose $\mathfrak{P}_{-1}\overset{f}{\to}\mathfrak{P}_{0}{\to}M{\to} 0$ is the minimal projective presentation of $M$, where, say, $\mathfrak{P}_{-1}=\oplus_{s=1}^{m} P_{a_{s}}$ and $\mathfrak{P}_{0}=\oplus_{t=1}^{n} P_{b_{t}}$, and $P_{a_{s}}$ and $P_{b_{t}}$ are the indecomposable projective $A$-modules at vertices $a_{s}$ and $b_{t}$, respectively, of quiver $Q$.

After fixing a basis, we can assume that $f$ is given by a $n\times m$ matrix, and we know that the $(i,j)$-th entry of $f$ is given by linear combination of paths from vertex $b_{i}$ to vertex $a_{j}$, for $i\in\lbrace1,2,\dotsc,n\rbrace$ and $j\in\lbrace1,2,\dotsc,m\rbrace$.

Do we know anything more specific about these entries? In particular, is it true that each entry of the matrix $f$ can be transformed via action of $\mathrm{Aut}(\mathfrak{P}_-1)\times\mathrm{Aut}(\mathfrak{P}_0)$ to be either $0$ or just a scalar multiple of a single path (i.e., not a linear combination of two or more paths)?

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the indecomposable projective $A$-module at vertex $l$ of quiver $Q$.

Let $M$ be a $\tau$-rigid $A$-module, where $\tau$ is the Auslander-Reiten translate. Suppose $\mathfrak{P}_{-1}\overset{f}{\to}\mathfrak{P}_{0}{\to}M{\to} 0$ is the minimal projective presentation of $M$, where, say, $\mathfrak{P}_{-1}=\oplus_{s=1}^{m} P_{a_{s}}$ and $\mathfrak{P}_{0}=\oplus_{t=1}^{n} P_{b_{t}}$, and $P_{a_{s}}$ and $P_{b_{t}}$ are the indecomposable projective $A$-modules at vertices $a_{s}$ and $b_{t}$, respectively, of quiver $Q$.

After fixing a basis, we can assume that $f$ is given by a $n\times m$ matrix, and we know that the $(i,j)$-th entry of $f$ is given by linear combination of paths from vertex $b_{i}$ to vertex $a_{j}$, for $i\in\lbrace1,2,\dotsc,n\rbrace$ and $j\in\lbrace1,2,\dotsc,m\rbrace$.

Do we know anything more specific about these entries? In particular, is it true that each entry of the matrix $f$ can be transformed via action of $\mathrm{Aut}(\mathfrak{P}_{-1})\times\mathrm{Aut}(\mathfrak{P}_0)$ to be either $0$ or just a scalar multiple of a single path (i.e., not a linear combination of two or more paths)?