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Let $f: V \to W$ be a linear transformation. Then we understand that $V\cong \ker(f) \oplus \text{im}(f)$ and $W\cong \text{im}(f) \oplus \text{coker}(f).$

With these identifications, $f$ is $\begin{bmatrix} 0 & \text{Id}\\ 0 & 0\end{bmatrix}.$ It says that any two linear transformations $f, g$ are equivalent (that is there is automorphisms $\phi$ of $V$ and $\psi$ of $W$ such that $\psi\circ f= g\circ \phi$) if and only if $\dim(\ker(f)) = \dim(\ker(g)).$

Now consider the sequence of linear maps $\begin{array}[cccccc] &V_1 & \xrightarrow{f_1} & V_2 &\xrightarrow{f_2} & \cdots\xrightarrow{f_{n-1}} & V_n\end{array}.$ We call

$\begin{array}[cccccc] &V_1 & \xrightarrow{f_1} & V_2 &\xrightarrow{f_2} & \cdots\xrightarrow{f_{n-1}} & V_n\end{array}.$ and $\begin{array}[cccccc] &V_1 & \xrightarrow{g_1} & V_2 &\xrightarrow{g_2} & \cdots\xrightarrow{g_{n-1}} & V_n\end{array}$ are equivalent if and only if there exist automorphisms $\phi_i: V_i \to V_i$ for $1\leq i \le n$ such that $\phi_{i+1}\circ f_i = g_i\circ \phi_i.$

For $n=3$ we understand that $\dim(\ker(f_1)), \dim(\ker(f_2)),$ and $\dim(\text{Im}(f_1)\cap \ker(f_2))$ classify the above diagrams up to above equivalence.

What kind of condition one should put to classify the above general diagrams?

Thank you in advance. Any help will be appreciated.

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    $\begingroup$ These are quiver representations for the path quivers: en.m.wikipedia.org/wiki/… $\endgroup$ Sep 26, 2020 at 17:10
  • $\begingroup$ @QiaochuYuan: Thank you so much for the comment. Do you know any reference for the classification of representations of path quivers up to isomorphism? Also, any classification theorems for the representations of simple quivers (that is without loops or multiple edges)? $\endgroup$
    – Surojit
    Sep 26, 2020 at 18:14
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    $\begingroup$ You can see, for example, here: math.uni-bielefeld.de/~ringel/opus/h-f.pdf $\endgroup$ Sep 26, 2020 at 23:32

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