= Interval representations of an equioriented three-vertex quiver
{title2=$I[a,b],\quad1\le a\le b\le3$}
For $1\to2\to3$, $I[a,b]$ has $k$ on the interval of vertices $a,\ldots,b$, zero elsewhere and identity maps along its internal arrows. These six representations are exactly the indecomposables. To decompose $V_1\xrightarrow fV_2\xrightarrow gV_3$, split $\operatorname{im}f\cap\ker g$, its complements inside each of $\operatorname{im}f$ and $\ker g$, and a complement to their sum. Lift the image basis through $f$ and transport the complementary nonkernel basis through $g$. This yields interval blocks directly, without the <Gabriel theorem>.
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