Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
2026-03-25 06:09:00.1774418940
Indecomposable quiver representations
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From your comments I'm assuming you are talking about an indecomposable representation of a representation finite quiver (i.e. Dynkin quiver).
Probably not the most efficient way, but one way to do it is to use the knitting algorithm for the Auslander-Reiten quiver. For this you start with the simple projective modules for your quiver (for which you have the representations), then in each step you first radical inclusions $\operatorname{rad} P\to P$, which you also have. And as a second step you compute the cokernel of the direct sum of all the maps starting in the earliest module you have constructed so far. Since you are in the representation finite case, you will have constructed every indecomposable projective after a finite number of steps (for $E_8$ unfortunately $120$ steps are necessary, to arrive at the maximal indecomposable representation you probably need half of them, since I assume it will be in the middle of the Auslander-Reiten quiver).