Given a quiver $Q$, we can compute its (absolutely) indecomposable representations over a finite field using the Hua formula. We use its formulation from arXiv:math/0608321.
For any $d\in\Gamma^+=\mathbb N^{Q_0}$, we define its multi-partition to be a finite sequence of (nonzero) vectors $\lambda=(\lambda_1,\dots,\lambda_m)$ in $\Gamma^+$ such that $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_m$ (meaning that $\lambda_k-\lambda_{k+1}\in\Gamma^+$) and $|\lambda|=\sum_k\lambda_k=d$. Define $$R_\lambda(q)=\prod_{k=1}^m\frac{q^{-\chi(\lambda_k,\lambda_k)}}{(q^{-1})_{\lambda_k-\lambda_{k+1}}},$$ where $(q)_d=\prod_{i}(q)_{d_i}$ for $d\in\Gamma^+$ and $(q)_n=(1-q)\dots(1-q^n)$ for $n\in\mathbb N$. Then the polynomials $A_d(q)$ counting absolutely indecomposable quiver representations having dimension vector $d$ are given by the formula $$\sum_d A_d(q)x^d=(q-1)Log\left(\sum_{\lambda}R_\lambda(q)x^{|\lambda|}\right),$$ where $Log$ is the plethystic logarithm. We use polynomials in $q=y^2$.
import os, sys
module_path = os.path.abspath(os.path.join('..'))
if module_path not in sys.path:
sys.path.append(module_path)
from msinvar import *
from msinvar.indecomposable import hua_formula
set_plots()
Q=JordanQuiver(1); show(Q)
Q.prec([3])
hua_formula(Q).dict()
{(1,): y^2, (2,): y^2, (3,): y^2}
Q=Quiver('1-2'); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()
{(1, 0): 1, (0, 1): 1, (1, 1): 1}
Q=KroneckerQuiver(2); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()
{(1, 0): 1, (0, 1): 1, (1, 1): y^2 + 1, (2, 1): 1, (1, 2): 1, (2, 2): y^2 + 1, (3, 2): 1, (2, 3): 1, (3, 3): y^2 + 1}
Q=KroneckerQuiver(2); show(Q)
Q.prec([3,3])
hua_formula(Q).dict()
{(1, 0): 1, (0, 1): 1, (1, 1): y^2 + 1, (2, 1): 1, (1, 2): 1, (2, 2): y^2 + 1, (3, 2): 1, (2, 3): 1, (3, 3): y^2 + 1}
Q=CyclicQuiver(3); show(Q)
Q.prec([3,3,3])
hua_formula(Q).dict()
{(1, 0, 0): 1, (0, 1, 0): 1, (1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 1): 1, (0, 1, 1): 1, (1, 1, 1): y^2 + 2, (2, 1, 1): 1, (1, 2, 1): 1, (2, 2, 1): 1, (1, 1, 2): 1, (2, 1, 2): 1, (1, 2, 2): 1, (2, 2, 2): y^2 + 2, (3, 2, 2): 1, (2, 3, 2): 1, (3, 3, 2): 1, (2, 2, 3): 1, (3, 2, 3): 1, (2, 3, 3): 1, (3, 3, 3): y^2 + 2}