Kac's cellular decomposition conjecture for indecomposable quiver representations

Fix a quiver $Q$ and a dimension vector $\alpha$, and write

$$a_\alpha(q)=\sum_{i=0}^n c_iq^i$$

for the polynomial counting absolutely indecomposable representations over finite fields. Consider the set of isomorphism classes of indecomposable representations of $Q$ of dimension vector $\alpha$.

Kac's cellular decomposition conjecture. This set admits a cellular decomposition by locally closed subvarieties isomorphic to affine spaces, with $c_i$ equal to the number of cells of dimension $i$.

The conjecture is a major motivation for constructing explicit cells of indecomposable representations. The paper notes that the set of isomorphism classes need not have a canonical variety structure and may depend on the base field, so the assertion is inspirational rather than literal in that generality.