Kac's polynomiality and root-multiplicity conjectures

Consider a quiver $Q$ and a dimension vector $d\in\mathbf{N}I$ that is a root. For each prime power $q$, let $a_d(q)$ be the number of isomorphism classes of absolutely indecomposable representations of $\mathbf{F}_qQ$ of dimension vector $d$. Let $\mathfrak{g}_Q$ be the Kac–Moody algebra associated to $Q$.

Kac's conjectures. For every root $d\in\mathbf{N}I$, there exists a polynomial $P_d(t)$ with nonnegative integer coefficients such that

$$P_d(q)=a_d(q)$$

for every prime power $q$, and

$$P_d(0)=\text{the multiplicity of the root }d\text{ in }\mathfrak{g}_Q.$$

These conjectures propose a precise link between counting absolutely indecomposable representations over finite fields and the root multiplicities of the associated Kac–Moody algebra. The source notes that the conjectures remained open, although two had recently been settled in many cases.