On the Number of Absolutely Indecomposable Representations of a Quiver

A conjecture of Kac states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac–Moody Lie algebra. In this paper we give a combinatorial reformulation of Kacʹs conjecture in terms of a property of q-multinomial coefficients. As a side result we give a formula for certain inverse Kostka–Foulkes polynomials.