On the number of absolutely indecomposable representations of a quiver (Q1818825)

A quiver \(Q\) consists of a set \(Q_0\) of vertices, a set \(Q_1\) of arrows and two maps \(t,h : Q_1 \rightarrow Q_0\) assigning to an arrow \(\phi\) its tail and its head. A representation of a quiver is a family of finite dimensional vector spaces over \(k\), one for each vertex, together with a linear map for each arrow. If \(k\) is a finite field with \(q\) elements then \textit{V. G. Kac} [ Lect. Notes Math. 996, 74-108 (1993; Zbl 0534.14004)] has shown that the number of absolutely indecomposable representations of a quiver over \(k\) is a polynomial in \(q\) with integral coefficients. He also conjectured that the constant term of this polynomial is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. The present paper is devoted to reformulate such a conjecture into a property of Gaussian multinomial coefficients.