On the number of absolutely indecomposable representations of a quiver

Abstract

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 Kac's conjecture in terms of a property of q-multinomial coefficients. As a side result we give a formula for certain inverse Kostka-Foulkes polynomials. (C) 1999 Academic Press.