Absolutely indecomposable representations and Kac-Moody Lie algebras

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Abstract
A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove the first half of this conjecture for indivisible dimension vectors and the second half for dimension vectors that are equal to one in some vertex.

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