Positivity of Kac polynomials and DT-invariants for quivers

Details Positivity of Kac polynomials and DT-invariants for quivers Positivity of Kac polynomials and DT-invariants for quivers

2012-04-11

arxiv.org/abs/1204.2375v1

Mathematics

Representation Theory

Scientific paper

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas- invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincar\'e polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.

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