Positivity of Kac polynomials and DT-invariants for quivers

2012/04/11 by Tamás Hausel, Hausel, Tamas, Emmanuel Letellier +3 · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1204.2375

openalex publication_date 2012/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Poincaré 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.

Cited by

• The skein partition function of the mapping torus
• Harder-Narasimhan Filtration on Moment Map for Quiver Representations
• The BPS decomposition theorem
• On the number of points of the Lusztig nilpotent variety over a finite field

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• Purity of critical cohomology and Kac's conjecture
• DT-invariants of quivers and the Steinberg character of GL_n
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