Hausel, TamasIST Austria; Letellier, Emmanuel; Rodríguez Villegas, Fernando
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. Thisis achieved by computing, via an arithmetic Fourier transform, the dimensions of the isotypical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincare 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.
Annals of Mathematics
The first author thanks the Royal Society for funding his research 2005-2012 in the form of a Royal Society University Research Fellowship as well as the Mathematical Institute and Wadham College in Oxford for a very productive environment. The second author is supported by Agence Nationale de la Recherche grant ANR-09-JCJC-0102-01. The third author is supported by the NSF grant DMS-1101484 and a Research Scholarship from the Clay Mathematical Institute.
1147 - 1168
Hausel T, Letellier E, Rodríguez Villegas F. Positivity for Kac polynomials and DT-invariants of quivers. Annals of Mathematics. 2013;177(3):1147-1168. doi:10.4007/annals.2013.177.3.8
Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2013). Positivity for Kac polynomials and DT-invariants of quivers. Annals of Mathematics. Princeton University Press. https://doi.org/10.4007/annals.2013.177.3.8
Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Positivity for Kac Polynomials and DT-Invariants of Quivers.” Annals of Mathematics. Princeton University Press, 2013. https://doi.org/10.4007/annals.2013.177.3.8.
T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Positivity for Kac polynomials and DT-invariants of quivers,” Annals of Mathematics, vol. 177, no. 3. Princeton University Press, pp. 1147–1168, 2013.
Hausel T, Letellier E, Rodríguez Villegas F. 2013. Positivity for Kac polynomials and DT-invariants of quivers. Annals of Mathematics. 177(3), 1147–1168.
Hausel, Tamás, et al. “Positivity for Kac Polynomials and DT-Invariants of Quivers.” Annals of Mathematics, vol. 177, no. 3, Princeton University Press, 2013, pp. 1147–68, doi:10.4007/annals.2013.177.3.8.