Kac’s conjecture from Nakajima quiver varieties
By Tamás Hausel. 2008
58 citations recorded by the source corpus.
We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/d1a2a87e379861cfc1e65d6a83830449a73a55a9
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
- O. Schiffmann
- Ben Davison
- E. Vasserot
- H. Nakajima
- Davesh Maulik
- Ben Webster
- I. Cherednik
- Andrew Mathas
In Statistical and Nonlinear Physics
Citations in the corpus
- The integrality conjecture and the cohomology of preprojective stacks (2016, background): “We explain how a specialisation of a special case of equation (9) yields Hausel’s formula for the Betti polynomials of Nakajima quiver varieties [18] in Section 7.”
- Indecomposable vector bundles and stable Higgs bundles over smooth projective curves (2014, methodology): “tant term A d(0) of the Kac polynomial attached to a quiver Qand a dimension vector d is equal to the multiplicity of the root P i diαi in the Kac-Moody Lie algebra gQ canonically associated to Q(see [H2] for details). In the context of a smooth projective curve one is therefore led to seek an analo”
- On cohomological Hall algebras of quivers: Generators (2017, background): “tured that the graded dimensions of the root spaces g[α] ⊂g are, after a suitable grading shift, precisely given by the Kac polynomial Aα(t). This is a generalization of the Kac conjecture, proved in [20], stating that for quivers with no 1-cycles the multiplicity of the root αin the Kac-Moody algeb”