# Stabilization of Kac polynomials: some conjectures

@article{Hennecart2020StabilizationOK, title={Stabilization of Kac polynomials: some conjectures}, author={Lucien Hennecart}, journal={arXiv: Representation Theory}, year={2020} }

We give some conjectures concerning the behaviour of Kac polynomials of quivers when increasing the number of arrows: they seem to converge in the ring of power series, with a linear rate of convergence. We prove the convergence for the Kronecker quiver in dimension $(1,d)$. It would be nice to find a geometric interpretation of this, either in terms of Nakajima quiver varieties, or in terms of Lusztig nilpotent varieties. All computations were made using SageMath.

#### References

SHOWING 1-10 OF 13 REFERENCES

Kac’s conjecture from Nakajima quiver varieties

- Mathematics
- 2008

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… Expand

Positivity for Kac polynomials and DT-invariants of quivers

- Mathematics, Physics
- 2012

We give a cohomological interpretation of both the Kac polynomial and the rened Donaldson-Thomas-invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a… Expand

Absolutely indecomposable representations and Kac-Moody Lie algebras

- Mathematics
- 2001

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… Expand

On the number of points of nilpotent quiver varieties over finite fields

- Mathematics
- 2017

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has… Expand

The integrality conjecture and the cohomology of preprojective stacks

- Mathematics
- 2016

By importing the compactly supported cohomology of various stacks of representations of the preprojective algebra $\Pi_Q$, for $Q$ an arbitrary quiver, into the theory of cohomological… Expand

Counting Representations of Quivers over Finite Fields

- Mathematics
- 2000

Abstract By counting the numbers of isomorphism classes of representations (indecomposable or absolutely indecomposable) of quivers over finite fields with fixed dimension vectors, we obtain a… Expand

MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS

- Mathematics
- 1994

IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian… Expand

Counting absolutely cuspidals for quivers

- Mathematics
- Mathematische Zeitschrift
- 2018

For an arbitrary quiver $$Q=(I,\Omega )$$Q=(I,Ω) and dimension vector $$\mathbf {d} \in \mathbb {N}^I$$d∈NI we define the dimension of absolutely cuspidal functions on the moduli stacks of… Expand

Generalized Kac-Moody algebras

- Mathematics
- 1988

On etudie une classe d'algebres de Lie qui ont une forme bilineaire contravariante qui est presque definie positive. Ces algebres generalisent celles de Kac-Moody et peuvent etre considerees comme… Expand

Infinite-dimensional Lie algebras

- Mathematics
- 1974

1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8.… Expand