Motivic Donaldson–Thomas invariants and the Kac conjecture

Mozgovoy, Sergey

doi:10.1112/s0010437x13007148

Cited by 11 publications

(13 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similarly, our proof of the general case will follow by interpreting the coefficients of A v (q) as the dimensions of the sign isotypical component of cohomology groups of a smooth generic quiver variety Qṽ attached to an extended quiver (see (1.9)). Recently, Mozgovoy [26] proved Conjecture 1.1 for any dimension vector for quivers with at least one loop at each vertex. His approach uses Efimov's proof [10] of a conjecture of Kontsevich-Soibelman [20] which implies positivity for certain refined DT-invariants of symmetric quivers with no potential.…”

Section: The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Positivity for Kac polynomials and DT-invariants of quivers

2013

View full text Add to dashboard Cite

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomasinvariants 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. The main resultsLet Γ = (I, Ω) be a quiver: that is, an oriented graph on a finite set I = {1, . . . , r} with Ω a finite multiset of oriented edges. In his study of the representation theory of quivers, Kac [17] introduced A v (q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q of dimension v = (v 1 , . . . , v r ) and showed they are polynomials in q. We call A v (q) the Kac polynomial for Γ and v. Following ideas of Kac [17], Hua [16] proved the following generating function identity:where P denotes the set of partitions of all positive integers, Log is the plethystic logarithm (see [14, §2.3.3]), , is the pairing on partitions defined bywith m j (λ) the multiplicity of the part j in the partition λ, which implies positivity for certain refined DT-invariants of symmetric quivers with no potential.The goal of Kontsevich-Soibelman's theory is to attach refined (or motivic, or quantum) DonaldsonThomas invariants (or DT-invariants for short) to Calabi-Yau 3-folds X. The invariants should only depend on the derived category of coherent sheaves on X and some extra data; this raises the possibility of defining DTinvariants for certain Calabi-Yau 3-categories which share the formal properties of the geometric situation, but are algebraically easier to study. The simplest of such examples are the Calabi-Yau 3-categories attached to quivers (symmetric or not) with no potential (c.f. [12]).Denote by Γ = (I, Ω) the double quiver, that is Ω = Ω Ω opp , where Ω opp is obtained by reversing all edges in Ω. The refined DT-invariants of Γ (a slight renormalization of those introduced by Kontsevich and Soibelman [19]) are defined by the following combinatorial construction.In fact, as a consequence of Efimov's proof [10] of [20, Conjecture 1], DT v (q) actually has non-negative coefficients. We will give an alternative proof of this in (1.10) by interpreting its coefficients as dimensions of cohomology groups of an associated quiver variety. Remark 1.2. We should stress that we have restricted to double quivers for the benefit of exposition; our results extend easily to any symmetric quiver. We outline how to treat the general case in §3.2. The technical starting point in this paper is a common generalization of (1.3) and Hua'...

show abstract

Section: The Main Resultsmentioning

confidence: 99%

“…Acknowledgements. We would like to thank Sergey Mozgovoy for explaining his papers [26,27] and for useful comments. We thank William Crawley-Boevey, Bernard Keller, Maxim Kontsevich, Andrea Maffei, Hiraku Nakajima, Philippe Satgé, Yan Soibelman and Balázs Szendrői for useful comments.…”

Section: The Main Resultsmentioning

confidence: 99%

Positivity for Kac polynomials and DT-invariants of quivers

2013

View full text Add to dashboard Cite

show abstract

“…This was achieved by giving a cohomological interpretation of A Γ,v (q). A more recent work by Mozgovoy [26] proves Conjecture 3.1.2 for any dimension vector for quivers with at least one loop at each vertex. His proof is accomplished via work of Kontsevich-Soibelman [18] and Efimov [4] on motivic Donaldson-Thomas invariants associated to quivers.…”

Section: Conjecture 312 the Polynomial A γV (T ) Has Non-negativementioning

confidence: 96%

Arithmetic harmonic analysis on character and quiver varieties II

Hausel¹,

Letellier²,

Rodríguez-Villegas³

2013

Advances in Mathematics

View full text Add to dashboard Cite

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on C × × C × , modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.Relation between Hilbert schemes and modular forms was first investigated by Göttsche [8].

show abstract

“…Recall that Kac conjectured that the coefficients of Kac polynomials (for any finite quiver) are non-negative [14]. This conjecture was proved in in the case of an indivisible dimension vector by Crawley-Boevey and van den Bergh [3] with further case proved by Mozgovoy [17]; it was proved in full generality by Hausel-Letellier-Villegas [12]. The proofs all give a cohomological interpretation of the coefficients of the Kac polynomial.…”

Section: Graph Polynomialsmentioning

confidence: 99%

Torus orbits on homogeneous varieties and Kac polynomials of quivers

2018

View full text Add to dashboard Cite

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients. Contents

show abstract

Motivic Donaldson–Thomas invariants and the Kac conjecture

Abstract: We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.

Cited by 11 publications

References 13 publications

Positivity for Kac polynomials and DT-invariants of quivers

Positivity for Kac polynomials and DT-invariants of quivers

Arithmetic harmonic analysis on character and quiver varieties II

Torus orbits on homogeneous varieties and Kac polynomials of quivers

Contact Info

Product

Resources

About