PURITY OF CRITICAL COHOMOLOGY AND KAC’S CONJECTURE BEN DAVISON Abstract. We provide a new proof of the Kac conjecture for an arbitrary quiver Q. We use sp e W ) built the fact that the (nilpotent) critical CoHA HQ,W for the quiver with potential (Q, e from Q is supercommutative and free, and an easy purity result, which implies purity of the sp space of generators of HQ,W . e

Contents 1. Introduction 2. Critical Cohomological Hall algebra 3. Purity 4. Kac polynomials References

1 3 5 8 10

1. Introduction The first purpose of this short paper is to demonstrate that Kac’s conjecture, recently proved by Hausel, Letellier and Rodriguez-Villegas in [4], is naturally connected to a purity statement in the critical cohomological ZQ0 -graded Hall algebra Hsp of an associated quiver with potential e Q,W e W ). The secondary purpose is to use a couple of basic results in this subject introduced in (Q, [1] to prove this purity statement. In a little more detail, via the results of [1] we are able to relate the compactly supported critical cohomology Hsp to the ordinary compactly supported e Q,W

cohomology of a stack Z of pairs (ρ, f ) where ρ is a representation of Q and f is a nilpotent endomorphism of ρ. We give an explicit stratification of this stack Zγ0 ⊂ . . . ⊂ Z n = Z such that each Z i \ Z i−1 has pure compactly supported cohomology, from which it follows that Z does too. This in turn implies the positivity of the associated weight polynomial, as odd cohomology vanishes. The idea of relating polynomials arising in positivity conjectures to weight polynomials of mixed Hodge structures is utilised also in [3] where the coefficients arising in quantum cluster mutation are related to the weight polynomials of the critical cohomology of spaces of framed representations. This connection was exploited in [2] to prove the quantum cluster positivity conjecture for quivers admitting a nondegenerate quasihomogeneous potential. There is however an important difference between that work and this, which is explained in terms of the difference

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between ‘DT invariants’ and the coefficients occurring in DT partition functions. That difference is easiest to explain in the case of the stacky partition function of all representations, without any stability condition or framing, which, happily for us, is the only case we need to consider in this paper. In this case, given a quiver with potential (Q0 , W 0 ) we form the partition function X ZQ0 ,W 0 (x) := χq (Hc,Gγ (MQ0 ,W 0 ,γ , ϕtr(W 0 )γ ))xγ q χ(γ,γ)/2 γ∈ZQ0

where for a complex of mixed Hodge structures L one defines as usual XX i m/2 χq (L) := (−1)i dim(Grm W (H (L)))q i∈Z m∈Z

and the other constituent terms are introduced in the body of the text. Conjecturally there is an alternative description Y 1/2 1/2 ZQ0 ,W 0 (x) := (1 − xγ )Ωγ (q )q /(1−q) γ∈ZQ0 \{0}

where the Ωγ (q 1/2 ) are polynomials in q 1/2 , and the order in which we take the product (in the general case, for nonsymmetric Q0 , the xi don’t commute), and the polynomials Ωγ (q 1/2 ), are determined by a stability condition (even though the spaces MQ,W,γ are not). These Ωγ (q 1/2 ) are the refined DT invariants, and one should note that the positivity of the coefficients in the partition function ZQ0 ,W 0 (x) do not imply positivity of these invariants, even if one can show that they exist. The Kac polynomials aγ (q) turn out to be equal, up to a factor of a power of q 1/2 and a substitution q 7→ q −1 , to the polynomials Ωγ (q 1/2 ) for our associated quiver with potential e W ), and so proving positivity for the terms of the partition function via purity is not sufficient (Q, to deduce positivity of the Kac polynomials, by the comment above. The extra ingredient needed is the main result of [1], which states that the algebra Grwt (Hsp ) is a free supercommutative e Q,W

algebra (here we have an extra grading given by the weight filtration of Hodge structures). The invariants Ωγ then acquire a new interpretation, as the weight polynomials of the space of generators in grade γ. Purity of Grwt (Hsp ), in the sense that the weight degree equals the e Q,W cohomological degree for every homogeneous element, then implies purity of the generators, and in this way we do indeed recover a new proof of Kac’s theorem. Acknowledgements. I would like to thank Sven Meinhardt for originally bringing my attention to this problem and the link to quivers with potential. I was supported by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG (German Research Foundation) during the research contained in this paper, and I would like to thank Daniel Huybrechts for getting me to Bonn. While completing the writing of the paper I was employed at the EPFL, supported by the Advanced Grant “Arithmetic and physics of Higgs moduli spaces” No. 320593 of the European Research Council. I would also like to thank Northwestern University for providing excellent working conditions during the completion of this paper, and Ezra Getzler and Kevin Costello for very helpful suggestions while I was there. This work was partly supported by the NSF RTG grant DMS-0636646.

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2. Critical Cohomological Hall algebra Let Q be a quiver, that is, a directed graph with finitely many arrows and finitely many vertices. We define the Ringel form χ(−, −) by setting X X χ(γ, γ 0 ) = γ(i)γ 0 (i) − γ(s(a))γ 0 (t(a)) i∈Q0

a∈Q1

e with superpotential for dimension vectors γ, γ 0 ∈ ZQ0 . As in [7], from Q we build a new quiver Q e e e W ∈ CQ/[CQ, CQ] by the following procedure (1) For every arrow a ∈ Q1 we add a new arrow e a with s(e a) = t(a) and t(e a) = s(a). (2) For every vertex i ∈ Q we add a new loop ω based at the vertex i. 0 i P P (3) We set W = ( i∈Q0 ωi )( a∈Q1 [a, e a]). e We denote by χ e the Ringel form associated to Q. e W )T by reversing all the arrows of Q, e and Now define a new quiver with superpotential (Q, reversing the order of all the cyclic words in W . Define the isomorphism of quivers ρ : Q → QT by sending a to e aT , e a to aT , and ωi to ωiT . e W) Proposition 2.1. There is an equality ρ∗ (W T ) = −W , and so in the terminology of [1], (Q, has a self-duality structure such that all dimension vectors are self-dual. L Q s(a) , Ct(a) ), and G = We define MQ,W,γ = γ e e 1 Hom(C i∈Q0 GLC (γ(i)), which acts on a∈Q MQ,W,γ via change of basis. e Definition 2.2. Define M sp e

Q,W,γ

⊂ MQ,W,γ to be the subspace of representations which send e

each ωi to a nilpotent endomorphism of Cγ(i) . e W ) to be a choice of edges S ⊂ Q e 1 such that every term Definition 2.3. We define a cut of (Q, of W contains exactly one member of S. Given a cut S we form a quiver QS by deleting all the e and define IS ⊂ CQS by IS := h ∂W |a ∈ Si. In fact we will always make arrows of S from Q, ∂a the assumption that S contains none of the arrows ωi . Given a cut S, we define ZQ,W,S,γ ⊂ e

M

Hom(Cγ(s(a)) , Cγ(t(a)) )

a∈(QS )1

to be the closed subspace of representations of QS satisfying the relations defined by IS , and define M Z Q,W,S,γ := ZQ,W,S,γ ⊕ Hom(Cγ(s(a)) , Cγ(t(a)) ). e e a∈S

We consider this space as a Gγ -equivariant subspace of MQ,γ e . The following is trivial, and holds precisely because we have chosen S not to contain any of the arrows ωi . It is required in order to prove Theorem 2.7

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Proposition 2.4. Let (1)

π:

M

Hom(Cs(a) , Ct(a) ) →

Hom(Cs(a) , Ct(a) )

e S )1 a∈(Q

e1 a∈Q

be the natural projection, then

M

M sp e

Q,W,γ

= π −1 π(M sp e

Q,W,γ

)

Next define  ∗ HQ,W,γ = (Hc,Gγ (MQ,W,γ , ϕtr(W )γ ) ⊗ Q(e χ(γ, γ)/2)[−e χ(γ, γ)]) , e e the dual of the equivariant critical cohomology with coefficients in the sheaf of vanishing cycles tensored with a power of a root of the Tate motive, as in [6]. If the reader has not met square roots of the Tate motive before, then they should ignore it as we will only work with Grwt (HQ,W ), the associated graded with respect to the weight filtration of Hodge structures. e The space Grwt (HQ,W ) is an algebra object in the category of ZQ0 ⊕ Zsc ⊕ Zwt graded Q vector e spaces, and Q(1/2) may be treated as the unique 1-dimensional vector space concentrated in grade (0, 0, 1) ∈ ZQ0 ⊕ Zsc ⊕ Zwt . Here Zsc ∼ = Zwt ∼ = Z – the subscripts are there to serve as a reminder that one copy of Z is keeping track of the grading induced by cohomological degree, and the other is keeping track of the grading induced by weights of Hodge structures. Similarly ∗  sp , ϕ ) ⊗ Q(e χ (γ, γ)/2)[−e χ (γ, γ)] Hsp := H (M . c,G tr(W ) γ γ e e Q,W,γ

Q,W,γ

We define the Hall algebra multiplications HQ,W ⊗HQ,W → HQ,W and Hsp e e e e

Q,W

⊗Hsp e

Q,W

as → Hsp e Q,W

in [1]. Each degree i piece with respect to the cohomological grading HQ,W,γ carries a mixed e Hodge structure, and the Hall algebra operations are morphisms of mixed Hodge structures, ) can be considered as a ZQ0 ⊕ Zsc ⊕ Zwt graded which is why Grwt (HQ,W ) and Grwt (Hsp e e Q,W algebras. Remark 2.5. For a general quiver with potential (Q0 , W 0 ), for HQ0 ,W 0 to be considered as an algebra object in the category of mixed Hodge structures, we must work in the category of monodromic mixed Hodge structures as defined in [6]. The claim that HQ0 ,W 0 is an algebra object in this category then rests on unpublished work of Saito, proving a Thom-Sebastiani type theorem for such mixed Hodge structures. In the case considered in this paper, however, we may use Theorem 2.7, or an extended version of it identifying the algebra structure on the critical cohomology on the left hand side of (2) or (3) with an algebra structure on the cohomology on the right hand side to deduce that we are working with an algebra object in the category of ordinary mixed Hodge structures. Then, all of the constituent maps of the multiplication morphism are known to belong to the category of mixed Hodge structures. See [1] for details, and in particular [1, Prop.A.5]. We recall two fundamental facts about the Hall algebras HQ,W and Hsp . e e Q,W

Theorem 2.6. The algebra Grwt (HQ,W ) is a free supercommutative algebra object in the catee Q 0 gory of Z ⊕ Zsc ⊕ Zwt graded vector spaces, where the supercommutativity is with respect to the

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Zsc -grading, and the generators of Grwt (HQ,W ) are given by a ZQ0 ⊕ Zsc ⊕ Zwt -graded subspace e Vprim ⊗ C[y], where y is placed in degree (0, 2, 2). Similarly, Grwt (Hsp ) is freely generated as e Q,W

nilp a supercommutative algebra by a ZQ0 ⊕ Zsc ⊕ Zwt -graded subspace Vprim ⊗ C[y].

e W ) has a self-duality structure, This is a special case of the main theorem of [1], using that (Q, by Proposition 2.1. Theorem 2.7. There are isomorphisms of graded mixed Hodge structures  ∗  ∗ •−l •−l0 • 0 ∼ ∼ HQ,W,γ H , Q) ⊗ Q(l/2) (Z H (2) (Z , Q) ⊗ Q(l /2) = = e e e c,Gγ c,Gγ Q,W,S,γ Q,W,S,γ ∗  ∗  0 nilp nilp •−l •−l 0 ∼ ∼ (3) , Q) ⊗ Q(l) H (Z , Q) ⊗ Q(l /2) Hsp,• H (Z = = e c,Gγ c,Gγ e e Q,W,S,γ Q,W,S,γ

Q,W,γ

where l=χ e(γ, γ) and l0 = χ e(γ, γ) + 2

X

γ(s(a))γ(t(a)).

a∈Q1

This is a special case of the theorem proved in the appendix of [1], which applies directly to (2), and applies to (3) on account of Proposition 2.4. P The extra shift comes from the fact that the relative (complex) dimension of π (from (1)) is a∈S γ(s(a))γ(t(a)). We have the following identity for all dimension vectors γ ∈ ZQ0 X (4) χ e(γ, γ) + 2 γ(s(a))γ(t(a)) = 0, a∈S

and so we finally deduce that the cohomological Hall algebra is given by the unshifted equi, i.e. we have the variant compactly supported cohomology of the spaces ZQ,W,S,γ and Z nilp e e Q,W,S,γ isomorphisms  ∗ ∼ (5) HQ,W,γ , Q) = Hc,Gγ (ZQ,W,S,γ e e  ∗ nilp ∼ (6) Hsp H (Z , Q) . = c,G γ e e Q,W,γ

Q,W,S,γ

3. Purity Let L be an object in the bounded derived category of mixed Hodge structures. We say that L is pure if the ith cohomology of L is pure of weight i. We begin this section with a conjecture. , Q) are pure. Conjecture 3.1. The mixed Hodge structures on Hc,Gγ (ZQ,W,S,γ e e W ) is a quiver with potential satisfying Example 3.2. Assume that Q has no arrows. Then (Q, the extra condition that W = 0. This implies that each space HQ,W,γ carries a pure mixed e Hodge structure, since it is the compactly supported cohomology of the Gγ -equivariant affine P γ(i)2 i∈Q 0 space A , with the trivial shift since in this case χ e = 0.

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Proposition 3.3. Let X be a G-equivariant variety, such that X admits a stratification ∅ = X0 ⊂ X1 ⊂ . . . Xt = X by G-equivariant subvarieties, satisfying the condition that for each Yi := Xi \ Xi−1 , there is a subgroup Ni ⊂ G and an inclusion gi : Asi → Yi such that the morphism of stacks gi0 : [Asi /Ni ] → [Yi /G] is an isomorphism. We assume further that the equivariant cohomology Hc,Ni (pt, Q) is pure. Then Hc,G (X, Q) is pure. Proof. The isomorphism gi0 induces an isomorphism in compactly supported cohomology Hc,Ni (Asi , Q) → Hc,G (Yi , Q).

(7)

n−2si There is a Gysin map Hnc,Ni (Asi , Q) → Hc,N (pt, Q) which shifts weights by 2si , from which we i deduce that the left hand side of (7) is pure, and so the right hand side is. The proposition then follows from the fact that the long exact sequence in compactly supported cohomology

→ Hnc,G (Xi , Q) → Hnc,G (Xi+1 , Q) → Hnc,G (Yi , Q) → is a complex in the category of mixed Hodge structures, and induction on t. Theorem 3.4. The mixed Hodge structures on Hc,Gγ (Z nilp e

Q,W,S,γ



, Q) are pure.

Proof. Since the left hand side of (3) doesn’t depend on S we may pick whichever cut S we like, as long as it contains none of the arrows ωi , so that we may use Proposition 2.4 to prove (3) in the first place. We assume that S consists of the arrows e a for a ∈ Q1 . We consider a different description of the representation varieties ZQ,W,S,γ . For e a ∈ S we have e

and we deduce that ZQ,W,S,γ e

∂W = ωt(a) a − aωs(a) ∂e a is the space of pairs (ρ, f ) where M ρ∈ Hom(Cγ(s(b)) , Cγ(t(b)) ) b∈Q1

and f∈

M

End(Cγ(i) )

i∈Q0 nilp Ze is Q,W,S,γ

is an endomorphism of ρ. Similarly, the space of pairs (ρ, f ), where f is a nilpotent endomorphism. A nilpotent endomorphism of the space Cr is defined, up to isomorphism, by its Jordan normal form, which is defined in turn by the partition of r induced by taking the sizes of the Jordan normal blocks. We let P(γ) be the set of partitions of γ, i.e. the set of assignments of partitions γ(i) = π1 (i) + . . . + πki (i), where πs (i) ≥ πs+1 (i) for all s, to each of the entries γ(i). We let P be the union of the P(γ) for γ ∈ ZQ0 . Then the set of choices of f , up to the action of the gauge group Gγ , is in natural correspondence with P(γ). The stabiliser Nπ of such an f is then an affine bundle over a product of linear groups, and in particular has pure Hc,Nπ (pt, Q) and is good, in the sense that principle Ni -bundles are Zariski locally trivial; we will more explicitly describe the Nπ in the next section. We define Yπ to be the space of pairs (ρ, f ) such that f belongs to the isomorphism class corresponding to π. We define a partial ordering on the set P(γ) by the prescription that π < π 0 if

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there is a map, for each i, from the multiset {π1 (i), . . . , πki (i)} to the multiset {π10 (i), . . . , πk0 0 (i)} i such that each element of the target is equal to the sum of the elements of its preimage. We complete this to a total ordering on P(γ), and define a Zγπ = Yπ 0 , π 0 ≤π

giving a stratification of Z nilp e

Q,W,S,γ

.

Given a multipartition π ∈ P(γ), we write π(i) = (1Ψi,1 , 2Ψi,2 , . . .). Given a pair (ρ, f ) belonging to the space Yπ , we consider it as a representation of the quiver Q in the category L Ψi,j of nilpotent C[x]-modules, where each i ∈ Q0 is sent to j (C[x]/xj )⊕ . Given two numbers 0 0 j, j 0 ∈ Z≥0 , there is an isomorphism Hom(C[x]/xj , C[x]/xj ) ∼ = Amin(j,j ) , and so we deduce that M 0 [Yπ /Gγ ] ∼ Amin(j,j )Ψs(a),j Ψt(a),j 0 /Nπ ]. =[ a∈Q01 ,j,j 0 ∈Z

The theorem then follows from Proposition 3.3.



We now consider the generating function X X sp,(γ,m,n) γ n/2 (8) χ(Grwt (Hnilp )) := (−1)m dim(H e )x q ∈ Z[[xγ , q 1/2 ]]. e Q,W

Q,W

(γ,n)∈ZQ0 ⊕ZGr m∈Zsc

, ϕtr(W )γ ), Fixing γ ∈ ZQ0 , the coefficient of xγ is given by the weight polynomial of Hc,Gγ (M sp e Q,W

and in particular the infinite alternating sum given by fixing a power of q 1/2 is well defined – in fact by Theorem 2.7 and Theorem 3.4 all Hodge structures are pure and we deduce X sp,(γ,n,n) γ n/2 χ(Grwt (Hnilp x q . )) := (−1)n dim H e e Q,W

Q,W

(γ,n)∈ZQ0 ⊕Zsc

By (6) and the proof of Theorem 3.4 the terms in the above sum for odd n are all zero, i.e. the partition function X sp,(γ,2n,2n) γ n )) := dim H e x q χ(Grwt (Hnilp e Q,W

Q,W

(γ,n)∈ZQ0 ⊕Z

has only positive coefficients. Definition 3.5. Given a formal power series f =

P

f x Q (γ,n)∈Z≥00 ⊕Z γ,n

γ q n/2

∈ Z[xγ , q ±1/2 |γ ∈

ZQ0 ] with fγ,n = 0 if γ = 0 or γi < 0 for any i ∈ Q0 , and such that for fixed γ, the series P n/2 is a Laurent power series in q 1/2 , we define n fγ,n q   X Y n −fγ,n Sym(f ) = (−1) xh(γ,n)γ q h(γ,n)n h(γ, n) Q h (γ,n)∈Z

0 ⊕Z

where the sum is over functions h : ZQ0 ⊕ Z → Z≥0 sending all but finitely many elements to zero, and by convention, the term in the sum corresponding to h = 0 is 1. There is an inverse to the function Sym, and so in particular it is injective.

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The function Sym has a more illuminating description. Let V be a ZQ0 ⊕ Zwt ⊕ Zsc -graded vector space, and let f be its characteristic function X X i )xγ q n/2 , f := (−1)i dim(V(γ,m) (γ,n)∈ZQ0 ⊕Zwt i∈Zsc

which we assume satisfies the conditions of Definition 3.5. Then Sym(f ) is the characteristic function of the supercommutative algebra generated by V , where supecommutativity means that for x, x0 of cohomological degree i and i0 , 0

x · x0 = (−1)ii x0 · x. In terms of this operation, we may write, using Theorem 2.6:   X nilp )) = Sym  )(−1)m xγ q n/2 (1 − q)−1  . χ(Grwt (Hnilp dim(Vprim,γ,m,n e Q,W

(γ,m,n)∈ZQ0 ⊕Zsc ⊕Zwt

The following is trivial. Lemma 3.6. If A is a free supersymmetric Z ⊕ Zsc -graded algebra generated by the graded subspace V , and Am,n = 0 unless m = n, then Vm,n = 0 unless m = n. We deduce from the lemma that we may write   X nilp (9) χ(Grwt (Hnilp )) = Sym  dim(Vprim,(γ,m,m) )xγ (−q)m/2 (1 − q)−1  , e Q,W

(γ,m)∈ZQ0 ⊕Z

nilp i.e. Vprim,(γ,m,n) vanishes for m 6= n.

4. Kac polynomials Associated to the quiver Q are the Kac polynomials aγ (q) counting the number of isomorphism classes of absolutely indecomposable representations of Q over a field of order q. These are indeed polynomials by a theorem of Kac. In terms of the Sym operation of Definition 3.5, we have the following theorem of Hua. First we’ll need somePnotation. For two partitions π(i), π(j) of numbers γ(i) and γ(j) we define hπ(i), π(j)i := of the number m which we write in the n≥1 πn (i)πn (j). If π is a partition Q ψ ψ 1 2 notation π = (1 , 2 , . . .) we define bπ (q) = j (1 − q) . . . (1 − q ψj ). If π ∈ P, then π ∈ P(γ) for some γ and we define |π| := γ. Theorem 4.1. [5, Thm.4.9] There is an equality of generating functions    X γ  X Sym  x aγ (q −1 )/(q −1 − 1) cπ (q)x|π| .  = γ∈ZQ0 , γ6=0

π∈P

PURITY OF CRITICAL COHOMOLOGY AND KAC’S CONJECTURE

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where hπ(s(a)),π(t(a))i a∈Q1 q Q . hπ(i),π(i)i b −1 π(i) (q ) i∈Q0 q

Q

cπ (q) :=

We will need the following lemma of Hua: Lemma 4.2. [5, Lem.3.1] Let π and π 0 be two partitions of the numbers m and m0 , which we write in the notation π = (1ψ1 , 2ψ2 , . . .). Then there is an identity X hπ, π 0 i = min(n, n0 )ψn ψn0 0 . n,n0

From the proof of Theorem 3.4 we may alternatively write, using Lemma 4.2, and the fact that each Nπ is good, Q X a∈Q q hπ(s(a)),π(t(a))i nilp 1 x|π| , (10) χ(Grwt (HQ,W )) = χq (Hc (Nπ , Q)) π∈P

where χq (Hc (Nπ , Q)) is the ordinary weight polynomial of the compactly supported cohomology of the group Nπ , which we now calculate. Lemma 4.3. There is an equality χq (Hc (Nπ , Q)) =

Y

q hπ(i),π(i)i bπ(i) (q −1 ).

i∈Q0

Proof. It’s sufficient to prove the claim under the assumption that Q0 has only one vertex, which will ease the notation somewhat. Elements of Nπ are the invertible elements of M (fm,m0 ) ∈ Hom(Nm , Nm0 ) m,m0

where we define Nm := (C[x]/xm )⊕ψm . We claim that an element fm,m0 is invertible if and only 0 if each fm,m is. The result will then follow as χq (Hom(Nm , Nm0 )) = q min(m,m )ψm ψm0 for m 6= m0 2 and χq (Aut(Nm )) = q mψm bψm (q −1 ) by a standard calculation of the points of the general linear group. −1 If (fm,m0 ) satisfies the condition that each fm,m is invertible, put gm,m0 = −fm0 ,m0 fm,m0 fm,m −1 – this gives an explicit inverse to (f for m 6= m0 and gm,m = fm,m the other hand, m,m0 ). OnL assume that (fm,m0 ) is invertible. Let V be the underlying vector space of m Nm , and let V m := ker(·xm : V → V ). Then the V m give a filtration of V , by C[x] modules, and so (fm,m0 ) defines an automorphism of the associated graded object. One may check that the operation defined by (fm,m0 ) on the kth graded piece is given by   f k,k ? 0 ? where f k,k is the morphism induced by fk,k on the space Nk,k /x · Nk,k . So f k,k is invertible, and so is fk,k . The result follows. 

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Putting together (10), Lemma 4.3 and Hua’s Theorem 4.1 we recover a variant of [7, Thm.5.1] X Sym( xγ aγ (q −1 )/(q −1 − 1)) = χ(Grwt (Hnilp )) e Q,W

γ∈ZQ0

and equating coefficients with (9), using injectivity of Sym, we deduce X nilp (11) qaγ (q −1 ) = dim(Vprim,(γ,m,m) )q m/2 m∈2Z

where the right hand sum is over the even numbers as the right hand side contains no odd powers of q. We recover the Theorem of Hausel, Lettellier and Rodriguez-Villegas: Theorem 4.4. The Kac polynomials aγ (q) have positive coefficients. Remark 4.5. We also recover a result of Kac, which states that the polynomials aγ are independent of the orientation of Q. In our framework, this is explained by the fact that the e W ) is independent of the orientation of Q. isomorphism class of the pair (CQ, Remark 4.6. By the usual manipulations involving power structures, one can show that nilp )))q . χ(Grwt (HQ,W )) = (χ(Grwt (HQ,W

It follows that the refined Donaldson Thomas invariants for the category of representations of e W , with the trivial stability condition, are given by Ωγ = aγ (q)q 1/2 . the Jacobi algebra for Q, Remark 4.7. By the previous remark, the generating function χ(Grwt (HQ,W )) contains only e integral powers of q, and all its coefficients are positive. This is some evidence towards the truth of Conjecture 3.1. References 1. B. Davison, The critical CoHA of a self dual quiver with potential, preprint. 2. B. Davison, D. Maulik, J. Sch¨ urmann, and B. Szendr˝ oi, Purity for graded potentials and quantum cluster positivity, http://arxiv.org/abs/1307.3379. 3. A. Efimov, Quantum cluster variables via vanishing cycles, http://arxiv.org/abs/1112.3601. 4. T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Positivity for kac polynomials and dt-invariants of quivers, Ann. Math. 177 (2013), 1147–1168. 5. J. Hua, Counting representations of quivers over nite elds, J. Alg 226 (2000), 1011–1033. 6. M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, http://arxiv.org/abs/1006.2706, June 2010. 7. S. Mozgovoy, Motivic Donaldson-Thomas invariants and McKay correspondence, http://arxiv.org/abs/ 1107.6044. B. Davison: EPFL E-mail address: [email protected]