ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

arXiv:math/0611074v2 [math.RT] 9 Feb 2007

PHILIPPE CALDERO AND MARKUS REINEKE
Abstract. Let A be the path algebra of a quiver Q with no oriented cycle. We study
geometric properties of the Grassmannians of submodules of a given A-module M . In
particular, we obtain some sufficient conditions for smoothness, polynomial cardinality
and we give different approaches to Euler characteristics. Our main result is the positivityof Euler characteristics when M is an exceptional module. This solves a conjecture of
Fomin and Zelevinsky for acyclic cluster algebras.

0. Introduction
Let M be a finite dimensional space on a field k. The Grassmannian Gre (M, k) of M
is the set of subspaces of dimension e. It is well known that Gre (M, k) is an algebraic
variety with nice properties. For instance, the linear group GLe (M, k) acts transitively on
Gre (M, k) with parabolic stabilizer, hence the variety Gre (M, k) is smooth and projective.
Suppose now that M has a structure of A-module, where A is a finitely generated kalgebra. It is natural to define the Grassmannian Gre (M, A) of A-submodules of M ofgiven dimension e. It is a closed subvariety of Gre (M, k), hence it is a projective variety.
But in general this variety is not smooth.
Let Q be a finite quiver with no oriented cycle and let A := kQ be the path algebra of
Q. Then, A is a finite dimensional algebra with minimal idempotents ei , i ∈ Q0 , where
Q0 is the set of vertices of Q. The A-module M has a dimension vector dim(M ) :=(dim(ei M ))i ∈ NQ0 which refines the notion of dimension. The Grassmannian Gre (M, A)
splits into a disjoint union of closed
P subvarieties Gre (M, A) of submodules of M with given
dimension vector e such thati e i = e. Hence it is natural to study the geometricproperties of the so-called quiver Grassmannian Gre (M ) := Gre (M, A).
The quiver Grassmannian can be found in [8, 22] in connection with the dimension of
spaces of morphims of kQ-modules. It plays an important role in Fomin-Zelevinsky’s theoryof cluster algebras, [10]. Indeed, as proved in [3] and [5], Euler characteristics χc (Gre (M ))of quiver Grassmannians define sets of generators of cluster algebras in the acyclic type.
To be more precise, consider the ring of Laurent polynomial Z[x±i , i ∈ Q0 ] and for any(f.d.) kQ-module M with dimension vector m = (mi ), set:
Y mt(h) −et(h) es(h)Y −m Xxt(h) ,xs(h)χc (Gre (M ))
xi i
XM :=
i∈Q0

e

h∈Q1

where Q1 is the set of arrows of Q, and s(h), t(h) are respectively the source and thetarget of the arrow h. Then, it is known that a set of generators of the cluster algebra
is given by {XM } where M runs over the set of exceptional modules, i.e. modules withno self-extension. These generators are called cluster variables. Moreover, the set {XM },where M runs over the set of f.d. modules is (at least conjecturally) a Z-base of the cluster
algebra which is an analogue of Lusztig’s dual semicanonical base, [6].
The aim of the paper is to study geometric properties of quiver Grassmannians, and inparticular to understand their Euler characteristics.
1

2

PHILIPPE CALDERO AND MARKUS REINEKE

In the first section with give an inductive formula to compute χc (Gre (M )) when M
is a preprojective (or postinjective) module. This uses the Auslander-Reiten algorithm ofconstruction of the preprojective representations and in particular the notion of almost split
sequences. Unfortunately, it is not clear from the inductive formula that the characteristicsχc (Gre (M )) are positive.
The second section is devoted to positivity. On the one hand, a well-known result of J. A.
Green and C. Ringel (see [12, 21]) asserts that quantum groups can be realized as deformed
Hall algebras of the category mod kQ and so χc (Gre (M )) can be realized as coefficients ofsome convolution product in a PBW-basis. On the other hand, we know that Lusztig’s
canonical basis of quantum groups has nice positivity properties. We obtain two resultsfrom these remarks:
First, proposition 3 asserts that if the quiver Grassmannian Gre (M ) |Fq is counted by apolynomial of q, then its characteristics is positive.
Our second positivity result, theorem 1, solves a conjecture of Fomin and Zelevinsky
for acyclic cluster algebras. It states that χc (Gre (M )) is positive if M is an exceptional
module.
Recall that the positivity in the Dynkin case was proved in [4]. With the help of proposition 3, we generalize this result for extended Dynkin quivers in section 3: the Euler
characteristics is positive for any representation M on an extended Dynkin quiver. Actually, what we need is a polynomiality property of the quiver Grassmannians Gre (M )and we prove that we can deduce it from a polynomial property of varieties of representations which are easier to handle, namely the varieties of representations X such thatdim Hom(X, M ) is equal to a fixed integer a.
In section 4, we give some results on smoothness. We realize the quiver Grassmannianas a geometric quotient of an affine variety and this enables us to obtain a description of
the tangent space of Gre (M ) at a point U ; we deduce the following
he , m − e i ≤ dim TU (Gre (M )) ≤ he , m − e i + dim Ext1 (M, M ),where h , i is the Euler form on ZQ0 identified with the Grothendieck group of mod kQ, see
4.2. This inequality implies clearly that the quiver Grassmannian is smooth if the module
M is exceptional. This also provides a dimension estimate of the variety.
We introduce in section 5 some material on cluster algebras in order to explain the main
application of the positivity theorem. As explained above, cluster variables for acyclic
quivers are positive, which is a particular case of a conjecture of Fomin and Zelevinsky.
We end with an open problem tightly linked with the Euler characteristics and its
positivity. Can we find a cellular decomposition of the quiver Grassmannian? Solving thisproblem can lead to some combinatorics for the calculus of χc (Gre (M )).
Acknowledgements: The first author wants to thank Gus Lehrer and Andrei Zelevinsky for stimulating conversations.
1. Quiver Grassmannians.
1.1. Let k be a field. We fix a finite quiver Q such that the path algebra H = kQ is finite
dimensional. Since this property is equivalent to the fact that Q has no oriented cycle,
such a quiver will be called acyclic.
Let Q0 be the set of vertices and Q1 the set arrows of Q. Set n = #Q0 . For any arrow
h of Q1 , we denote by s(h) its source and t(h) its target. We fix an ordering of Q0 = [1, n]such that if there exists a (non trivial) path from j to i, then j < i. Since Q has no oriented
cycles, it is possible to construct such an ordering.

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

3

A representation (V, x) of Q over k is a Q0 -gradedQ k-vector space V = ⊕i∈Q0 Vi together with an element x = (xh )h∈Q1 in EV := h∈Q1 Hom(Vs(h) , Vt(h) ). A morphism
between two representations (V, x) and (V ′ , x′ ) is a Q0 -uple of (k-spaces) morphisms in
Q′′
i∈Q0 Homk (Vi , Vi ) compatible with x and x , i.e. inducing commuting square diagram.
We denote by Repk Q the category of (finite dimensional) representations of Q over k.
For each vertex i, let Si be the associated simple representation. The Grothendieck groupof the category Repk Q can be identified with ZQ0 via the dimension vector map dim :dim (M ) = (dim(Mi ))i∈Q0 .
There is a well known equivalence between Repk Q and the category mod(kQ) of finite
dimensional kQ-modules; hence a representation of Q will be naturally considered as a
kQ-module.
Q−1). Clearly, theThe group GV := i∈Q0 GL(Vi ) acts on EV by (gi ).(xh ) = (gt(h) xh gs(h)isoclasses of a finite-dimensional H-module M are naturally identified with the points of a
GV -orbit OM of representations of Q.
For any finite dimensional H-module M , and any e in ZQ0 , we denote by Gre (M ) the
Grassmannian of submodules of M with dimension vector e :Gre (M ) = {N, N ∈ H -mod, N ⊂ M, dim (N ) = e }.
It is a closed subvariety (hence projective) of the classical Grassmannian of the vector spaceM.
We know that this variety is obtained by base change from a variety defined over Z. So
the field k will sometimes be omitted. Let χc be the Euler-Poincaré characteristic of l-adic
cohomology with proper support defined byχc (Gre (M )) =∞X(−1)i dim Hci (Gre (M ), Ql ).
i=01.2. We give here some basic properties for the computation of the characteristics ofquiver Grassmannians. This first proposition asserts that we can restrict ourselves to theindecomposable case.
Proposition 1. [3, Proposition 3.6] Let M and N be two Q-representations and let g bea dimension vector, thenXχc (Grg (M ⊕ N )) =χc (Gre (M ))χc (Grf (N )).
e +f =g

Proof. It is sufficient to remark, see [3, Equation (26)], that the mapaπ : Grg (M ⊕ N ) →Gre (M )) × Grf (N ), X 7→ (X ∩ M, X + N/N ),e +f =g

is a surjective morphism of algebraic varieties with affine fibers.

Now, we can calculate these characteristics for some particular Q-representations.
(
1 if e = dim (Si ) or 0
(1.1)
χc (Gre (Si )) =0 otherwise
Let Pi be the indecomposable projective module with top Si . It is known that if M is asubmodule of Pi , then either M = Pi or M ⊂RadPi = ⊕i→j Pj . By this argument and the
proposition above, we obtain

4

PHILIPPE CALDERO AND MARKUS REINEKE(1.2)

χc (Gre (Pi )) = 1 +X YPχc (Gre j (Pj )).

e j =e j→iThis together with equation 1.1 provides an inductive formula for the ordering of Q0 ,which enables to compute the characteritics of projective modules. For injective modules,
which are dual to projective modules, it is sufficient to see that
opp

Q
GrQ
e N ≃ Grm −e DN,

where D is the duality functor from RepkQ to RepkQopp .
1.3. We now provide an inductive formula for the Euler characteristics of quiver Grassmannians of a wider class of modules called preprojective and postinjective modules. For
this, we need to recall the definition of an almost split sequence and the Auslander-Reiten
theorem, [1].
A short exact sequence
(1.3)

σ

0→M →E→N →0in RepkQ is called almost split if the following conditions are satisfied(i) M and N are indecomposable(ii) The map σ does not split(iii) Given any representation N ′ and any morphism ρ : N ′ → N which is not a split
epimorphism, then ρ factors through σ.
The proposition that follows is a slight variation of proposition 1. Actually, the hypothesis passes from ”split” to ”almost split”.
Proposition 2. Let 0 → M → E → N → 0 be an almost split sequence and let g be adimension vector. Then,(P
if g 6= dim N
e +f =g χc (Gre (M ))χc (Grf (N ))χc (Grg (E)) = P
if g = dim N
e +f =g χc (Gre (M ))χc (Grf (N )) + 1

Proof. As in the proof of proposition 1, it is sufficient to remark, see [3, Lemma 3.11], thatthe almost split property implies that the mapa
Gre (M )) × Grf (N ), X 7→ (X ∩ M, X + M/M ),π : Grg (E) →is a morphism of algebraic varieties such that
1. case g = dim N : π −1 (M ′ , N ′ ) is empty if (M ′ , N ′ ) = (0, N ), and is an affine space if
not.
2. case g 6= dim N : π −1 (M ′ , N ′ ) is an affine space for any (M ′ , N ′ ).

The Auslander-Reiten theorem asserts that for any non injective indecomposable representation M , there exists a unique almost split sequence beginning with M . Dually, for
any non projective indecomposable representation N , there exists a unique almost split
sequence terminating in N .
For any non injective indecomposable representation M , let τ− M be the unique indecomposable representation (up to isomorphism) N such that 1.3 is almost split. By
convention set τ− I = 0 if I is injective. The indecomposable representations of the form
τ−m (Pi ), m ∈ N, i ∈ Q0 are called (indecomposable) preprojective representations.
Dually, for any non projective indecomposable representation N , let τ+ N be the (unique)indecomposable representation M such that 1.3 is almost split. By convention set τ+ P = 0

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

5

if P is projective. The indecomposable representations of the form τ+m (Ii ), m ∈ N, i ∈ Q0
are called (indecomposable) postinjective representations.
Let P, resp. I, be the set of indecomposable preprojective, resp. postinjective, representations. By the Auslander-Reiten algorithm, P can be endowed with an ordering such that:(i) {Pi , i ∈ Q0 } is the set of minimal elements for ,(ii)  is generated by M  τ− M , M ∈ P(iii) For any M in P, and any summand Ej of E in the almost split sequence 0 → M →E → τ− (M ) → 0, we have M  Ej  τ− M .
Hence, Proposition 2 gives a recursive formula to calculate the Euler characteristics of any
quiver Grassmannian of a preprojective or postinjective representation. Namely, with thenotation above, χc (Grf (τ− M )) =(P
χc (Grf (E)) − e +f ′ =f χc (Gre (M ))χc (Grf ′ (τ− M )) if f 6= dim τ− (M )(1.4)
1
if f = dim τ− (M )Note that the positivity of Euler characteristics of quiver Grassmannians is not clear fromthis formula. It will be the subject of the two next sections.
2. Positivity
2.1. Let X be a complex quasi-projective variety, and let XR be a model of X over somefinitely generated subring Z ⊂ R ⊂ C, that is, C ×R XR ≃ X. Given a maximal idealm ⊂ R, we can form the reduction Xk of XR to a scheme over the finite field k = R/m.
We say that X has the polynomiality property if there exists a polynomial P (t) ∈ Z[t]such that, for infinitely many maximal ideals m of R, the number of k-rational points of
Xk equals P (|k|) (note that this property in fact depends on the chosen model XR !). By
a straightforward generalization of [20, Proposition 6.1], it is sufficient to require existenceof a rational function P (t) ∈ Q(t) with this property.
In particular, we can consider the polynomiality property for the quiver Grassmannian,since it is defined over Z. We begin with a proposition which claims that the polynomialityof the quiver Grassmannian implies its ”positive polynomiality”.
Proposition 3. Let M be a finite-dimensional Q-representation and let e be in ZQ0 .
Suppose that the cardinality of Gre (M )Fq is a non zero integral polynomial Pe ,M evaluated in q. Then, the polynomial Pe ,M has positive coefficients. In particular, we haveχc (Gre (M )) > 0.
Proof. The last assertion comes from the classical result χc (Gre (M ) ) = Pe ,M (1), see forexample [3, Lemma 3.5].
So, we just need to prove that Pe ,M has positive coefficients. Actually, we will show that
this is a consequence of Lusztig’s theorems on canonical bases of quantum groups, [18] and[19].
i
For m in NQ0 , and Vm = ⊕Fm
q , let Hm (q) be the C-vector space of all GVm -invariant
functions on Em := EVm .
Let H(q) be the vector space ⊕m ∈NQ0 Hm (q). We can define on H(q) a C-algebra
structure byX(f1 ∗ f2 )(M ) =f1 (M/N )f2 (N ),N ⊂M

where N runs over the (finite) set of submodules of M . This defines an associative unital
NQ0 -graded algebra, the Hall algebra of Q (see [21]).

6

PHILIPPE CALDERO AND MARKUS REINEKE

For any isoclass X of Fq Q-modules, let 1X ∈ H(q) be the characteristic function of thecorresponding orbit OX ∈ Edim X .
We have(1X ∗ 1Y )(M ) = #{N ⊂ M, N ≃ Y, M/N ≃ X}.
For any dimension vector m , let 1m be the constant function with value 1 on Em . Set
m = dim M . Then it is easily seen that
(2.1)

(1m −e ∗ 1e )(M ) = #{N, N ⊂ M, dim N = e } = # Gre (M ) = Pe ,M (q),Recall that we fixed an ordering of Q0 = [1, n] such that if there exists a (non trivial)
path from j to i, then j < i.
Consider the divided power
1 ei
i)
1 ,
1(e
αi =
[ei ]q ! αiPi
where [ei ]q ! = [ei ]q [ei − 1]q . . . [1]q , and [a]q := a−1
i=0 q for any integer a. Then we have
(e2 )
(en )
1)
1(e
α1 ∗ 1α2 . . . ∗ 1αn = 1e .

(e )It is known that 1αii is an element of Lusztig’s canonical basis B of H(q). Hence, 1e is
a product of elements of the canonical basis, and so is 1m −e ∗ 1e .
By Lusztig’s positivity theorem, [18, Theorem 14.4.13], the product of elements of thecanonical basis has positive polynomial coefficients in the canonical basis. We can write:
XX
Qb (q)b, Qb (q) =(2.2)
1m −e ∗ 1e =cb,m q m , cb,m ∈ Z≥0 .
b∈B

b,m

Now, in order to apply 2.1, we need a description of the evaluation b(M ) of the function
b on the module M .
A theorem of Lusztig, [19, Theorem 5.2], gives such a description. Actually, the functionb is obtained by taking an alternating sum of traces of the Frobenius map on the cohomology
of a simple perverse sheaf Pb which is pure of weight 0. More precisely, we have
i

(2.3)

b(M ) =(Pb )M
X dim HXi(−1)i aij,b,M , with | aij,b,M |= q i/2 ,j=1

where aij,b,M , 1 ≤ j ≤ dim H i (Pb )M , is the set of eigenvalues of the Frobenius map on the
stalk at M of the i-th cohomology of the l-adic complex Pb .
Now, we can use 2.1, 2.2, 2.3 to obtain(2.4)

Pe ,M (q) =X(−1)i cb,m q m aij,b,M .

b,i,j,mFor any nonnegative integer s, we can replace q by q s and obtain(2.5)

Pe ,M (q s ) =X(−1)i cb,m q ms asij,b,M .

b,i,j,mAs M and e are fixed, we now omit e and M in the formulas. We set(2.6)

P (q) = Pe ,M (q) =Xkpk q k .

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

Let t be an indeterminate, we calculate by 2.5 the formal serie exp
pk ∈ Z (by hypothesis), cb,m ∈ Z≥0 , we obtain:
Y

(2.7)

k

7

P

s ts
n≥1 P (q ) s .

As,

Y
1
1
=
.
k
p
(−1)i cb,mma
(1 − q t) k
(1
−
q
t)
ij,b
i,j,b,m

We conclude that any aij,b which is not cancelled in the right hand product is an integral
power of q.
This implies that any aij,b which is not cancelled in the right hand sum of 2.4 is an
integral power of q. In particular, by the last equality of 2.3, we obtain that if aij,b is notcancelled in the right hand sum of 2.4, then i is even.
Using 2.4 again, we see that P has nonnegative coefficients.

We will see in section 3 that this proposition implies the positivity of χ(Gre (M )) in the
Dynkin and extended Dynkin case. Recall that the Dynkin case was also proved in [4].
2.2. We study here the case where the module M has no self-extension. We will provethat the polynomiality, and so the positivity, is implied by this hypothesis. This example
will be fundamental in section 5.
In the sequel, a Q-representation will be called exceptional if Ext1Q (M, M ) = 0. For
example, any indecomposable preprojective or postinjective module is exceptional.
Theorem 1. Let Q be an acyclic quiver and let M be an exceptional Q-representation.
Then, for any dimension vector e such that Gre (M ) is non empty, the cardinality ofGre (M ) |Fq is a positive polynomial in q. In particular, χc (Gre (M )) > 0.
Proof. Set m := dim M and Gm := GVm .
From the equations 2.1 and 2.2, we have
(2.8)

# Gre (M ) |Fq = 1m −e ∗ 1e (M ) =X

Qb (q)b(M ), Qb (q) ∈ Z≥0 [q].

b∈BNow, we need to prove that there exists a unique b such that b(M ) 6= 0 in the sum. Indeed,
let b be an element of B which is non zero on M .
The corresponding perverse sheaf Pb is by construction a Gm -invariant perverse sheaf,[19]. By construction, see [7, 8.4], its support is a closed Gm -subvariety Sb of Em and its
restriction on a open Zariski dense Ob of Sb is a Gm -invariant local system.
Now, the restriction of Pb on OM is a local system. As Sb contains M , it contains the
Gm -orbit OM which is an open dense orbit since M is exceptional. So, Sb = Em . As theorbit OM has a connected stabilizer (on C), then [7, Lemma 8.4.11] implies that the local
system is the trivial sheaf on OM . Now, a perverse sheaf whose support is a vector space
and which is trivial on a Zariski open subset is the constant sheaf.
This implies that Pb is the constant sheaf and so b(M ) = 1.
Equation 2.8 gives(2.9)# Gre (M ) |Fq = Qb (q).

This ends the proof.
Corollary 1. If M is a preprojective or a postinjective module, then any non empty quiver
Grassmannian of M has positive Euler characteristics.

8

PHILIPPE CALDERO AND MARKUS REINEKE

3. A reformulation of the polynomiality property and its applications
3.1. Now, let’s consider a completed version of the Hall algebra (compare [20]) and perform some computations with certain generating functions in it. By applying an evaluation map, this yields an identity involving cardinalities of quiver Grassmannians over finite
fields.
QbDefine H(q):= m ∈NQ0 Hm (q). The convolution product on H(q) of section 2 prolongsb We also consider the skew commutative formal power series ringto a productQon H.
′′′
Q
Cq [[N 0 ]] = m ∈NQ0 Ctm with product tm · tm = q −hm ,m i tm +m . We define the integral
RR1b· tdim X for all Fq Qas the linear map: H(q)→ Cq [[NQ” 0 ]] given by 1X =|Aut(X)|

representations X. The integral is in fact a C-algebra homomorphism by [20].
b
We consider the following special elements of H(q):X
XX|Hom(X, M )|1X , gM :=|Hom0 (X, M )|1X ,
1m , eM :=e :=m ∈NQ0[X][X]

where the last two sums run over all isomorphism classes of finite dimensional representations of Fq Q. We write Hom(X, M ) for the space of Fq Q-homomorphism, and Hom0 (X, M )for the subset of monomorphisms.
b
Lemma 1. The identity gM · e = eM holds in H(q),for all representations M of Fq Q.

Proof. By the definitions, we haveXXX
gM · e =|Hom0 (X, M )| · 1X 1Y =|Hom0 (Z/Y, M )| · 1Z .
[Z] Y ⊂Z

[X],[Y ]For each representation Z, the inner sum on the right hand side is the cardinality of the setof pairs (Y, g) consisting of a subrepresentation Y of Z and an injective map g : Z/Y → M .
But this set is in bijection to Hom(Z, M ) by assigning to a map f : Z → M the pair(ker(f ), f : Z/ker(f ) → M ). Thus,X
gM · e =|Hom(Z, M | · 1Z = eX .
[Z]We integrate the three elements:
Z
X
X |Em |
1e=
· tdim X =tm ,|Aut(X)|
|G
|
m
m
[X]

since 1/|Aut(X)| can be rewritten as |OX |/|Gdim X |. Similarly,
Za,M
X X |Em
| m
eM =
t ,
|G
|
m
m
a∈N

a,M
where Emdenotes the subset of Em consisting of representations X such that the dimension of Hom(X, M ) equals a. Finally,Z
X
X |Hom0 (X, M )||Grm (M )|tm .
tdim X =(gM ) =
|Aut(X)|m[X]

To see this, we consider the set of pairs ([X], g) consisting of an isomorphism class of arepresentation X of dimension vector m , together with an injective map g : X → M .
Sending such a pair to the image of g induces a surjection to Grm (M ), whose fibres are

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

9

precisely the Aut(X)-orbits of injective maps.
Since the integral is an algebra map, the lemma implies the following identity in Cq [[NQ0 ]]:X′|Grm ′ (M )|tm ·X |Em ′′ |mm′

|Gm ′′ |
′′

′′

tm =

a,M
X X |Em
|
m a∈N

|Gm |

tm .

Comparing coefficients, this yields the following recursive formula:

Proposition 4. The cardinalities of the quiver Grassmannians are given by:
a,M
X |Em
X
|Em −m ′ ||′′.
q −hm ,m −m i |Grm ′ (M )| ·|Grm (M )| =−|Gm ||Gm −m ′ |
′
a∈N

m <m

The above recursive formula now yields:
Corollary 2. The quiver Grassmannians Grm (M ) have the polynomiality property if alla,Mhave the polynomiality property.
the varieties Em
In particular, this yields the polynomiality property of quiver Grassmannians in the caseof Dynkin quivers without reference to the existence of Hall polynomials: in this case, eacha,MEm
is a union of finitely many Gm -orbits in Em , all of which have the polynomiality
property by identifying OX with the homogeneous space Gm /Aut(X).
3.2.

We can apply corollary 2 and proposition 3 to the extended Dynkin type.

Proposition 5. Let Q be a quiver of extended Dynkin type, let M be a kQ-module. Then,for any dimension vector e the quiver Grassmannian Gre (M ) has the polynomiality property. If Gre (M ) is non empty, then its Euler characteristics is positive.
Proof. The last assertion is a consequence of the first one by proposition 3. To prove
the first one, we use proposition 1 to reduce to the case of indecomposable M . We will
construct a decomposition of Em into a disjoint union of strata, such that
1. The number of strata is finite,
2. each stratum has the polynomiality property,
3. dim Hom( , M ) is constant along the stratum.
a,M
has the polynomial property; hence Corollary
Points 1, 2, 3 imply that each variety Em
2 provides the first assertion.
Recall the classification [9] of the indecomposable kQ-representations into preprojective,
postinjective, regular homogeneous and regular non-homogeneous ones, reflecting the classification of the Auslander-Reiten components. All indecomposables of dimension vector amultiple of the minimal imaginary root δ are represented in these last two classes. Moreover, any regular homogeneous component contains a unique module with given dimension
vector nδ, n > 0.
As M is indecomposable, it belongs to one of the A.R. components. We denote by C0 thiscomponent if M is regular homogeneous. If not, we can just set C0 = ∅ in the sequel.
Fix isomorphism classes P , I, R, C of a preprojective, resp. postinjective, resp. regular
non-homogeneous representation, resp. C0 -component representation, and fix finitely many
partitions λ1 , . . . , λk . We define the stratum S = SP,I,R,C,λ1,...,λk ⊂ Em as consisting of all
representations of the form
P ⊕I ⊕R⊕C⊕

k M∞
M
i=1 j=1(λ )Ui,j i j ,10

PHILIPPE CALDERO AND MARKUS REINEKE

where the Ui,j are regular homogeneous indecomposables of dimension vector j ·δ belonging
to k pairwise different homogeneous components C1 , . . . , Ck (different from C0 ), that is, Ui,jbelongs to Ci for all i, j. Since there are at most finitely many isomorphism classes ofpreprojective or postinjective or regular non-homogeneous indecomposables of any given
dimension vector, this defines a finite stratification of Em . Hence, point 1 is verified.
Since the number of isomorphism classes of indecomposable representations of any given
dimension vector behaves polynomially in the cardinality of the finite base field by [15],the number of isomorphism classes in each stratum S does, too. So, let PS (q) be thepolynomial counting the number of isomorphism classes of indecomposable representations
in S.
By standard facts on morphisms between indecomposables over extended Dynkin quivers [9] (in particular, the standard directedness properties and orthogonality of regularcomponents), the isomorphism type of the automorphism group of a representation is constant along each stratum. Thus, the cardinality of orbits is constant, say #OS , along eachstratum. From these facts, we conclude that #S = PS (q)#OS . Moreover, an orbit has thepolynomial property, so point 2 is proved.
Again by [9], dim Hom( , M ) is constant along each of these refined strata; this gives
point 3 and we are done.

4. The tangent space of the quiver Grassmannian.
This section adopts some methods from [22] (see also [17, section 4.6]) and is devoted
to the calculation of the tangent space of the quiver Grassmannian Gre (M ) considered asa scheme. As a corollary, we obtain the smoothness of the variety Gre (M ) when M is an
exceptional module.
4.1. Let M = ((Mi )i , (Mα )α ) be a representation of Q over k of dimension vector m , andlet e be another dimension vector. We still denote by Ee the variety of representations of
Q over k of dimension vector e , and by Ge the group acting on it.
Fix k-vector spaces UiQof dimension ei for all i ∈ I and define Hom(e , m ) as the spaceof I graded linear maps i∈I Homk (Ui , Mi ).
Consider the subvariety Hom0 (e , M ) of Re × Hom(e , m ) consisting of pairs of tuples((Uα )α , (fi )i ) such that all fi are injective, and Mα fi = fj Uα for all arrows α : i → j. Thus,
Hom0 (e , M ) parametrizes representations of dimension vector e together with an injective
kQ-morphism to M . The group Ge acts freely on Hom0 (e , M ) by (gi )i ((Uα ), (fi )) =((gj Uα gi−1 )α:i→j , (fi gi−1 )).
Lemma 2. The quiver Grassmannian Gre (M ) is isomorphic to the geometric quotient
Hom0 (e , M )/Ge .
Proof. Consider the map φ : Hom0 (e , M ) → Gre (M ), which maps an element ((Uα ), (fi ))to ⊕i fi (Ui ) ⊂ M .
It is well defined since the relations Mα fi = fj Uα assert that ⊕i fi (Ui ) is a submodule of
M . Moreover φ is clearly surjective.
Suppose that ((Uα ), (fi )) and ((Uα′ ), (fi′ )) have the same image by φ. Hence, for any i,fi (Ui ) = fi′ (Ui ) and by injectivity of fi , fi′ for any x in Ui there exists a unique y in Uisuch that fi′ (y) = fi (x). The maps x 7→ y define an element gi of Aut(Ui ). This gives(gi )i ((Uα ), (fi )) = ((Uα )′ , (fi′ )). Hence each fiber of φ is a (free) Ge -orbit. This implies thelemma.

Note that the following proposition generalizes a well-known result for classical Grassmannian of vector spaces.

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

11

Proposition 6. Let M be a kQ-module of dimension vector m , and let U be a submoduleof M of dimension vector e . Then, the tangent space TU (Gre (M )) at U of the quiver
Grassmannian Gre (M ) is isomorphic to HomkQ (U, M/U ).
Proof. To compute the tangent space TU (Gre (M )), we fix a point ((Uα ), (fi )) in Hom0 (e , M )in the fiber φ−1 (U ), compute the tangent space T of Hom0 (e , M ) at this point, and factor
it by the image of the differential of the action of Ge .
To compute T , we perform a calculation with dual numbers: since Re × Hom(e , m ) is
just an affine space, an element of the tangent space at the point ((Uα ), (fi )) looks like((Uα + ǫζα ), (fi + ǫAi )), with ((ζα ), (Ai )) ∈ Re × Hom(e , m ). The condition for this tobelong to the tangent space T is therefore:Mα (fi + ǫAi ) = (fj + ǫAj )(Uα + ǫζα )for all α : i → j, which yields the condition(4.1)

Mα Ai = fj ζα + Aj Uα ,for all α : i → j.
The differential of the action of Ge is computed by applying the definition of the action
to a point ((1 + ǫxi )i ) of T1 Ge :(1 + ǫxi )i ((Uα + ǫζα ), (fi + ǫAi )) == (((1 + ǫxj )(Uα + ǫζα )(1 − ǫxi ))α:i→j , ((fi + ǫAi )(1 − ǫxi ))) == ((Uα + ǫ(ζα + xj Uα − Uα xi ))α:i→j , (fi + ǫ(Ai − fi xi ))).
By the above, we arrive at the following formula for the tangent space to the quiver
Grassmannian:{((ζα ), (Ai )) | Mα Ai = fj ζα + Aj Uα }TU (Grm (M )) =.
{((xj Uα − Uα xi )α:i→j , (−fi xi )) | (xi )i ∈ T1 Ge }To understand this condition better, we can assume without loss of generality that Mα
is given as a 2 × 2-block matrix


Uα ξα
Mα =
0 Nα
1
for allh α,iand that fi = 0 for all i (1 representing the identity matrix). We also writeiAi = BCi for all i ∈ I. Then the condition 4.1 reads   
 
Bi
1
Bj
Uα ξα
=
ζα +
Uα ,
0 Nα
Ci
0
Cj
yielding the two conditions
Uα Bi + ξα Ci = ζα + Bj UαandNα Ci = Cj Uαfor all α : i → j. The subspace to be factored out reads−xi{((xj Uα − Uα xi )α:i→j ), ())}.
0
Hence, choosing xi = −Bi , we can assume each Bi to be zero. We also see that each ζα isuniquely determined by Bi and Ci by the above formula. So the only remaining choices are
the Ci , subject to the condition Nα Ci = Cj Uα for all α : i → j. This condition just meansthat C = (Ci ) defines a kQ-morphism from U to N = (Nα )α , and N can be identified with
M/U . This proves the proposition.
12

PHILIPPE CALDERO AND MARKUS REINEKE

4.2. We give here some consequences of the tangent space formula.
First recall that, since kQ is hereditary, the Euler form hM, M ′ i := dim HomkQ (M, M ′ )−dim Ext1kQ (M, M ′ ) passes to the Grothendieck group.
Corollary 3. Let M be a kQ-module of dimension vector m , and let U be a submodule of
M of dimension vector e . Then,he , m − e i ≤ dim TU (Gre (M )) ≤ he , m − e i + dim Ext1 (M, M ).
In particular,
he , m − e i ≤ dim Gre (M ) ≤ he , m − e i + dim Ext1 (M, M ).
Proof. Applying Hom( , M ) to the short exact sequence 0 → U → M → M/U → 0, we geta surjective map Ext1 (M, M ) → Ext1 (U, M ), since kQ is hereditary.
Applying Hom(U, ) to the same sequence yields a surjection
Ext1 (U, M ) → Ext1 (U, M/U ).
This implies
dim Ext1 (U, M/U ) ≤ dim Ext1 (U, M ) ≤ dim Ext1 (M, M ).
Hence, the equality dim HomkQ (U, M/U ) = he , m −e i+dim Ext1 (U, M/U ) gives the proposition.

Corollary 4. Let M be an exceptional module and let e be such that Gre (M ) is non empty,
then Gre (M ) is a smooth projective variety of dimension he , m − e i.
Qi
Proof. Consider the closed subscheme Gre (m ) of Em × i Grm
ei consisting of pairs (M, (Ui )i )of a representation M of dimension vector m and a collection Ui ⊂ Mi of ei -dimensional
subspaces such that
Mh (Us(h) ) ⊂ Ut(h)for all arrows
Q h. iThis condition defines a subbundle of the trivial vector bundle with fibre
Em over i Grm
ei , thus Gre (m ) is smooth and reduced. Its dimension is easily calculatedasXXX
ei (mi − ei ) +es(h) et(h) +(ms(h) − es(h) )mt(h) = dim Em + he , m − e i.
ihh

The projection π : Gre (m ) → Em is proper, and its fibre π −1 (M ) is isomorphic to thescheme Gre (M ). Now assume that Gre (M ) is non-empty, and that M is exceptional. Thismeans that the image of π contains the orbit of M , which is open. But since π is proper, its
image is closed. Thus, π is surjective. By the theorem on generic smoothness [13, III.10.7.],
this implies that a generic fibre of π is smooth, thus reduced, and of dimension
dim Gre (m ) − dim Em = he , m − e i.
Using again the fact that the orbit of M is open in Em , we conclude that Gre (M ) is asmooth variety of dimension he , m − e i.
Note that this corollary does not imply the last part of theorem 1 since smooth projective
varieties may have non positive characteristics.

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

13

5. Application to acyclic cluster algebras
5.1. We recall some terminology on cluster algebras. The reader can find more precise
and complete information in [10] and [11].
Let n be a positive integer. We fix the ambient field F := Q(x1 , . . . , xn ), where the xi ’s
are indeterminates. Let x be a free generating set of F over Q and let B = (bij ) be ann × n antisymmetric matrix with coefficients in Z. Such a pair (x, B) is called a seed.
Let (u, B) be a seed and let uj , 1 ≤ j ≤ n, be in u. We define a new seed as follows.
Let u′j be the element of F defined by the exchange relation:
Y bij
Y −bij(5.1)
uj u′j =ui +ui .
bij >0bij <0Set u′ = u ∪ {u′j }\{uj }. Let B ′ be the n × n matrix given by(−bik
b′ik =
bik + 12 ( |bij | bjk + bij |bjk | )if i = j or k = jotherwise.

By a result of Fomin and Zelevinsky, (u′ , B ′ ) = µj (u, B) is a seed. It is called the mutationof the seed (u, B) in the direction uj (or j). We consider all the seeds obtained by iterated
mutations from the seed (x, B). The free generating sets occurring in the seeds are called
clusters, and the variables they contain are called cluster variables. By definition, thecluster algebra A(x, B) associated to the seed (x, B) is the Z-subalgebra of F generated
by the set of cluster variables. The graph whose vertices are the seeds and whose edge are
the mutations between two seeds is called the mutation graph of the cluster algebra.
Let Q be an acyclic finite quiver, we can associate a matrix B Q = (bQij )i,j∈Q0 defined by
if Q1,ij non empty
#Q1,ijQ
bij = −#Q1,ji if Q1,ji non empty
0
otherwise
where Q1,ij is the set of arrows from i to j. The cluster algebra associated to the seed(x, BQ ) will be denoted by A(Q).

5.2. We denote by EQ the set of exceptional indecomposable Q-representations. According
to [5], see also [3], there exists a bijection β from the set EQ to the set ClQ of cluster variablesof the cluster algebra A(Q).
The bijection is given by
Y −m X
Y mt(h) −et(h) es(h)xi iβ(M ) = XM =xt(h) .
xs(h)χc (Gre (M ))
i∈Q0

e

h∈Q1

Note first that this formula precises, in the acyclic case, the Laurent phenomenon, see[10], true in general, and which asserts that the cluster variables are Laurent polynomials
with integer coefficients in the xi , 1 ≤ i ≤ n.
This theorem answers a positivity conjecture of Fomin and Zelevinsky in the acycliccase. It is a direct consequence of Theorem 1
Theorem 2. Let Q an acyclic quiver and A(Q) be the associated cluster algebra. Then,±1
the cluster variables are in Z≥0 [x±11 , . . . , xn ]We deduce another proof of the denominator property in the acyclic case, [5], [2], [14].

Corollary 5. Let M be an exceptional indecomposable kQ-module with dimension vecof XM as an irreducible fraction of integral
tor dim M = (mi ). Then the denominatorQipolynomials in the variables xi is i xm
.
i

14

PHILIPPE CALDERO AND MARKUS REINEKE

Proof. By the positivity
theorem, XM is a linear combination with positive integer coeffiQ
cients of terms xnl l , nl ∈ Z. These terms are indexed by the set of dimension vectors of
submodules N of M , and for each submodule N , we have thatXX
nl = −dl +ei +(dj − ej ).
i→l

l→j

Let us put d = dim M and e = dim N . Then it is sufficient to prove that
1. for all l, we have nl ≥ −dl and2. for all l, there exists a submodule N such that the equality holds.
Since we have 0 ≤ ei ≤ di for all i, the first assertion is clear. For the second one, we
simply choose N to be the submodule of M P
generated by all the spaces Mj such that there
exists an arrow l P→ j. Then the terms in i→l ei vanish since Q has no oriented cycles
and the terms in l→j (dj − ej ) vanish by the construction of N .

6. Remarks on constructivity.

6.1. We add here some remarks on constructive calculus of the Euler characteristics of thequiver Grassmannian in the acyclic case. Indeed, the use perverse sheaves gives the positivity of χc (Gre (M )) but one can’t expect any combinatorial formula from this approach.
It is natural to ask if the quiver Grassmannian Gre (M ) has a cellular decomposition, inparticular, a decomposition induced by the Schubert cell decomposition of the (subspaces-)
Grassmannian of M .
In [6], the authors follow this idea for the Kronecker quiver and obtain closed formulas
for the cluster variables and even for a Z-base of the corresponding cluster formula.
Let’s sketch a general approach to the problem. Let Q be any acyclic quiver and set
H := kQ. We consider the multiplicative group H ∗ of invertible elements of H. Let M bean H-module with dimension vector dim (M ) = m and let φ be the group homomorphism
H ∗ → GL(M ) resulting from the module structure of M .
As Q is acyclic, we obtain that H ∗ is a solvable, so is φ(H ∗ ), and this implies that there
exists a Borel subgroup B of GL(M ) which contains φ(H ∗ ).
Let GrPε (M ) be the (classical) Grassmannian of subspaces of M with dimension ε. If| e |:= i ei = ε, then Gre (M ) is a closed subvariety of Grǫ (M ). The group homomorphism`∗
φ induces an action of H ∗ on Gre (M ). We claim that |e |=ε Gre (M ) = Grǫ (M )H .
as H ∗ is Zariski
Indeed, it is clear that any element of Gre (M ) is fixed by H ∗ . Conversely, `∗dense in H, any subspace fixed by H is stable by H and so, it belongs to |e |=ε Gre (M ).
Now consider the Schubert cell decomposition relative to B:aXj ,Grε (M ) =jwhere Xj is an affine space endowed with a left B-action.
Hence, there comes:
Proposition 7. We have the following decompositionaa∗
Gre (M ) =XjH ,|e |=εjwhere Xj are affine spaces with a regular H ∗ -action.

The problem is that this decomposition has no reason to be a cellular decomposition
since the action of B on the affine cell Xj is in general not affine. But the method may
afford nice formulas when the Q-representation M is well understood.

ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

15

6.2. Let us describe some results of [6], obtained by using a Schubert decomposition.
Let Q be the Kronecker quiver with vertices 1 and 2 and two arrows α, β from 1 to 2.
The isoclasses of indecomposable Q-representations where found by Kronecker, [16], andare described a follows
Let M n be the Q-representation defined by α(ui ) = vi , β(ui ) = vi+1 , 1 ≤ i ≤ n, where(u1 , . . . , un ), resp. (v1 , . . . , vn+1 ), is a base of M1n , resp. M2n .
Let DM n be the dual Q-representation of M n , ie the spaces are the dual spaces (M2n )∗ ,and (M1n )∗ , and the arrows are just the adjoint morphisms.
Let Mreg (λ)n , λ ∈ k, be the Q-representation defined by α(ui ) = vi , β(ui ) = λvi + vi+1 ,1 ≤ i ≤ n, where (u1 , . . . , un ), resp. (v1 , . . . , vn ), is a base of Mreg (λ)n1 , resp. Mreg (λ)n2 .
Then, M n , DM n , n ≥ 0; Mreg (λ)n , n > 0, λ ∈ k, is the complete set of isoclasses ofindecomposable Q-representations.
Using the method above one finds,
Proposition 8. [6]χc (Gre (Mn(P e1 −1  n+1−e1 pp e2 −n−1+p n−p−e1
p
)) =
n+1
e2
χc (Gr(e1 ,e2 ) (DM n )) = χc (Gr(e2 ,e1 ) (M n )),nχc (Gre (Mreg (λ) )) =Xppe2 − n + p

e1
n − p − e1

if e1 6= 0,
if e1 = 0n − e1.
p

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Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex,
FranceE-mail address: caldero@math.univ-lyon1.fr
Fachbereich C, Bergische Universität Wuppertal, D - 42097 Wuppertal, Germany
E-mail address: reineke@math.uni-wuppertal.de