GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ

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1 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstract. Let Q be a finite quiver without oriented cycles, and let Λ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w W, there is associated a unipotent cell N w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring C[N w ] of N w is a cluster algebra in a natural way. A central role is played by generating functions ϕ X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C w of the category of nilpotent Λ-modules. The first aim of this article is to compare the function ϕ X with the so-called cluster character of X, which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every X in C w, ϕ X coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with X using any cluster-tilting object T of C w. A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of X and certain quiver Grassmannians. This isomorphism is obtained in a very general setup and should be of interest in itself. Another important tool of the proof is a representation-theoretic version of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky, adapted to Kac-Moody groups. As an application, we get a new description of a generic basis of the cluster algebra A(Γ T ) obtained from C[N w ] via specialization of coefficients to. Here generic refers to the representation varieties of a quiver potential arising from the cluster-tilting module T. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont. Contents. Introduction and main results 2. Reminder on cluster algebras 8 3. Partial flag varieties and quiver Grassmannians 4. Categorification of the Chamber Ansatz Monomials of twisted minors Cluster character identities E-invariant and Ext Generic bases for cluster algebras Categorification of the twist automorphism 5 References 54. Introduction and main results.. In the recent literature on cluster algebras, calculations of Euler characteristics of certain varieties related to quiver representations play a prominent role. In [GLS2, GLS3, GLS5], cluster variables of coordinate rings of unipotent cells of algebraic groups and Kac- Moody groups were shown to be expressible in terms of Euler characteristics of varieties Mathematics Subject Classification (200): 3F60, 4M5, 4M99, 6G20, 20G44.

2 2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER of flags of submodules of preprojective algebra representations. In another direction, starting with a formula of Caldero and Chapoton [CC], the coefficients of the Laurent polynomial expansions of cluster variables of some cluster algebras were described as Euler characteristics of Grassmannians of submodules of quiver representations. This was first achieved for acyclic cluster algebras [CK], later for cluster algebras admitting a 2-Calabi- Yau categorification [P, FK], and more recently for general antisymmetric cluster algebras of geometric type [DWZ2]. There is a posterior but essentially different proof in [Pl], see also [N]. The first aim of this paper is to compare these two types of formulas for the large class of cluster algebras which can be realized as coordinate rings of unipotent cells of Kac-Moody groups. To do this, we will return to the very source of cluster algebras, namely to the Chamber Ansatz of Berenstein, Fomin and Zelevinsky [BFZ, BZ], which describes parametrizations of Lusztig s totally positive parts of unipotent subgroups and Schubert varieties. The second aim of this paper is to provide a new understanding of the Chamber Ansatz formulas in terms of representations of preprojective algebras, together with a generalization to the Kac-Moody case. In particular the mysterious twist automorphisms of the unipotent cells needed in these formulas turn out to be just shadows of the Auslander-Reiten translations of the corresponding Frobenius categories of modules over the preprojective algebras. Our treatment of the Chamber Ansatz shows that the numerators of the twisted minors of [BFZ, BZ] form a cluster, and that the Laurent expansions with respect to these special clusters have coefficients equal to Euler characteristics of varieties of flags of submodules of preprojective algebra representations. This provides the desired link between the two types of Euler characteristics mentioned above, and it allows us to show that the cluster characters of Fu and Keller [FK] coincide after an appropriate change of variables with the ϕ-functions of [GLS2, GLS5]. Finally, our third aim is to exploit these results for studying natural bases of cluster algebras containing the cluster monomials. We consider the class of coefficient-free cluster algebras obtained by specializing to the coefficients of the cluster algebra structures on unipotent cells. In [GLS5, Section 5.6] we have found such bases, consisting of appropriate subsets of Lusztig s dual semicanonical bases. Here, using the above connection with Fu- Keller cluster characters, we give a new description of the same bases in terms of module varieties of endomorphism algebras of cluster-tilting modules. In the special case when the cluster algebra is acyclic, this proves Dupont s generic basis conjecture [D]. In general, the elements of these bases are generating functions of Euler characteristics of quiver Grassmannians, at generic points of some particular irreducible components of the module varieties. These special irreducible components can be characterized in terms of the new E-invariant introduced by Derksen, Weyman and Zelevinsky [DWZ2] for representations of quivers with potential, and one may therefore conjecture that a similar description of a generic basis can be extended to any antisymmetric cluster algebra..2. To state our results more precisely, we need to introduce some notation. Let Q be a finite quiver with vertex set {,..., n} and without oriented cycles. Denote by Λ the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be the Weyl group of g. The graded dual U(n) gr of the universal enveloping algebra U(n) of the positive part n of g can be identified with the coordinate ring C[N] of an associated pro-unipotent pro-group N with Lie algebra n. For w W, let N w := N (B wb ) be the corresponding unipotent cell in N, where B denotes the standard negative Borel subgroup of the Kac-Moody group G attached to g. Here we use the same notation as in [GLS5]. For details on Kac-Moody groups we refer

3 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 3 to [Ku, Sections 6 and 7.4]. Let x i (t) denote the one-parameter subgroup of N associated to the simple root α i. For each reduced expression i = (i r,..., i ) of w, the map x i : (t r,..., t 2, t ) x ir (t r ) x i2 (t 2 )x i (t ) gives a birational isomorphism from C r to N w. In [GLS5] we have described a cluster algebra structure on C[N w ] in terms of the representation theory of the preprojective algebra Λ. For a nilpotent Λ-module X and a = (a r,..., a ) N r let F i,a,x be the projective variety of flags X = (0 = X r X X 0 = X) of submodules of X such that X k /X k = S a k i k for all k r, where S j denotes the one-dimensional Λ-module supported on the vertex j of Q. The varieties F i,a,x were first introduced by Lusztig [L] for his Lagrangian construction of U(n). Dualizing Lusztig s construction, we can associate with X a regular function ϕ X C[N] satisfying ϕ X (x i (t)) = a N r χ(f i,a,x )t a., and χ denotes the topological Euler charac- Here t = (t r,..., t ) C r, t a := t ar r t a 2 teristic. 2 ta Buan, Iyama, Reiten, and Scott [BIRS] have attached to w a 2-Calabi-Yau Frobenius subcategory C w of the category of finite-dimensional nilpotent Λ-modules. (The same categories were studied independently in [GLS4] for special elements w called adaptable.) In [GLS5] we showed that the C-span of {ϕ X X C w } is a subalgebra of C[N], which becomes isomorphic to C[N w ] after localization at the multiplicative subset {ϕ P P is C w -projective-injective}. Moreover, we showed that C[N w ] carries a cluster algebra structure, whose cluster variables are of the form ϕ X for indecomposable modules X in C w without self-extension. In Section 2 we explain this in more detail. The category C w comes with a remarkable module V i for each reduced expression i of w (see [BIRS, Section III.2], [GLS5, Section 2.4]). The ϕ-functions of the indecomposable direct summands of V i are some generalized minors on N which form a natural initial cluster of C[N w ]. We introduce the new module W i := I w Ω w (V i ), where Ω w = τw is the inverse Auslander-Reiten translation of C w, and I w is the direct sum of the indecomposable C w -projective-injectives. For a Λ-module X, the set Ext Λ (W i, X) is in a natural way a left module over the stable endomorphism algebra E := End Cw (W i ) op = EndCw (V i ) op. Denote by Gr E d (Ext Λ (W i, X)) the projective variety of E-submodules of Ext Λ (W i, X) with dimension vector d, a so-called quiver Grassmannian. Our first main result is Theorem. For X C w and all a N r, there is an isomorphism of algebraic varieties F i,a,x = Gr E d i,x (a) (Ext Λ(W i, X)), where d i,x is an explicit bijection from {a F i,a,x } to {d Gr E d (Ext Λ (W i, X)) }.

4 4 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER It follows easily that the set {a χ(f i,a,x ) 0} has a unique element if and only if Ext Λ (W i, X) = 0. Now by construction, W i is a cluster-tilting module of C w, that is, Ext Λ (W i, X) = 0 if and only if X belongs to the additive hull add(w i ) of W i. Moreover, in this case F i,a,x is reduced to a point. Hence Theorem has the following important consequence: Theorem 2. For X C w, the polynomial function t ϕ X (x i (t)) is reduced to a single monomial t a if and only if X add(w i )..3. Let W i,,..., W i,r denote the indecomposable direct summands of W i. The r-tuple of regular functions (ϕ Wi,,..., ϕ Wi,r ) is a cluster of C[N w ], and it follows from Theorem 2 that the ϕ Wi,k (x i (t)) are monomials in the variables t,..., t r. Inverting this monomial transformation yields expressions of the t k s as explicit rational functions on N w, a result originally called the Chamber Ansatz by Berenstein, Fomin and Zelevinsky [BFZ] in type A n, because of a convenient description of these formulas in terms of chambers in a wiring diagram. To present these formulas in the general Kac-Moody setting, we need more notation. By construction, the summands V i,k of V i are related to the modules W i,k by short exact sequences 0 W i,k P (V i,k ) V i,k 0 where for X C w, P (X) denotes the projective cover in C w. We set ϕ V i,k := ϕ W i,k ϕ P (Vi,k ), a Laurent monomial in the ϕ Wi,k (since add(w i ) contains all C w -projectives). As will be explained in Section.6 below, the regular functions ϕ V i,k on N w are the twisted generalized minors of [BZ] corresponding to i (in the Dynkin case). Denote by q(i, j) the number of edges between two vertices i and j of the underlying unoriented graph of the quiver Q. For k r, put n ( ) (.) C i,k := ϕ V i,k ϕ ϕ q(ik,j) V i,k, (j) V i,k (ik ) j= where k (j) := max{0, s k i s = j} and V i,0 is by convention the zero module. Theorem 3. For k r and t = (t r,..., t ) we have C i,k (x i (t)) = t k. Therefore, for X C w we get an equality in C[N w ]: (.2) ϕ X = a N r χ(f i,a,x )C ar i,r Ca 2 i,2 Ca i,..4. Using Theorem, we now want to compare Equation (.2) with similar formulas of Fu and Keller. To simplify our notation, we define R := {, 2,..., r}, R max := {k R there is no k < s r with i s = i k }, R := R \ R max. Let T = T T r be a basic cluster-tilting module in C w, where the numbering is chosen so that T k is C w -projective-injective for k R max. Assume that (ϕ T,..., ϕ Tr ) is a cluster of C[N w ], i.e. that it can be obtained from (ϕ Vi,,..., ϕ Vi,r ) by a sequence of mutations. In this case, T is called V i -reachable. (One conjectures that this is always the

5 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 5 case.) The endomorphism algebra E T := End Λ (T ) op has global dimension 3, see [GLS5, Proposition 2.9]. Thus we may consider B (T ) := (B (T ) k,l ) k,l R := ((dim Hom Λ (T k, T l )) k,l R ) t, the matrix of the Ringel bilinear form for E T. (For a matrix B, we denote the inverse of its transpose by B t.) For a general 2-Calabi-Yau Frobenius category C with a cluster-tilting object, Fu and Keller [FK, Section 3] (extending previous work of Palu [P]) have attached to every object of C a Laurent polynomial called its cluster character. When applied to the category C w and the cluster-tilting object T, the formula for this cluster character can be written as (.3) θ T X := ϕ (dim Hom Λ(T,X)) B (T ) T d N R χ(gr E T d (Ext Λ(T, X))) ˆϕ d T (X C w ). Here we use the abbreviations ϕ g T := k R ϕg k T k for g = (g, g 2,..., g r ) Z r, ˆϕ T,k := ϕb(t ) l,k l R T l for k R, ˆϕ d T := k R ˆϕ d k T,k for d = (d k ) k R N R. By [FK, Theorem 4.3] and [GLS5, Theorem 3.3], the cluster variables of C[N w ] are of the form θ T X for indecomposable rigid modules X of C w, and (.3) gives therefore a representation-theoretic description of their cluster expansions with respect to the cluster (ϕ T,..., ϕ Tr ). However, for an arbitrary X C w not much is known about the function θ T X. For instance it is a priori only a rational function on N w. Using Theorem and the Chamber Ansatz Theorem 3, we prove our next main result: Theorem 4. For every X C w we have θ T X = ϕ X. In particular, θ T X is a regular function on N w for every X C w, that is, the image of the cluster character X θ T X is in the cluster algebra C[N w ]..5. In the last part of this paper, we deduce from Theorem 4 a new description of a generic basis for the coefficient-free cluster algebra obtained from C[N w ] by specializing to the functions ϕ P for all C w -projective-injectives P. (This algebra can be seen as the coordinate ring of the subvariety N (N wn ) of N w, but we will not use it.) In [GLS5, Section 5.6] we have already described such a basis in terms of generic modules over the preprojective algebra Λ. Here we want to express it in terms of generic modules over the stable endomorphism algebra E T of the cluster-tilting module T. The quiver Γ T of E T has the set R as vertices, with k R corresponding to T k, and it has [B (T ) l,k ] + arrows from k to l, where we write for short [z] + = max(z, 0). We consider the cluster algebra A(Γ T ) C((x k ) k R ) with initial seed ((x k ) k R, Γ T ). We have a unique ring homomorphism Π T : C[N w ] C((x k ) k R ) such that Π T (ϕ Tk ) = x k for k R, and Π T (ϕ Tk ) = for k R max. The homomorphism Π T restricts to an epimorphism C[N w ] A(Γ T ), which we also denote by Π T. Following Palu [P], for an E T -module Y we put (.4) ψ Y := x gy χ(gr E T d (Y )) ˆxd T, d N R

6 6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER where and g Y := (g k ) k R := x g Y := x gk k, k R ( ) dim Ext E T (S k, Y ) dim Hom ET (S k, Y ) k R ˆx T,k := x B (T ) l,k l R l, ˆx d T := ˆx dk T,k. k R (Here S k, k R are the simple E T -modules.) In fact, if Y = Ext Λ (T, X) for some X C w, in view of Theorem 4 we have ψ Y = Π T (ϕ X ). For d N R let mod(e T, d) be the affine variety of representations of E T with dimension vector d. It will be convenient to consider mod(e T, d) with the right action of GL d := k R GL d(k) (C) by conjugation. For each irreducible component Z of mod(e T, d) there is a dense open subset U Z such that for all U, U U we have ψ U = ψ U. Define ψ Z := ψ U, where U U. An irreducible component Z of mod(e T, d) is called strongly reduced if there is a dense open subset U Z such that ( ) codim Z (U. GL d ) = dim Hom ET τ E (U), U T for all U U, where τ ET denotes the Auslander-Reiten translation of mod(e T ). It follows from Voigt s Lemma [G, Proposition.] that strongly reduced components are (schemetheoretically) generically reduced, hence the name. But contrary to what the terminology might suggest, being strongly reduced is not a property of the scheme Z equipped with its GL d -action, since the definition uses additionally the representation theory of the algebra E T. Note also that E T is given by a quiver with potential [BIRSm], and that is the E-invariant defined in [DWZ2]. dim Hom ET (τ E T (U), U) = E inj (U) Let Irr(mod(E T, d)) be the set of irreducible components of mod(e T, d), and set Irr(E T ) := Irr(mod(E T, d)). d N R Let Irr sr (E T ) denote the set of all strongly reduced irreducible components in Irr(E T ). For Z Irr(E T, d) define Null(Z) := {m N R m(k) = 0 if d(k) 0}. Finally, let us denote by S w the dual semicanonical basis of C[N w ] constructed in [GLS5]. We can now state Theorem 5. The set G T w := {x m ψ Z Z Irr sr (E T ), m Null(Z)} is a basis of the cluster algebra A(Γ T ). It is equal to the image of the dual semicanonical basis S w under Π T : C[N w ] A(Γ T ). Each finite-dimensional path algebra is isomorphic to E T for some appropriate Λ, w and T, see [GLS5, Section 6]. In this case, mod(e T, d) is an (irreducible) affine space for all d, and it is easy to see that mod(e T, d) is strongly reduced. Thus Theorem 5 implies Dupont s conjecture [D, Conjecture 6.]. On the other hand, even if E T is not hereditary but mutation equivalent to an acyclic quiver, it is quite easy to find examples of irreducible components of varieties mod(e T, d) which are not strongly reduced.

7 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 7 Since Theorem 5 gives a description of the generic basis G T w of A(Γ T ) entirely in terms of the varieties of representations of the algebra E T, it is natural in view of [DWZ2] to ask if the first statement of Theorem 5 generalizes to other classes of cluster algebras..6. The paper closes with our categorical interpretation of the twist automorphisms of the unipotent cells, introduced by Berenstein, Fomin and Zelevinsky in connection with the Chamber Ansatz. For x N w, the intersection N (B wx T ) consists of a unique element, which, following [BFZ, BZ, GLS5], we denote by η w (x). (The anti-automorphism g g T of the Kac-Moody group is defined in [GLS5, Section 7.]. For more details on η w we refer to [GLS5, Section 8].) The map η w is in fact a regular automorphism of N w, and we denote by (η w) the C-algebra automorphism of C[N w ], defined by Theorem 6. For every X C w, we have ((η w) f)(x) = f(η w (x)) (f C[N w ]). (η w) (ϕ X ) = ϕ Ω w(x) ϕ P (X). Moreover, η w preserves the dual semicanonical basis S w of C[N w ] and permutes its elements. Thus, the regular functions ϕ V i,k occurring in Theorem 3 are obtained by twisting the generalized minors ϕ Vi,k with ηw, in agreement with [BFZ, BZ] in the Dynkin case. We believe that Theorem 6 provides a conceptual explanation of the existence of the automorphism η w, and of its compatibility with total positivity [BZ, Proposition 5.3]..7. The article is organized as follows: In Section 2 we give a short reminder on cluster algebras and some previous results. In Section 3 we construct isomorphisms between flag varieties and quiver Grassmannians in a very general setup. The isomorphisms stated in Theorem turn out to be special cases. Section 4 contains the proofs of Theorems and 2 and of the Chamber Ansatz Theorem 3 together with some illustrating examples. The proof of the cluster character identities stated in Theorem 4 and a detailed example are in Sections 5 and 6. The proof of Theorem 5 is in Sections 7 and 8. Finally, Section 9 contains the proof of Theorem Notation. Throughout, we work over the field C of complex numbers. For a C- algebra A let mod(a) be the category of finite-dimensional left A-modules. By an A- module we always mean a module in mod(a), unless stated otherwise. Often we do not distinguish between a module and its isomorphism class. Let D := Hom C (, C) be the usual duality functor. For a quiver Q let rep(q) be the category of finite-dimensional representations of Q over C. It is well known that we can identify rep(q) and mod(cq). By a subcategory we always mean a full subcategory. For an A-module M let add(m) be the subcategory of all A-modules which are isomorphic to finite direct sums of direct summands of M. A subcategory U of mod(a) is an additive subcategory if any finite direct sum of modules in U is again in U. By Fac(M) (resp. Sub(M)) we denote the subcategory of all A-modules X such that there exists some t and some epimorphism M t X (resp. monomorphism X M t ). For an A-module M let Σ(M) be the number of isomorphism classes of indecomposable direct summands of M. An A-module is called basic if it can be written as a direct sum

8 8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER of pairwise non-isomorphic indecomposable modules. An A-module M is called rigid if Ext A (M, M) = 0. For an A-module M and a simple A-module S let [M : S] be the Jordan-Hölder multiplicity of S in a composition series of M. Let dim(m) := dim A (M) := ([M : S]) S be the dimension vector of M, where S runs through all isomorphism classes of simple A-modules. For a set U we denote its cardinality by U. If f : X Y and g : Y Z are maps, then the composition is denoted by gf = g f : X Z. If U is a subset of a C-vector space V, then let Span C U be the subspace of V generated by U. Let N = {0,, 2,...} be the natural numbers, including 0, and let Z be the ring of integers. For a domain R let R(X,..., X r ), R[X,..., X r ] and R[X ±,..., X± r ] be the field of rational functions, the polynomial ring, and the ring of Laurent polynomials in the variables X,..., X r with coefficients in R, respectively. 2. Reminder on cluster algebras 2.. Let F := Q(X,..., X r ) be the field of rational functions in r variables. We fix a subset F {,..., r}. A seed in F is a pair (x, Γ), where Γ = (Γ 0, Γ, s, t) is a finite quiver without loops and without 2-cycles with set of vertices Γ 0 = {,..., r}, and x = (x,..., x r ) with x,..., x r algebraically independent elements in F. The vertices in {,..., r} \ F are called mutable, and the ones in F are frozen. Given a seed (x, Γ) in F and a mutable vertex k of Γ, we define the mutation of (x, Γ) at k as µ k (x, Γ) := (x, Γ ). The quiver Γ is obtained from Γ by applying the Fomin-Zelevinsky quiver mutation at k, which is defined as follows: For i, j r let γ ij := number of arrows j i in Γ number of arrows i j in Γ. (Recall that there are no 2-cycles in Γ. So at least one of the numbers on the right-hand side is 0.) By definition also Γ has no loops and no 2-cycles, and the corresponding numbers γ ij for Γ are γ ij if i = k or j = k, γ ij := γ ij + γ ik γ kj + γ ik γ kj otherwise. 2 Finally, x = (x,..., x r) is defined by { ( x x k k i s := x i + j k j) x x s if s = k, otherwise where the products are taken over all arrows of Γ which start, respectively end, in k. Set µ (x,γ) (x k ) := x k. It is easy to check that (x, Γ ) is again a seed in F and that (x, Γ) = (x, Γ). µ 2 k

9 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 9 Two seeds (x, Γ) and (y, Σ) are mutation equivalent if there is a sequence (k,..., k t ) with k i {,..., r} \ F for all i such that In this case, we write (y, Σ) (x, Γ). For a seed (x, Γ) in F let µ kt... µ k2 µ k (x, Γ) = (y, Σ). X (x,γ) := (y,σ) (x,γ) {y,..., y r } where the union is over all seeds (y, Σ) with (y, Σ) (x, Γ). By definition, the cluster algebra A(x, Γ) associated to (x, Γ) is the subalgebra of F generated by X (x,γ). We call (y, Σ) a seed in A(x, Γ) if (y, Σ) (x, Γ). In this case, y is a cluster in A(x, Γ), the elements y,..., y r are cluster variables and y m yr mr with m i 0 for all i are cluster monomials in A(x, Γ). For any seed of the form (y, Γ) in F we obtain an isomorphism A(x, Γ) A(y, Γ) given by x i y i for all i r. So one sometimes writes just A(Γ) instead of A(x, Γ). Note that for any cluster y in A(x, Γ) we have y i = x i for all i F. These cluster variables are also called coefficients of A(x, Γ). Localizing A(x, Γ) at i F x i yields an algebra A(x, Γ, F ± ), which we also call a cluster algebra. There are algebra epimorphisms defined by A(x, Γ) A(x, Γ) and A(x, Γ, F ± ) A(x, Γ) x i { if i F, x i otherwise, where A(x, Γ) Q((x i ) i {,...,r}\f ) is again a cluster algebra with x := (x i ) i {,...,r}\f, and the quiver Γ is obtained from Γ by deleting all vertices in F and all arrows starting or ending in one of the vertices in F. We say that the cluster algebra A(x, Γ) is obtained from A(x, Γ) by specialization of coefficients to, and the two epimorphisms defined above are called specialization morphisms. Clearly, the specialization morphisms induce a surjective map X (x,γ) \ {x i i F } X (x,γ). Using the identification C[N w ] A(Γ T ), the epimorphism Π T defined in Section.5, can be seen as a specialization morphism. Thus the cluster algebra A(Γ T ) is obtained from C[N w ] by specialization of coefficients to Cluster algebra structures for coordinate rings of unipotent cells. In a series of papers [GLS, GLS2, GLS5] we constructed a map ϕ: nil(λ) C[N] which maps a nilpotent Λ-module X to a function ϕ X C[N]. This map satisfies the following properties: (i) For all X, Y nil(λ) we have ϕ X ϕ Y = ϕ X Y. (ii) Let X, Y nil(λ) with dim Ext Λ (X, Y ) = dim Ext Λ (Y, X) =, and let 0 X E Y 0 and 0 Y E X 0

10 0 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER be non-split short exact sequences. Then we have (iii) Restriction yields a map ϕ X ϕ Y = ϕ E + ϕ E. ϕ: C w C[N w ]. (Again we identified C[N w ] with the localization of the C-span of {ϕ X X C w } at {ϕ P P is C w -projective-injective}.) (iv) Let i = (i r,..., i ) be a reduced expression of w, and let Γ := Γ i and F := R max. (The definitions of Γ i and R max can be found in Sections 3.6 and.4, respectively.) Then there is an algebra isomorphism with η i (x k ) = ϕ Vi,k for all k r. η i : A(x, Γ, F ± ) C[N w ] Using the isomorphism η i one can now speak of cluster variables and cluster monomials in C[N w ]. For example, an r-tuple (ϕ T,..., ϕ Tr ) is a cluster in C[N w ] if and only if there is a seed (y, Σ) in A(x, Γ, F ± ) with η i (y i ) = ϕ Ti for all i. In this case, let T := T T r. The vertices of the quiver Γ T of the endomorphism algebra End Λ (T ) op are naturally parametrized by,..., r and the following hold: (v) With the exception of arrows between coefficients c, d F, the quivers Σ and Γ T coincide. The seed (y, Σ) in A(x, Γ) is already determined by y. (vi) The module T is a basic cluster-tilting module in C w. For any mutable vertex k there is a unique indecomposable T k C w with T k = T k such that µ k (T ) := T k T/T k is a basic cluster-tilting module in C w. For y k := µ (y,σ)(y k ) we have η i (y k ) = ϕ T k. We say that (ϕ T,..., ϕ T k,..., ϕ Tr ) is obtained from (ϕ T,..., ϕ Tk,..., ϕ Tr ) by mutation in direction k. We also say that µ k (T ) is obtained from T by mutation in direction k. (vii) We have dim Ext Λ (T k, T k ) = dim Ext Λ (T k, T k) =, and there are short exact sequences 0 T k j k T j T k 0 and 0 T k k i T i T k 0, where we sum over all arrows in Γ T ending and starting in k, respectively. Furthermore, the identity y k y k = k i y i + j k y j in A(x, Γ) corresponds to the identity ϕ Tk ϕ T k = k i ϕ Ti + j k ϕ Tj in C[N w ]. For i, j R, the number of arrows k i in Γ T equals [B (T ) i,k ] + and the number of arrows j k is [ B (T ) j,k ] +.

11 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ (viii) The cluster monomials in C[N w ] are ϕ m T ϕ mr T r where m i 0 for all i, and T := T T r runs through the set of V i -reachable cluster-tilting modules in C w. (ix) All cluster monomials in C[N w ] belong to the dual semicanonical basis of C[N w ]. 3. Partial flag varieties and quiver Grassmannians 3.. Basic algebras and nilpotent modules. Let Γ = (Γ 0, Γ, s, t) be a finite quiver with set of vertices Γ 0 = {,..., n}, and set of arrows Γ. For an arrow a: i j in Γ let s(a) := i and t(a) := j be its start vertex and terminal vertex, respectively. A path of length m in Γ is an m-tuple p = (a,..., a m ) of arrows in Γ such that s(a i ) = t(a i+ ) for all i m. We define s(p) := s(a m ) and t(p) := t(a ). Additionally, for each vertex i Γ 0 there is a path e i of length 0 with s(e i ) = t(e i ) = i. An arrow a in Γ is a loop if s(a) = t(a). A path p = (a, a 2 ) is a 2-cycle if s(p) = t(p). The path algebra CΓ of Γ has the paths in Γ as a C-basis, and the multiplication of two paths p and q is defined by (a,..., a m, b,..., b l ) if s(p) = t(q), p = (a,..., a m ) and q = (b,..., b l ), p if q = e pq := s(p), q if p = e t(q), 0 if s(p) t(q). Extending this rule linearly turns CΓ into an associative C-algebra with unit element. For m 0 let CΓ m be the ideal in CΓ generated by all paths of length m. An algebra A is called basic if A = CΓ/J, where J is an ideal in CΓ with J CΓ 2. For the rest of this section, we assume that A = CΓ/J is a basic algebra. Let S,..., S n be the -dimensional A-modules associated to the vertices of Γ. (If A is finite-dimensional, then S,..., S n are all simple A-modules up to isomorphism.) We focus on A-modules having only S,..., S n as composition factors. These modules are called nilpotent. The category of all nilpotent A-modules is denoted by nil(a). (If A is finite-dimensional, then nil(λ) = mod(a).) Let Î,..., În be the injective envelopes of S,..., S n, respectively. (The modules Îj are in general infinite-dimensional A-modules.) Let J i be the maximal ideal of A spanned by all residue classes p := p + J of paths, where p runs through all paths except e i. Thus A/J i is -dimensional and (as an A- module) isomorphic to S i. (In the following, we sometimes do not distinguish between a path p in CΓ and its residue class p.) Each (not necessarily finite-dimensional) A-module X can be interpreted as a representation X = (X(i), X(a)) i Γ0,a Γ of the quiver Γ, where the vector space X(i) is defined by e i X, and the linear map X(a): X(s(a)) X(t(a)) is defined by x ax. Recall that a subrepresentation of X is given by U = (U(i)) i Γ0, where U(i) is a subspace of X(i) for all i, and for all a Γ we have X(a)(U(s(a))) U(t(a)). When passing from modules to representations, the submodules obviously correspond to the subrepresentations. The dimension vector of a representation X = (X(i), X(a)) i Γ0,a Γ is by definition dim Γ (X) := (dim X(i)) i Q0. Definition 3.. For a dimension vector d, let Gr A d (X) be the projective variety of subrepresentations Y of X with dim Γ (Y ) = d. Such a variety is called a quiver Grassmannian.

12 2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER If X is nilpotent, then dim X(i) = [M : S i ] for all i Q 0. We study Grassmannians Gr A d (X) only for nilpotent A-modules X, so there is no danger of confusing the two types of dimension vectors dim Γ ( ) and dim A ( ) associated to X and its submodules Refined socle and top series. For an arbitrary (not necessarily finite-dimensional) A-module X and a simple A-module S, let soc S (X) be the sum of all submodules U of X with U = S. (If there is no such U, then soc S (X) = 0.) Similarly, let top S (X) = X/V, where V is the intersection of all submodules U of X such that X/U = S. (If there is no such U, then V = X and top S (X) = 0.) Define rad S (X) := V. Let us interpret X as a representation X = (X(i), X(a)) i Γ0,a Γ of Γ, and let j n. Then soc Sj (X) can be seen as a subrepresentation (X (i)) i Γ0 of X, where { X 0 if i j, (i) = a Γ,s(a)=j Ker(X(a)) if i = j. Similarly, rad Sj (X) can be seen as a subrepresentation (X (i)) i Γ0 { X X(i) if i j, (i) = a Γ,t(a)=j Im(X(a)) if i = j. of X, where It follows that soc Sj (X) and top Sj (X) are isomorphic to (possibly infinite) direct sums of copies of S j. Now fix some sequence i = (i r,..., i ) with i k n for all k. There exists a unique chain (0 = X r X X 0 X) of submodules X k of X such that X k /X k = soc Sik (X/X k ) for all k r. We define soc i (X) := X 0, X + k := Xi,+ k := X k for all 0 k r, and X + := X i,+ := (X r + X + X+ 0 ). If soc i(x) = X, then we call this chain the refined socle series of type i of X. Similarly, there exists a unique chain (0 X r X X 0 = X) of submodules X k of X such that X k /X k = top Sik (X k ) for all k r. Set top i (X) := X/X r, rad i (X) := X r, and X k := Xi, k := X k for all 0 k r. Define X := X i, := (X r X X 0 ). If rad i(x) = 0, then X is called the refined top series of type i of X. The following lemma is straightforward: Lemma 3.2. For arbitrary (not necessarily finite-dimensional) A-modules X and Y and every A-module homomorphism f : X Y the following hold: (i) f(soc i (X)) soc i (Y ) and f(rad i (X)) rad i (Y ). (ii) If f is a monomorphism (resp. epimorphism), then the induced maps X/ soc i (X) Y/ soc i (Y ) and rad i (X) rad i (Y ) are both monomorphisms (resp. epimorphisms). (iii) If soc i (Y ) = Y, then f(rad i (X)) = 0.

13 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 3 For k, s r define J k,s := Also the next lemma is easy to show: { J ik J ik J is if k s, A otherwise. Lemma 3.3. For an arbitrary (not necessarily finite-dimensional) A-module X and k r we have J k, X = X k = rad (i k,...,i )(X). Corollary 3.4. The algebra A/J k, is finite-dimensional for all k r. Proof. Use Lemma 3.3 and the fact that the quiver Γ of A is finite. Let D i be the category of all A-modules X in mod(a) such that soc i (X) = X. Lemma 3.5. For an A-module X the following are equivalent: (i) X D i. (ii) soc i (X) = X. (iii) rad i (X) = 0. Proof. By definition, (i) and (ii) are equivalent. The equivalence of (ii) and (iii) follows by an obvious induction on the length r of the sequence i. Let A i := A/J r,. We identify the category mod(a i ) of finite-dimensional A i -modules with the category of all X in nil(a) such that J r, X = 0. Under this identification we obviously get the following: Lemma 3.6. We have D i = mod(a i ) Partial composition series. Definition 3.7. For X D i and a = (a r,..., a ) with a j 0 let F i,a,x be the (possibly empty) set of chains X = (0 = X r X X 0 = X) of submodules X k of X such that X k /X k = S a k i k for all k r. We call (0 = X r X X 0 = X) a partial composition series of type i of X. Clearly, F i,a,x is a projective variety. The weight of X F i,a,x is defined by wt(x ) := (a r,..., a 2, a ). If X = X (resp. X = X + ), we define a (X) := wt(x ) and a k (X) := a k (resp. a + (X) := wt(x + ) and a + k (X) := a k) for all k r. Lemma 3.8. For X D i and (X r X X 0 ) F i,a,x we have for all k r. X k X k X + k Proof. For k r we show that X k X + k by decreasing induction on k. Clearly, we have X r X r +. (By definition, X r = X r + = 0.) Next, assume that X s X s + for some s r. Thus, there is an epimorphism π : X/X s X/X s +. We have X s /X s soc Sis (X/X s ), and by definition X s + /X+ s = soc Sis (X/X s + ). This implies that π(x s /X s ) X s + /X+ s. In other words, x + X s + X s + /X+ s for all x X s. For each such x there exists some y X s + with x + X+ s = y + X s +. This implies that x y

14 4 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER is in X s +. Since X s + X s + we get x X+ s. Thus we have proved that X k X + k for all k r. Similarly, one shows by induction on k that X k X k for all k r. The next lemma follows from the uniqueness of refined socle and top series. Lemma 3.9. Let X D i. If a is equal to wt(x ) or wt(x + ), then χ (F i,a,x ) =. Corollary 3.0. For every X D i there exists some a such that χ (F i,a,x ) The modules V i. Let A = CΓ/J be a basic algebra, and let i = (i r,..., i ) with i k n for all k. Without loss of generality we assume that for each j n there exists some k with i k = j. For k r and j n let k := max{0, s k i s = i k }, k + := min{k + s r, r + i s = i k }, k max := max{ s r i s = i k }, k min := min{ s r i s = i k }, k j := max{ s r i s = j}. For k r define V k := V i,k := soc (ik,...,i )(Îi k ) and V i := V V r. We also set V 0 := 0. For every j n let I i,j := V kj I i := I i, I i,n. The modules in add(i i ) are called i-injective. and Lemma 3.. For k r we have V k = D(eik (A/J k, )). indecomposable injective A/J k, -module. In particular, V k is an Proof. Clearly, J k, V k = 0. Thus V k is an A/J k, -module. We have soc Sik (V k ) = S ik. Thus V k can be embedded into the indecomposable injective A/J k, -module D(e ik (A/J k, )). We have soc Sik (D(e ik (A/J k, ))) = S ik. Therefore D(e ik (A/J k, )) can be embedded into Îi k. Thus we get two monomorphisms ι V k D(eik (A/J k, )) ι 2 Îi k. Since soc (ik,...,i )(D(e ik (A/J k, ))) = D(e ik (A/J k, )), we can apply Lemma 3.2(i) and get ι 2 (D(e ik (A/J k, ))) soc (ik,...,i )(Îi k ). Since D(e ik (A/J k, )) is finite-dimensional by Corollary 3.4, this implies that V k = D(eik (A/J k, )). Corollary 3.2. V i D i. Proof. We have soc i (V i ) = V i, and V i is finite-dimensional by Corollary 3.4 and Lemma 3.. Lemma 3.3. An A i -module X is injective if and only if X add(i i ). Proof. One easily checks that e j J r, = e j J kj,. This implies D(e j (A/J r, )) = D(e j (A/J kj,)). But D(e j (A/J kj,)) = I i,j by Lemma 3.. Thus the modules in add(i i ) are the injective A i -modules. Lemma 3.4. For every k r there is a monomorphism V k V k.

15 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 5 Proof. We have J k, J k,. Thus there is a short exact sequence 0 J k,/j k, A/J k, A/J k, 0. Applying e ik and then the duality D yields a short exact sequence 0 D(e ik (A/J k,)) D(e ik (A/J k, )) D(e ik (J k,/j k, )) 0. Now the result follows from Lemma 3.. The following lemma is well known and easy to prove: Lemma 3.5. For any A-module X and any idempotent e in A the following hold: (i) There is an isomorphism of (eae) op -modules D(eX) = Hom A (X, D(eA)) defined by η f η := [x (ea η(eax))]. (ii) Assume that X is finite-dimensional. Then there is an isomorphism of eaemodules ex = DHom A (X, D(eA)) defined by ex [f f(x)(e)]. The vector space DHom A (X, D(eA)) is an End A (D(eA)) op -module in an obvious way, and we have eae = End A (D(eA)) op. Under the isomorphisms ex = DHom A (X, D(eA)) and eae = End A (D(eA)) op, the action of End A (D(eA)) op on DHom A (X, D(eA)) turns into the action eae ex := eaex of eae on ex. Lemma 3.6. For any A-module X we have Hom A (X, V k ) = Hom A (X/X k, V k) = Hom A/Jk, (X/X k, V k). Proof. We have soc (ik,...,i )(V k ) = V k, and rad (ik,...,i )(X) = X k. By Lemma 3.2(iii) this implies f(x k ) = 0 for every f Hom A(X, V k ). This yields the identification Hom A (X, V k ) = Hom A (X/X k, V k). Now X/X k and V k are annihilated by J k,. Thus X/X k and V k are A/J k, -modules. This implies Hom A (X/X k, V k) = Hom A/Jk, (X/X k, V k). Corollary 3.7. For any finite-dimensional A-module X we have DHom A (X, V k ) = e ik (X/X k ). Proof. The A-modules X/X k and V k can be regarded as an A/J k, -module, since both are annihilated by J k,, and V k is injective as an A/J k, -module. Now we apply Lemma Balanced modules. An A-module X is called i-balanced if X D i and X = X +. Thus, X is i-balanced if and only if X k = X+ k for all 0 k r. Proposition 3.8. Let X D i. Then the following are equivalent: (i) X is i-balanced. (ii) There is a unique b such that F i,b,x. (iii) There is a unique b such that χ (F i,b,x ) 0.

16 6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Proof. (i) = (ii): Since X D i, we know that soc i (X) = X and rad i (X) = 0. This implies that F i,wt(x ),X and F i,wt(x + ),X are both non-empty. Set b := wt(x+ ). Since X is i-balanced, we have X k = X+ k for all k. In other words, b = wt(x ) = wt(x + ). The uniqueness of b follows now from Lemma 3.8. (ii) = (iii): This follows directly from Lemma 3.9. (iii) = (i): Since X D i, Lemma 3.9 implies χ(f i,wt(x ),X ) = χ(f i,wt(x + ),X ) =. Since we assume b to be unique, we get wt(x ) = wt(x + ). Now (i) follows from Lemma 3.8. Lemma 3.9. Let X and Y be A-modules. Then the following hold: (i) If X and Y are i-balanced, then X Y is i-balanced. (ii) If X is i-balanced, then each direct summand of X is i-balanced. Proof. One easily checks that for every direct sum decomposition M = M M 2 of an A-module M and every sequence j = (j t,..., j ) with j s n for all s, we have soc j (M) = soc j (M ) soc j (M 2 ) and rad j (M) = rad j (M ) rad j (M 2 ). This implies both (i) and (ii). We say that the pair (A, i) is balanced, if for each k r the A-module V k = V i,k is (i k,..., i )-balanced. The following lemma follows directly from the definitions: Lemma Assume that (A, i) is balanced. For k r and 0 s < k we have rad (is,...,i )(V i,k ) = (V i,k ) i, s = (V i,k ) (i k,...,i ), s = (V i,k ) (i k,...,i ),+ s = soc (ik,...,i s+ )(V i,k ). Lemma 3.2. Assume that (A, i) is balanced. Then the modules V i,,..., V i,r are pairwise non-isomorphic. Proof. Assume V i,k = Vi,s with k > s. By definition V i,k = soc (ik,...,i s,...,i )(Îi k ) and V i,s = soc (is,...,i )(Îi s ). Clearly, rad (is,...,i )(V i,s ) = 0. Since V i,k = Vi,s we also get rad (is,...,i )(V i,k ) = 0. But V i,k is (i k,..., i )-balanced. By Lemma 3.20 this implies soc (ik,...,i s+ )(V i,k ) = rad (is,...,i )(V i,k ) = 0. But we have soc Sik (V i,k ) = S ik. This implies soc (ik,...,i s+ )(V i,k ) 0, a contradiction. Proposition Assume that (A, i) is balanced. For k, s r we have Hom A (V k, V s ) = e ik (J k,s+ /J k, ) e is. Proof. Recall that V k = D(e ik (A/J k, )) and V s = D(e is (A/J s, )). By Lemma 3.6 we have Hom A (V k, V s ) = Hom A (V k /(V k ) s, V s ). We have (V k ) s = J s, V k = D(e ik (A/J k,s+ )). For the second equality we used that V k is (i k,..., i )-balanced. Note that (V k ) s k s. We get = 0 if Hom A (V k /(V k ) s, V s ) = D(e is (V k /(V k ) s )) = D (e is (D(e ik (A/J k, ))/D(e ik (A/J k,s+ )))). For the first isomorphism we used Lemma 3.5. duality D to the short exact sequence and we obtain 0 J k,s+ /J k, A/J k, A/J k,s+ 0, Now we first apply e ik and then the D(e ik (A/J k, ))/D(e ik (A/J k,s+ )) = D(e ik (J k,s+ /J k, )).

17 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 7 Now D(e is D(e ik (J k,s+ /J k, ))) = D(D(e ik (J k,s+ /J k, )e is )) = e ik (J k,s+ /J k, )e is Hom A (V k, V s ) = e ik (J k,s+ /J k, )e is. implies Using Lemma 3.5, the isomorphism e ik J k,s+ /J k, e is Hom A (V k, V s ) can be described more precisely: Let e ik be is e ik (J k,s+ /J k, )e is. Then e ik be is is mapped to the homomorphism V k V s, which maps a linear form η : e ik (A/J k, ) C in D(e ik (A/J k, )) to the linear form ψ D(e is (A/J s, )) defined by ψ(e is a) := η(e ik be is a). For A-modules X and Y let I i (X, Y ) be the subspace of Hom A (X, Y ) consisting of the morphisms factoring through a module in add(i i ). Define Hom A (X, Y ) := Hom A (X, Y )/I i (X, Y ). Lemma Assume that (A, i) is balanced. Then for each X D i and k r we have I i (X, V k ) = Hom A (X/X + k, V k). Proof. There is a short exact sequence 0 X + k X X/X+ k 0. Applying the functor Hom A (, V k ) we can identify Hom A (X/X + k, V k) with a subspace of Hom A (X, V k ). Suppose that f : X V k is a homomorphism. Assume first that f = h g with g : X I and I add(i i ). It follows from Lemma 3.6 and Lemma 3.3 that we can assume without loss of generality that g is a monomorphism. By Lemma 3.2(i) we know that g(x + k ) I+ k. By definition I k = rad (ik,...,i )(I) and soc (ik,...,i )(V k ) = V k. Thus Lemma 3.2(iii) implies h(i k ) = 0. Since (A, i) is balanced, we get I k = I+ k. This shows that f(x+ k ) = 0. In other words, f Hom A(X/X + k, V k). So we proved that I i (X, V k ) Hom A (X/X + k, V k). To show the other inclusion, let f : X V k be a homomorphism with f(x + k ) = 0. Thus there is a factorization f = h g, where g : X X/X + k is the projection. Let u : X I be a monomorphism with I add(i i ), and let u 2 : I I/I + k be the projection. By Lemma 3.2(ii) we get a monomorphism g 2 : X/X + k I/I+ k such that u 2 u = g 2 g. Now X/X + k and I/I+ k are A/J k,-modules, V k is an injective A/J k, -module, and g 2 is a monomorphism. Thus there exists a homomorphism u 3 : I/I + k V k such that u 3 g 2 = h. The following commutative diagram illustrates the situation: It follows that I u 2 I/I + k u g 2 X g X/X + k f V k h f = h g = u 3 g 2 g = u 3 u 2 u. u 3 Thus we have proved that Hom A (X/X + k, V k) I i (X, V k ). Note that for the proof of this inclusion we did not use the assumption that (A, i) is balanced.

18 8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Proposition Assume (A, i) is balanced, and let X D i. For k r we have DHom A (X, V k ) = e ik (X + k /X k ). Proof. There is a short exact sequence η : 0 X + k /X k X/X k X/X+ k 0. As noted in Lemma 3.6 we have Hom A (X, V k ) = Hom A (X/X k, V k), and by Lemma 3.23 we know that I i (X, V k ) = Hom A (X/X + k, V k). Note that X/X + k and X+ k /X k are both annihilated by J k,. Thus they are A/J k, -modules, and V k = D(e ik (A/J k, )) is an injective A/J k, -module. Now we apply Hom A (, V k ) to η and obtain Hom A (X, V k ) = Hom A (X + k /X k, V k). By Lemma 3.5 we get Hom A (X + k /X k, V k) = Hom A (X + k /X k, D(e i k (A/J k, ))) = D(e ik (X + k /X k )). Thus we have proved that DHom A (X, V k ) = e ik (X + k /X k ) The quiver of E i. Again, let A = CΓ/J be a basic algebra, and let us fix some sequence i = (i r,..., i ). Define E i := End A (V i ) op. Since we work over an algebraically closed field, Lemma 3.2 and a result by Gabriel (see for example [DK, Theorem combined with Theorem 3.6.6]) imply that E i is a finite-dimensional basic algebra. We want to determine the quiver Γ Ei of E i. The vertices of Γ Ei correspond to the indecomposable direct summands V,..., V r of V i. Define a quiver Γ i as follows: The set of vertices of Γ i is just {, 2,..., r}. For each pair (k, s) with s, k r and k + s + k > s and each arrow a: i s i k in the quiver Γ of A, there is an arrow γa k,s : s k in Γ i. These are called the ordinary arrows of Γ i. Furthermore, for each k r there is an arrow γ k : k k provided k > 0. These are the horizontal arrows of Γ i. Proposition Assume that (A, i) is balanced. Then there is a quiver isomorphism Γ i Γ Ei with k V k for all k r. Proof. One can almost copy the proof of [BIRS, Theorem III.4.]. One only has to replace the ideals I j used in [BIRS] by our ideals J j. (We have I j = J j if and only if Γ has no loop at the vertex j.) Furthermore, everything has to be dualized. In Proposition 3.25 we identify the vertex of Γ Ei corresponding to V k with the vertex k of Γ i. Some examples can be found in Section The E i -module DHom A (X, V i ). Using Lemma 3.5 together with Propositions 3.22 and 3.24, we arrive at the following conclusion: Assume (A, i) is balanced, and let X D i. Using the identifications Hom A (V k, V s ) = e ik (J k,s+ /J k, )e is, Hom A (V s, V k ) = e is (A/J s, )e ik, DHom A (X, V k ) = e ik (X + k /X k ),

19 GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 9 the algebra E i acts on Y := DHom A (X, V i ) as follows: Assume s k r. e ik (J k,s+ /J k, )e is e ik (X + k /X k ) e is (X s + /Xs ) e is (A/J s, )e ik For e ik be is e ik (J k,s+ /J k, )e is and x s e is (X + s /X s ) = e is X + s /e is X s we have e ik be is x s = e ik bx s, and for e is be ik e is (A/J s, )e ik and x k e ik (X + k /X k ) = e i k X + k /e i k X k e is be ik x k = e is bx k. we have We consider Y as a representation Y = (Y (k), Y (γ)) k,γ of the quiver Γ i of E i. To describe Y, we just need to know how the maps Y (γ) act on the vector spaces Y (k) = e ik (X + k /X k ), where k r. Again using the description of Γ Ei based on [BIRS, Theorem III.4.] we obtain the following result: First, assume γ k : k k is a horizontal arrow of Γ i. Then Y (γ k ) acts as left multiplication with e ik : Next, let γa k,s with a: e ik (X + k /X k ) e ik e ik (X + k /X k ) : s k be an ordinary arrow of Γ i. Then Y (γa k,s ) acts as left multiplication e ik (X + k /X k ) a e i s (X s + /Xs ) Remark For X D i the following hold: (i) I i (X, V i ) is a submodule of the End A (V i )-module Hom A (X, V i ). This implies that DHom A (X, V i ) is a submodule of the E i -module DHom A (X, V i ). Clearly, DHom A (X, V i ) is also a module over the algebra B i := (End A (V i )) op. (ii) For X D i we have Hom A (X, V i ) = Hom Ai (X, V i ). Since add(i i ) are the injective A i -modules, we can apply the Auslander-Reiten formula to obtain an isomorphism of B i -modules DHom Ai (X, V i ) = Ext A i (τ A i (V i ), X), where τ Ai denotes the Auslander-Reiten translation of the finite-dimensional algebra A i An isomorphism between partial flag varieties and quiver Grassmannians. In this section we prove that the varieties F i,a,x of partial composition series of modules X D i are isomorphic to certain quiver Grassmannians G i,a,x. In the proof we first construct a (rather trivial) isomorphism between partial flag varieties F i,a,x of graded vector spaces and the image G i,a,x of the usual embedding of Fi,a,X into a product of classical subspace Grassmannians. Then we show that the restriction to the subvarieties F i,a,x F i,a,x and G i,a,x G i,a,x yields an isomorphism F i,a,x G i,a,x. Let X D i for some i = (i r,..., i ). We define a map d i,x : N r Z r by (a r,..., a ) (f,..., f r ), where f k := (a k a k) + (a k a k ) + + (a k min a kmin )

20 20 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER for all k r, and (a r,..., a ) := a (X). In the following theorem, if d i,x (a) / N r, then Gr E i d i,x (a)(y ) is by definition the empty set. Theorem Assume that (A, i) is balanced, and let X D i. Then for each a N r there exists an isomorphism of algebraic varieties F : F i,a,x Gr E i d i,x (a)(y ), where Y is the E i -module DHom A { (X, V i ). Furthermore, the map a d i,x (a) yields a bijection {a N r F i,a,x } f N r Gr E i f }. (Y ) Our proof of Theorem 3.27 will show that dim Ei (Y ) = d i,x (a + (X)). Furthermore, if F i,a,x, and X = (X r X X 0 ) F i,a,x, then f k = dim(e ik (X k /X k )) for all k r. Note that f k = 0 if k + = r Proof of Theorem Assume that (A, i) is balanced. For the rest of this section, besides i, we also fix some a = (a r,..., a ) N r and some X D i. With the same notation as in Theorem 3.27, we define G i,a,x := Gr E i f (Y ), where f := d i,x (a). We consider X as a representation X = (X(j), X(a)) j Γ0,a Γ of the quiver Γ of A, and the E i -module Y is considered as a representation Y = (Y (k), Y (γ)) k,γ of the quiver Γ i of E i. Given X = (0 = X r X X 0 = X) in F i,a,x we consider each X k as a subrepresentation of X. Thus we have X k = (X k (j)) j Γ0 such that for all arrows a of Γ. X(a)(X k (s(a))) X k (t(a)) Our aim is the construction of two mutually inverse isomorphisms of varieties F F i,a,x G i,a,x. G Recall that Γ 0 = {,..., n}. We need to work with the category of Γ 0 -graded vector spaces. Its objects are just tuples W = (W (j)) j Γ0 of C-vector spaces W (j). Set e j W := W (j) for all j Γ 0. The morphisms are defined in the obvious way. The degree of W is dim(w ) := (dim(w (j))) j Γ0. Let e,..., e n denote the canonical coordinate vectors of Z n. (Thus the jth entry of e j is, and all other entries are 0.) Each representation X = (X(j), X(a)) j Γ0,a Γ of Γ yields a Γ 0 -graded vector space gr(x) := (X(j)) j Γ0. Let F i,a,x be the projective variety of chains X = (0 = X r X X 0 = gr(x)) of Γ 0 -graded subspaces of gr(x) such that gr(x k ) X k gr(x + k ) and dim(x k /X k ) = a k e ik for all k r. For a vector space L let Gr d (L) be the projective variety of d-dimensional subspaces of L. Clearly, the variety of (f k + dim(e ik X k ))-dimensional subspaces U k of e ik X such that e ik X k U k e ik X + k is isomorphic to Gr f k (e ik (X + k /X k )). The isomorphism is given by U k U k := U k /e ik X k.