CALDERO-CHAPOTON ALGEBRAS

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1 CALDERO-CHAPOTON ALGEBRAS GIOVANNI CERULLI IRELLI, DANIEL LABARDINI-FRAGOSO, AND JAN SCHRÖER Abstract. Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a Caldero- Chapoton algebra A Λ to any (possibly infinite dimensional) basic algebra Λ. By definition, A Λ is (as a vector space) generated by the Caldero-Chapoton functions C Λ(M) of the decorated representations M of Λ. If Λ = P(Q, W ) is the Jacobian algebra defined by a -acyclic quiver Q with non-degenerate potential W, then we have A Q A Λ A up Q, where A Q and A up Q are the cluster algebra and the upper cluster algebra associated to Q. The set B Λ of generic Caldero-Chapoton functions is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra P(Q, W ) and was introduced by Geiss, Leclerc and Schröer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. We define B Λ for arbitrary Λ, and we conjecture that B Λ is a basis of the Caldero-Chapoton algebra A Λ. Thanks to the decomposition theorem, all elements of B Λ can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras. Contents. Introduction. Basic algebras and decorated representations 4. E-invariants and g-vectors of decorated representations 7 4. Caldero-Chapoton algebras 0 5. Strongly reduced components of representation varieties 6. Component graphs and CC-clusters 0 7. Caldero-Chapoton algebras and cluster algebras 8. Sign-coherence of generic g-vectors 4 9. Examples 5 References. Introduction.. Let A Q be the Fomin-Zelevinsky cluster algebra [FZ, FZ] associated to a finite -acyclic quiver Q. By definition A Q is generated by an inductively defined set of rational functions, called cluster variables. The cluster variables are contained in the set M Q of cluster monomials, which are by definition certain monomials in the cluster variables. Now let W be a non-degenerate potential for Q, and let Λ = P(Q, W ) be the associated Jacobian algebra introduced by Derksen, Weyman and Zelevinsky [DWZ, DWZ]. The Date: Mathematics Subject Classification. Primary F60; Secondary 6G0, 6G0.

2 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER category of decorated representations of Λ is denoted by decrep(λ). To any M decrep(λ) one can associate a Laurent polynomial C Λ (M), the Caldero-Chapoton function of M. It follows from [DWZ, DWZ] that the cluster monomials form a subset of the set C Λ of Caldero-Chapoton functions... The generic basis conjecture. One of the main problems in cluster algebra theory is to find a basis of A Q with favourable properties. As an important requirement, this basis should contain the set M Q of cluster monomials in a natural way. The concept of strongly reduced irreducible components of varieties of decorated representations of a Jacobian algebra Λ was introduced in [GLS]. To each strongly reduced component Z one can associate a generic Caldero-Chapoton function C Λ (Z). It was conjectured in [GLS] that the set B Λ of generic Caldero-Chapoton functions forms a C-basis of A Q. Using a non-degenerate potential defined by Labardini [L], Plamondon [P] found a counterexample and then conjectured that B Λ is a basis of the upper cluster algebra A up Q. This conjecture should also be wrong in general. We replace it by yet another conjecture. We study the Caldero-Chapoton algebra A Λ := C Λ (M) M decrep(λ) alg generated by all Caldero-Chapoton functions. We do not restrict ourselves to Jacobian algebras, but work with algebras Λ defined as arbitrary quotients of completed path algebras. In particular, we generalize the notation of a Caldero-Chapoton function to this general setup. One easily checks that the functions C Λ (M) do not only generate A Λ as an algebra but also as a vector space over the ground field C. Conjecture.. B Λ is a C-basis of A Λ. We show that the set B Λ of generic Caldero-Chapoton functions is linearly independent provided the kernel of the skew-symmetric incidence matrix B Q of Q does not contain any non-zero element in Q n 0. This generalizes [P, Proposition.9]. For Λ = P(Q, W ) a Jacobian algebra associated to a quiver Q with non-degenerate potential W we have A Q A Λ A up Q where A Q is the cluster algebra and A up Q is the upper cluster algebra associated to Q. (We refer to [BFZ, DWZ, FZ] for missing definitions.) For this special case, we have a list of conjectures, which hopefully will lead to a better understanding of the rather mysterious relation between A Q and A up Q... Parametrization of strongly reduced components. Plamondon [P, Theorem.] parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Let Λ = C Q /I be a basic algebra, where the quiver Q has n vertices. Let decirr(λ) be the set of irreducible components of all varieties decrep d,v (Λ) of decorated representations of Λ, where (d, v) runs through N n N n. By decirr s.r. (Λ) we denote the subset of strongly reduced components. (The definition is in Section 5.) Recall that decirr s.r. (Λ) parametrizes the elements in B Λ. Let G s.r. Λ : decirr s.r. (Λ) Z n be the map sending Z decirr s.r. (Λ) to the generic g-vector g Λ (Z) of Z. (The definition of a g-vector is in Section.) Using Plamondon s result for finite-dimensional algebras,

3 CALDERO-CHAPOTON ALGEBRAS and a long-path truncation argument, we get the following parametrization of strongly reduced components for arbitrary Λ. Theorem.. For a basic algebra Λ the following hold: (i) The map G s.r. Λ : decirr s.r. (Λ) Z n is injective. (ii) The following are equivalent: (a) G s.r. Λ is surjective. (b) Λ is finite-dimensional..4. A decomposition theorem for strongly reduced components. The notion of a direct sum of irreducible components of representation varieties was introduced in [CBS]. The Zariski closure Z := Z Z t of a direct sum of irreducible components Z,..., Z t of varieties of representations of Λ is always irreducible, but in general Z is not an irreducible component. It was shown in [CBS] that Z is an irreducible component provided the dimension of the first extension group between the components is generically zero. The following decomposition theorem is an analogue for strongly reduced components. Instead of extension groups, we work with a generalization E Λ (,?) of the Derksen-Weyman-Zelevinsky E-invariant [DWZ]. (We define E Λ (,?) in Section.) Theorem.. For Z,..., Z t decirr(λ) the following are equivalent: (i) Z Z t is a strongly reduced irreducible component. (ii) Each Z i is strongly reduced and E Λ (Z i, Z j ) = 0 for all i j. Based on Theorem., we show that all elements of B Λ can be seen as CC-cluster monomials. (The CC-cluster monomials generalize Fomin and Zelevinsky s notion of cluster monomials.).5. Sign-coherence of g-vectors. A subset U of Z n is called sign-coherent if for each i n we have either a i 0 for all (a,..., a n ) U, or we have a i 0 for all (a,..., a n ) U. The following theorem generalizes [P, Theorem.7()]. Theorem.4. Let Λ be a basic algebra, and let Z,..., Z t decirr s.r. (Λ) be strongly reduced components. Assume that Z Z t is a strongly reduced component. Then {g Λ (Z ),..., g Λ (Z t )} is sign-coherent..6. The paper is organized as follows. In Section we recall definitions and basic properties of basic algebras and their (decorated) representations. We also introduce truncations of basic algebras, which play a crucial role in some of our proofs. In Section we introduce and study g-vectors and E-invariants of decorated representations. Caldero- Chapoton functions and Caldero-Chapoton algebras are defined in Section 4. Our main results Theorem. and. are proved in Section 5. In Section 6 we introduce component graphs, component clusters and CC-clusters, and we show that the cardinality of loopcomplete subgraphs of a component graph is bounded by the number of simple modules. We also present several general conjectures on the structure of component graphs and on a generalization of the Fomin-Zelevinsky Laurent phenomenon. Section 7 explains the relation between Caldero-Chapoton algebras and cluster algebras, and it contains several

4 4 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER conjectures on the relation between cluster algebras and upper cluster algebras. Section 8 contains the proof of Theorem.4. Finally, in Section 9 we discuss several examples of Caldero-Chapoton algebras..7. Notation. We denote the composition of maps f : M N and g : N L by gf = g f : M L. We write U for the cardinality of a set U. A finite-dimensional module M is basic provided it is a direct sum of pairwise nonisomorphic indecomposable modules. For a module M and some m let M m be the direct sum of m copies of M. For a finite-dimensional basic algebra Λ let τ Λ be its Auslander-Reiten translation. For an introduction to Auslander-Reiten theory we refer to the books [ARS] and [ASS]. For n and a set S, depending on the situation, we identify S n with the set of (n )- or ( n)-matrices with entries in S. By N we denote the natural numbers, including zero. For d = (d,..., d n ) N n let d := d + + d n. For n N let M n (Z) be the set of (n n)-matrices with integer entries. For a ring R let R[x ±,..., x± n ] be the algebra of Laurent polynomials over R in n independent variables x,..., x n. For a = (a,..., a n ) Z n set x a := x a xan n.. Basic algebras and decorated representations.. Basic algebras and quiver representations. Throughout, let C be the field of complex numbers. A quiver is a quadruple Q = (Q 0, Q, s, t), where Q 0 is a finite set of vertices, Q is a finite set of arrows, and s, t: Q Q 0 are maps. For each arrow a Q we call s(a) and t(a) the starting and terminal vertex of a, respectively. If not mentioned otherwise, we always assume that Q 0 = {,..., n}. Let B Q = (b ij ) M n (Z), where b ij := {a Q s(a) = j, t(a) = i} {a Q s(a) = i, t(a) = j}. A path in Q is a tuple p = (a m,..., a ) of arrows a i Q such that s(a i+ ) = t(a i ) for all i m. Then length(p) := m is the length of p. Additionally, for each vertex i Q 0 there is a path e i of length 0. We often just write a m a instead of (a m,..., a ). A path p = (a m,..., a ) of length m is a cycle in Q, or more precisely an m-cycle in Q, if s(a ) = t(a m ). The quiver Q is acyclic if there are no cycles in Q, and for s the quiver Q is called s-acyclic if there are no m-cycles for m s. A representation of a quiver Q = (Q 0, Q, s, t) is a tuple M = (M i, M a ) i Q0,a Q, where each M i is a finite-dimensional C-vector space, and M a : M s(a) M t(a) is a C-linear map for each arrow a Q. We call dim(m) := (dim(m ),..., dim(m n )) the dimension vector of M. Let dim(m) := dim(m ) + + dim(m n ) be the dimension of M. For a path p = (a m,..., a ) in Q let M p := M am M a. The representation M is called nilpotent provided there exists some N > 0 such that M p = 0 for all paths p in Q with length(p) > N. For i Q 0 let S i := (M i, M a ) i,a be the representation of Q with M i = C, M j = 0 for all j i, and M a = 0 for all a Q. For a nilpotent representation M the ith entry dim(m i ) of its dimension vector dim(m) equals the Jordan-Hölder multiplicity [M : S i ] of S i in M. For m N let CQ[m] be a C-vector space with a C-basis labeled by the paths of length m in Q. Note that CQ[m] is finite-dimensional. We do not distinguish between a path p of length m and the corresponding basis vector in CQ[m].

5 CALDERO-CHAPOTON ALGEBRAS 5 The completed path algebra of a quiver Q is denoted by C Q. As a C-vector space we have C Q = m 0 CQ[m]. We write the elements in C Q as infinite sums m 0 a m with a m CQ[m]. The product in C Q is then defined as ( a i )( b j ) := a i b j. i 0 j 0 k 0 i+j=k A potential of Q is an element W = m w m of C Q, where each w m is a C-linear combination of m-cycles in Q. By definition, W = 0 is also a potential. The definition of a non-degenerate potential can be found in [DWZ, Section 7]. The category mod(c Q ) of finite-dimensional left C Q -modules can be identified with the category nil(q) of nilpotent representations of Q. By m we denote the arrow ideal in C Q, which is generated by the arrows of Q. An ideal I of C Q is admissible if I m. We call an algebra Λ basic if Λ = C Q /I for some quiver Q and some admissible ideal I of C Q. A representation of a basic algebra Λ = C Q /I is a nilpotent representation of Q, which is annihilated by the ideal I. We identify the category rep(λ) of representations of Λ with the category mod(λ) of finite-dimensional left Λ-modules. Up to isomorphism the simple representations of Λ are the -dimensional representations S,..., S n. The category of (possibly infinite dimensional) Λ-modules is denoted by Mod(Λ), we consider rep(λ) as a subcategory of Mod(Λ)... Decorated representations of quivers. Let Λ = C Q /I be a basic algebra. A decorated representation of Λ is a pair M = (M, V ), where M is a representation of Λ and V = (V,..., V n ) is a tuple of finite-dimensional C-vector spaces. Let dim(v ) := (dim(v ),..., dim(v n )) and dim(v ) := dim(v ) + + dim(v n ). We call dim(m) := (dim(m), dim(v )) the dimension vector of M. One defines morphisms and direct sums of decorated representations in the obvious way. Let decrep(λ) be the category of decorated representations of Λ. Let M = (M, V ) decrep(λ). We write M = 0 if all M i are zero, and V = 0 if all V i are zero. Furthermore, M = 0 if M = 0 and V = 0. For i n set S i := (S i, 0), and let Si := (0, V ), where V i = C and V j = 0 for all j i. The representations Si are the negative simple decorated representations of Λ... Varieties of representations. For d = (d,..., d n ) N n let rep d (Λ) be the affine variety of representations of Λ with dimension vector d. By definition the closed points of rep d (Λ) are the representations M = (M i, M a ) i Q0,a Q of Λ with M i = C d i for all i Q 0. One can regard rep d (Λ) as a Zariski closed subset of the affine space rep d (Q) := a Q Hom C (C ds(a), C dt(a) ). For d = (d,..., d n ) let G d := n i= GL(Cd i ). The group G d acts on rep d (Λ) by conjugation. More precisely, for g = (g,..., g n ) G d and M rep d (Λ) let g.m := (M i, g t(a) M ag s(a) ) i Q0,a Q.

6 6 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER For M rep d (Λ) let O(M) be the G d -orbit of M. The G d -orbits are in bijection with the isomorphism classes of representations of Λ with dimension vector d. For (d, v) N n N n let decrep d,v (Λ) be the affine variety of decorated representations M = (M, V ) with M rep d (Λ) and V = C v := (C v,..., C vn ), where v = (v,..., v n ). For M = (M, V ) decrep d,v (Λ) define g.m := (g.m, V ). This defines a G d -action on decrep d,v (Λ). The G d -orbit of M is denoted by O(M). We have () dim O(M) = dim O(M) = dim G d dim End Λ (M), see for example [G]..4. Quiver Grassmannians. Let Λ = C Q /I be a basic algebra. For a representation M = (M i, M a ) i Q0,a Q of Λ and e N n let Gr e (M) be the quiver Grassmannian of subrepresentations U of M with dim(u) = e. (By definition a subrepresentation of M is a tuple U = (U i ) i Q0 of subspaces U i M i such that M a (U s(a) ) U t(a) for all a Q.) So Gr e (M) is a projective variety, which can be seen as a closed subvariety of the product of the classical Grassmannians Gr ei (M i ) of e i -dimensional subspaces of M i, where e = (e,..., e n ). Let χ(gr e (M)) be the Euler-Poincaré characteristic of Gr e (M)..5. Truncations of basic algebras. For a basic algebra Λ = C Q /I and some p let Λ p := Λ/J p, where J p is the ideal of Λ generated by all (residue classes) of paths of length p in Q. We call Λ p the p-truncation of Λ. We get canonical surjective algebra homomorphisms C Q π Λ πp Λ p with Ker(π) = I and I p := Ker(π p π) = I + m p, where m p is the pth power of the arrow ideal m. Thus we can write Λ p = C Q /I p. As a vector space, Λ p is isomorphic to where V p /(V p (I + m p )) V p := 0 m p CQ[m]. Clearly, Λ p is a finite-dimensional basic algebra, and the canonical epimorphism π p : Λ Λ p induces embeddings rep(λ p ) rep(λ) and decrep(λ p ) decrep(λ) in the obvious way. Lemma.. Let Λ = C Q /I be a basic algebra. Then the following hold: (i) Let M = (M, V ) decrep(λ). If p dim(m), then M is in the image of the embedding decrep(λ p ) decrep(λ). (ii) Let M, N rep(λ). If p dim(m), dim(n), then dim Hom Λp (M, N) = dim Hom Λ (M, N). (iii) Let M, N rep(λ). If p dim(m) + dim(n), then dim Ext Λ p (M, N) = dim Ext Λ(M, N). (iv) Let (d, v) N n N n. If p d, then decrep d,v (Λ p ) = decrep d,v (Λ). Proof. Let a m a be a path of length m in Q, and let M be a representation of Λ. For any non-zero vector v 0 M set v i := a i a v 0 for i m. Assume that each of the vectors v,..., v m is non-zero. We claim that v 0, v,..., v m are pairwise different and linearly independent. Let b be a path of maximal length such that bv 0 0. Such a path b

7 CALDERO-CHAPOTON ALGEBRAS 7 exists, because M is nilpotent. By induction v,..., v m are linearly independent. Assume now that m v 0 = λ i v i for some λ i C. We have v i = a i a v 0. Therefore we get bv 0 = i= m λ i ba i a v 0. i= Since ba i a is either zero or a path of length length(b) + i, we have ba i a v 0 = 0 for all i m. Since bv 0 0, this is a contradiction. Therefore v 0, v,..., v m are linearly independent. It follows that for any M decrep(λ) with dim(m) = (d, v) and any path b with length(b) d we have bm = 0. This implies (i). Parts (ii) and (iv) are easy consequences of (i). Any extension of representations M and N of Λ is a representation of Λ of dimension dim(m) + dim(n). This implies (iii).. E-invariants and g-vectors of decorated representations.. Definition of E-invariants and g-vectors. Let Q be a quiver, and let W be a potential of Q. Let Λ = P(Q, W ) be the associated Jacobian algebra [DWZ, Section ]. For decorated representations M and N of Λ the g-vector g(m) and the invariants E inj (M) and E inj (M, N ) were defined in [DWZ], where E inj (M) is called the E-invariant of M. We define invariants g Λ (M), E Λ (M) and E Λ (M, N ) of decorated representations M and N of an arbitrary basic algebra Λ = C Q /I as follows. with For a decorated representation M = (M, V ) of Λ let be the g-vector of M. g Λ (M) := (g,..., g n ) g i := g i (M) := dim Hom Λ (S i, M) + dim Ext Λ(S i, M) + dim(v i ) For decorated representations M = (M, V ) and N = (N, W ) of Λ let E Λ (M, N ) := dim Hom Λ (M, N) + The E-invariant of M is defined as E Λ (M) := E Λ (M, M). n g i (N ) dim(m i ). Lemma.. Let Λ = P(Q, W ), where W is a potential of Q. For M, N decrep(λ) the following hold: (i) g Λ (M) = g(m). (ii) E Λ (M, N ) = E inj (M, N ). i= Proof. Part (i) follows from [P, Lemma 4.7, Proposition 4.8]. It can also be shown in a more elementary way by using the exact sequence displayed in [DWZ, Equation (0.4)]. Part (ii) is a direct consequence of (i) and the definition of E Λ (M, N ) and E inj (M, N ).

8 8 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER.. Homological interpretation of the E-invariant. For i n let I i Mod(Λ) be the injective envelope of the simple representation S i of Λ. One easily checks that the socle soc(i i ) of I i is isomorphic to S i, and that () dim Hom Λ (M, I i ) = dim(m i ) for all M rep(λ). Note that in general I i is infinite dimensional. For M rep(λ) let 0 M f I Λ 0 (M) I Λ (M) denote a minimal injective presentation of M. The modules I0 Λ(M) and IΛ (M) are up to isomorphism uniquely determined by M. We will need the following theorem due to Auslander and Reiten. Theorem. ([AR, Theorem.4 (b)]). Let M and N be representations of a finitedimensional basic algebra Λ. Then we have dim Hom Λ (τ Λ (N), M) = dim Hom Λ(M, N) dim Hom Λ (M, I Λ 0 (N)) + dim Hom Λ (M, I Λ (N)). Lemma.. Let Λ = C Q /I be a finite-dimensional basic algebra, and let M rep(λ). Let 0 M f I Λ 0 (M) I Λ (M) be a minimal injective presentation of M. Then for i n we have (i) [soc(i Λ 0 (M)) : S i] = [soc(m) : S i ] = dim Hom Λ (S i, M) and I Λ 0 (M) = I dim Hom Λ(S,M) I dim Hom Λ(S n,m) n. (ii) [soc(i Λ (M)) : S i] = [soc(coker(f)) : S i ] = dim Ext Λ (S i, M) and I Λ (M) = I dim Ext Λ (S,M) I dim Ext Λ (Sn,M) n. Proof. Since I0 Λ(M) is the injective envelope of M, we have soc(m) = soc(i0 Λ (M)). This implies (i). By the construction of injective presentations, I Λ (M) is the injective envelope of Coker(f). It follows that soc(coker(f)) = soc(i Λ (M)). We apply the functor Hom Λ (S i, ) to the exact sequence This yields an exact sequence 0 M f I Λ 0 (M) Coker(f) 0. 0 Hom Λ (S i, M) F Hom Λ (S i, I Λ 0 (M)) Hom Λ (S i, Coker(f)) G Ext Λ(S i, M) 0 Here we used that I0 Λ(M) is injective, which implies Ext Λ (S i, I0 Λ (M)) = 0. By (i) we know that F is an isomorphism. Thus G is also an isomorphism. This implies (ii). Combinining Lemma. and Lemma. yields the following result. Lemma.4. Let M = (M, V ) be a decorated representation of a basic algebra Λ, and let g Λ (M) = (g,..., g n ) be the g-vector of M. If p > dim(m), then for all i n. g i = [I Λp 0 (M) : S i] + [I Λp (M) : S i] + dim(v i )

9 CALDERO-CHAPOTON ALGEBRAS 9 The following result is a homological interpretation of the E-invariant in terms of Auslander-Reiten translations. This can be seen as a generalization of [DWZ, Corollary 0.9]. Proposition.5. Let M = (M, V ) and N = (N, W ) be decorated representations of a basic algebra Λ. If p > dim(m), dim(n), then n E Λ (M, N ) = E Λp (M, N ) = dim Hom Λp (τ Λ p (N), M) + dim(w i ) dim(m i ). In particular, we have and for all p, q > dim(m), dim(n). E Λp (M, N ) = E Λq (M, N ) i= dim Hom Λp (τ Λ p (N), M) = dim Hom Λq (τ Λ q (N), M) Proof. Since p > dim(m), dim(n) we can apply Lemma. and get Let dim Hom Λp (M, N) = dim Hom Λ (M, N), dim Hom Λp (S i, N) = dim Hom Λ (S i, N), dim Ext Λ p (S i, N) = dim Ext Λ(S i, N). 0 N I Λp 0 (N) IΛp (N) be a minimal injective presentation of N, where we regard N now as a representation of Λ p. It follows from Lemma. and Equation () that This implies n dim Hom Λp (M, I Λp 0 (N)) = dim Hom Λp (S i, N) dim(m i ), i= n dim Hom Λp (M, I Λp (N)) = dim Ext Λ p (S i, N) dim(m i ). E Λ (M, N ) = dim Hom Λ (M, N) + + i= n ( dim Hom Λ (S i, N) + dim Ext Λ(S i, N)) dim(m i ) i= n dim(w i ) dim(m i ) i= = dim Hom Λp (M, N) + + n ( dim Hom Λp (S i, N) + dim Ext Λ p (S i, N)) dim(m i ) i= n dim(w i ) dim(m i ) i= = dim Hom Λp (M, N) dim Hom Λp (M, I Λp 0 (N)) + dim Hom Λ p (M, I Λp (N)) n + dim(w i ) dim(m i ). i=

10 0 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER The first equality follows from Lemmas.,. and.4. The second equality says that E Λ (M, N ) = E Λp (M, N ). Applying Theorem. yields n E Λp (M, N ) = dim Hom Λp (τ Λ p (N), M) + dim(w i ) dim(m i ). This finishes the proof. Corollary.6. For decorated representations M and N of a basic algebra Λ we have E Λ (M, N ) 0. i= 4. Caldero-Chapoton algebras 4.. Caldero-Chapoton functions. To any basic algebra Λ = C Q /I we associate a set of Laurent polynomials in n independent variables x,..., x n as follows. The Caldero- Chapoton function associated to a decorated representation M = (M, V ) of Λ is defined as C Λ (M) := x g Λ(M) e N n χ(gr e (M))x BQe. Note that C Λ (M) Z[x ±,..., x± n ]. Let C Λ := {C Λ (M) M decrep(λ)} be the set of Caldero-Chapoton functions associated to Λ. For M = (M, 0) we sometimes write C Λ (M) instead of C Λ (M). The definition of C Λ (M) is motivated by the (different versions of) Caldero-Chapoton functions appearing in the theory of cluster algebras, see for example [Pa, Section ]. Such functions first appeared in work of Caldero and Chapoton [CC, Section ], where they show that the cluster variables of a cluster algebra of a Dynkin quiver are Caldero-Chapoton functions. Lemma 4.. For decorated representations M = (M, V ) and N = (N, W ) the following hold: (i) g Λ (M N ) = g Λ (M) + g Λ (N ). (ii) C Λ (M) = C Λ (M, 0)C Λ (0, V ). (iii) C Λ (M N ) = C Λ (M)C Λ (N ). Proof. Part (i) follows directly from the definitions and from the additivity of the functors Hom Λ (,?) and Ext Λ (,?). To prove (ii), let M = (M, V ) be a decorated representation of Λ. For the decorated representation (0, V ) we have n C Λ (0, V ) = where dim(v ) = (v,..., v n ). For the decorated representation (M, 0) we have C Λ (M, 0) := x g Λ(M,0) e N n χ(gr e (M))x BQe where g i (M, 0) = dim Hom Λ (S i, M) + dim Ext Λ (S i, M) for i n. Now one easily checks that C Λ (M) = C Λ (M, 0)C Λ (0, V ). Thus (ii) holds. Now (iii) follows from (i), (ii) and the well known formula χ(gr e (M N)) = χ(gr e (M))χ(Gr e (N)) (e,e ) i= x v i i

11 CALDERO-CHAPOTON ALGEBRAS where the sum runs over all pairs (e, e ) N n N n such that e + e = e. 4.. Definition of a Caldero-Chapoton algebra. In the previous section, we associated to a basic algebra Λ the set C Λ = {C Λ (M) M decrep(λ)} of Caldero-Chapoton functions. Clearly, C Λ is a subset of the integer Laurent polynomial ring Z[x ±,..., x± n ] generated by the variables x,..., x n. By definition the Caldero- Chapoton algebra A Λ associated to Λ is the C-subalgebra of C[x ±,..., x± n ] generated by C Λ. The following is a direct consequence of Lemma 4.(iii). Lemma 4.. The set C Λ generates A Λ as a C-vector space. 4.. Linear independence of Caldero-Chapoton functions. Let Λ = C Q /I be a basic algebra. Except in some trivial cases, the set C Λ of Caldero-Chapoton functions associated to decorated representations of Λ is linearly dependent. Often the Caldero- Chapoton functions satisfy beautiful relations, which should be studies more intensively. On the other hand, by Lemma 4., there are C-bases of A Λ consisting only of Caldero- Chapoton functions. Our aim is to provide a candidate B Λ for such a basis. Before constructing B Λ in Section 5, we prove the following criterion for linear independence of certain sets of Caldero-Chapoton functions. Let Q n 0 := {(a,..., a n ) Q n a i 0 for all i}, Q n >0 := {(a,..., a n ) Q n a i > 0 for all i}. Proposition 4.. Let Λ = C Q /I be a basic algebra. representations of Λ. Assume the following: (i) Ker(B Q ) Q n 0 = 0. (ii) The g-vectors g Λ (M j ), j J are pairwise different. Let M j, j J be decorated Then the Caldero-Chapoton functions C Λ (M j ), j J are pairwise different and linearly independent in A Λ. Proof. We treat B Q as a linear map Q n Q n. For a, b Z n define a b if there exists some e Q n 0 such that a = b + B Q e. We claim that this defines a partial order on Z n. Clearly, a a, so is reflexive. Furthermore, assume that a b and b a. Thus a = b + B Q e and b = a + B Q e for some e, e Q n 0. It follows that a = a + B Q(e + e ). Thus e + e Ker(B Q ). Our assumption (i) yields e = e = 0. Thus a = b. This shows that is antisymmetric. Finally, assume that a b c. Thus we have a = b + B Q e and b = c + B Q e for some e, e Q n 0. It follows that a = c + B Q(e + e ). In other words, we have a c. Thus is transitive. The partial order on Z n induces obviously a partial order on the set of Laurent monomials in the variables x,..., x n. Namely, set x a x b if a b. Let deg(x a ) := a be the degree of x a. Among the Laurent monomials x g Λ(M)+B Q e occuring in the expression C Λ (M) = x g Λ(M) χ(gr e (M))x BQe = χ(gr e (M))x gλ(m)+bqe e N n e N n

12 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER the monomial x g Λ(M) is the unique monomial of maximal degree. For e = 0 the Grassmannian Gr e (M) is just a point, and B Q e = 0. Thus, if e = 0, we have χ(gr e (M))x B Qe =. This shows that the Laurent monomial x g Λ(M) really occurs as a non-trivial summand of C Λ (M). In particular, we have C Λ (M) C Λ (N ) if g Λ (M) g Λ (N ). Now let M,..., M t be decorated representations of Λ with pairwise different g-vectors. Assume that λ C Λ (M ) + + λ t C Λ (M t ) = 0 for some λ j C. Without loss of generality we assume that λ j 0 for all j. There is a (not necessarily unique) index s such that x g Λ(M s) is maximal in the set {x g Λ(M j ) j t}. It follow that the Laurent monomial x g Λ(M s) does not occur as a summand of any of the Laurent polynomials C Λ (M j ) with j s. (Here we use that the g-vectors of the decorated representations M j are pairwise different.) This implies λ s = 0, a contradiction. Thus C Λ (M ),..., C Λ (M t ) are linearly independent. For the example, where Λ is the path algebra of an affine quiver of type A, the main argument used in the proof of Proposition 4. can already be found in [C, Section 6.]. Note that condition (d) in the following lemma coincides with condition (i) in Proposition 4.. Lemma 4.4. For the conditions (a) rank(b Q ) = n. (b) Each row of B Q has at least one non-zero entry, and there are n rank(b Q ) rows of B Q, which are non-negative linear combinations of the remaining rank(b Q ) rows of B Q. (c) Im(B Q ) Q n >0. (d) Ker(B Q ) Q n 0 = 0. the implications hold. (a) = (b) = (c) = (d) Proof. The implication (a) = (b) is trivial. Next, assume (b) holds. Let m := rank(b Q ). We denote the jth row of B Q by r j. By assumption there are pairwise different indices i,..., i m {,..., n} such that for each k n with k / {i,..., i m } we have for some non-negative rational numbers λ (k) j r k = λ (k) r i + + λ (k) m r im. Since r k is non-zero, at least one of the λ (k) j is posititve. Clearly, there is some e Q n such that r ij e = for all j m. (The (k n)-matrix with rows r i,..., r im has rank m. Thus, we can see it as a surjective homomorphism Q n Q m.) Now observe that the kth entry of B Q e is λ (k) + + λ (k) m for all k n with k / {i,..., i m } and that this entry is positive. It follows that Im(B Q ) Q n >0. Finally, to show (c) = (d) let b Im(B Q ) Q n >0. Thus there is some a Qn such that B Q a = b. Since B Q is skew-symmetric, we get ab Q = b. Now let e Ker(B Q ) Q n 0. We get B Q e = 0, and therefore ab Q e = b e = 0. Since b has only positive entries and e has only non-negative entries, we get e = 0. This finishes the proof.

13 CALDERO-CHAPOTON ALGEBRAS If we replace condition (i) by condition (a), Proposition 4. was first proved by Fu and Keller [FK, Corollary 4.4]. Essentially the same argument was later also used by Plamondon [P]. That the Fu-Keller argument can be applied under condition (b) was observed by Geiß and Labardini. To any triangulation T of a punctured Riemann surface with non-empty boundary, one can associate a -acyclic quiver Q T. It is shown in [GL] that there is always a triangulation T such that the matrix B QT satisfies condition (b). 5. Strongly reduced components of representation varieties 5.. Decomposition theorems for irreducible components. Let Λ = C Q /I be a basic algebra, and let (d, v) N n N n. By Irr d (Λ) and decirr d,v (Λ) we denote the set of irreducible components of rep d (Λ) and decrep d,v (Λ), respectively. For Z decirr d,v (Λ) we write dim(z) := (d, v). Let Irr(Λ) = Irr d (Λ) and decirr(λ) = decirr d,v (Λ). d N n (d,v) N n N n Note that any irreducible component Z decirr(λ) can be seen as an irreducible component in Irr(Λ dec ), where Λ dec := Λ C C is defined as the product of Λ with n copies of C. In fact, we can identify decrep(λ) and rep(λ dec ). Thus statements on varieties of representations can be carried over to varieties of decorated representations. By definition we have We have an isomorphism decrep d,v (Λ) = {(M, C v ) M rep d (Λ)}. decrep d,v (Λ) rep d (Λ) of affine varieties mapping (M, C v ) to M. Thus the irreducible components of decrep d,v (Λ) can be interpreted as irreducible components of rep d (Λ). For Z decirr d,v (Λ) let πz be the corresponding component in Irr d (Λ). Recall that the group G d acts on decrep d,v (Λ) by g.(m, C v ) := (g.m, C v ), and that the G d -orbit of a decorated representation M is denoted by O(M). For Z, Z, Z decirr(λ) define c Λ (Z) := min{dim(z) dim O(M) M Z}, e Λ (Z) := min{dim Ext Λ(M, M) M = (M, V ) Z}, E Λ (Z) := min{e Λ (M) M Z}, end Λ (Z) := min{dim End Λ (M) M = (M, V ) Z}, hom Λ (Z, Z ) := min{dim Hom Λ (M, M ) M i = (M i, V i ) Z i, i =, }, ext Λ(Z, Z ) := min{dim Ext Λ(M, M ) M i = (M i, V i ) Z i, i =, }, E Λ (Z, Z ) := min{e Λ (M, M ) M i Z i, i =, }. It is easy to construct examples of components Z decirr(λ) such that end Λ (Z) hom Λ (Z, Z), e Λ (Z) ext Λ (Z, Z) and E Λ(Z) E Λ (Z, Z). Note that for Z decirr d,v (Λ) we have c Λ (Z) = dim(z) dim(g d ) + end Λ (Z). This follows from Equation ().

14 4 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER By [CBS, Lemma 4.] the functions dim Hom Λ (,?) and dim Ext Λ (,?) are upper semicontinuous. Using this one easily shows the following lemma. Lemma 5.. For Z, Z, Z decirr(λ) the following hold: (i) The sets are open in Z. (ii) The sets {M Z dim(z) dim O(M) = c Λ (Z)}, {M = (M, V ) Z dim Ext Λ(M, M) = e Λ (Z)}, {M Z E Λ (M) = E Λ (Z)}, {M = (M, V ) Z dim End Λ (M) = end Λ (Z)} {((M, V ), (M, V )) Z Z dim Hom Λ (M, M ) = hom Λ (Z, Z )}, {((M, V ), (M, V )) Z Z dim Ext Λ(M, M ) = ext Λ(Z, Z )}, {(M, M ) Z Z E Λ (M, M ) = E Λ (Z, Z )} are open in Z Z. For Z decirr(λ) there is a dense open subset U of Z such that g Λ (M) = g Λ (N ) for all M, N U. This follows again by upper semicontinuity. For M U let be the generic g-vector of Z. g Λ (Z) := g Λ (M) Lemma 5.. For Z, Z, Z decirr(λ) we have c Λ (Z) e Λ (Z) E Λ (Z) and ext Λ(Z, Z ) E Λ (Z, Z ). Proof. Let d = dim(πz) and d i = dim(πz i ). Choose some p d, d + d. By Lemma. we can regard all the representations in Z, Z and Z as representations of Λ p. Thus we can interpret Z, Z and Z as irreducible components in decirr(λ p ). Now Proposition.5 allows us to assume without loss of generality that Λ = Λ p. Voigt s Lemma [G, Proposition.] implies that c Λ (Z) e Λ (Z). The Auslander-Reiten formula Ext Λ (M, N) = DHom Λ (τ Λ (N), M) yields dim Ext Λ(M, N) dim Hom Λ (τ Λ (N), M). This implies e Λ (Z) E Λ (Z) and ext Λ (Z, Z ) E Λ (Z, Z ). (Here we used again Proposition.5.) Following [GLS] we call an irreducible component Z decirr(λ) strongly reduced provided c Λ (Z) = e Λ (Z) = E Λ (Z). For example, if Λ is finite-dimensional, one can easily check that for any injective Λ-module I rep(λ) the closure of the orbit O(I, 0) is a strongly reduced irreducible component. Similarly, it follows directly from the definitions that for all decorated representations of the form M = (0, V ), the closure of O(M) is a strongly reduced component. (In this case, O(M) is just a point, and it is equal to its closure.) Let decirr s.r. d,v (Λ) be the set of all strongly reduced components of decrep d,v(λ), and let decirr s.r. (Λ) := d,v (Λ). (d,v) N n N n decirr s.r.

15 CALDERO-CHAPOTON ALGEBRAS 5 An irreducible component Z in Irr(Λ) or decirr(λ) is called indecomposable provided there exists a dense open subset U of Z, which contains only indecomposable representations or decorated representations, respectively. In particular, if Z decirr d,v (Λ) is indecomposable, then either d = 0 or v = 0. Given irreducible components Z i of decrep di,v i (Λ) for i t, let (d, v) := (d, v ) + + (d t, v t ) and let Z Z t be the points of decrep d,v (Λ), which are isomorphic to M M t with M i Z i for i t. The Zariski closure of Z Z t in decrep d,v (Λ) is denoted by Z Z t. It is quite easy to show that Z Z t is an irreducible closed subset of decrep d,v (Λ), but in general it is not an irreducible component. Theorem 5. ([CBS]). For Z,..., Z t decirr(λ) the following are equivalent: (i) Z Z t is an irreducible component. (ii) ext Λ (Z i, Z j ) = 0 for all i j. Furthermore, the following hold: (iii) Each irreducible component Z decirr(λ) can be written as Z = Z Z t with Z,..., Z t indecomposable irreducible components in decirr(λ). Suppose that Z Z t = Z Z s is an irreducible component with Z i and Z i indecomposable irreducible components in decirr(λ) for all i. Then s = t and there is a bijection σ : {,..., t} {,..., s} such that Z i = Z σ(i) for all i. The next lemma is an easy exercise. Lemma 5.4. For i n and any decorated representation M = (M, V ) of Λ we have E Λ (M, S i ) = dim(m i) and E Λ (S i, M) = 0. Corollary 5.5. For any Z decirr s.r. d,v (Λ) we have d iv i = 0 for all i n. Lemma 5.6. Let Z decirr d,v (Λ), and assume that p > d. Then the following are equivalent: (i) Z decirr s.r. (Λ). (ii) Z decirr s.r. (Λ p ). Proof. Since p > d, we can apply Lemma. and Proposition.5 and get c Λp (Z) = c Λ (Z) and E Λp (Z) = E Λ (Z). This yields the result. The additivity of the functor Hom Λ (,?) and upper semicontinuity imply the following lemma. Lemma 5.7. Let Z, Z, Z decirr(λ). Suppose that Z = Z Z. Then the following hold: (i) end Λ (Z) = end Λ (Z ) + end Λ (Z ) + hom Λ (Z, Z ) + hom Λ (Z, Z ). (ii) E Λ (Z) = E Λ (Z ) + E Λ (Z ) + E Λ (Z, Z ) + E Λ (Z, Z ).

16 6 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER For a = (a,..., a n ) and b = (b,..., b n ) Z n set a b := a b + + a n b n. The following lemma is obvious. Lemma 5.8. Let d, d, d N n with d = d + d. Then dim(g d ) dim(g d ) dim(g d ) = (d d ). Lemma 5.9. Let Z, Z, Z decirr(λ) with Z = Z Z. We have dim(z) = dim(z ) + dim(z ) + (dim(πz ) dim(πz )) hom Λ (Z, Z ) hom Λ (Z, Z ). Proof. For i =, let (d i, v i ) := dim(z i ), and let (d, v) := dim(z). We have dim(z) = dim(z ) + dim(z ) and dim(πz i ) = d i. The map f : G d Z Z Z defined by (g, (M, C v ), (M, C v )) (g.(m M ), C v ) is a morphism of affine varieties. For (M, M ) Z Z define f M,M : G d O(M ) O(M ) O(M M ) by (g, N, N ) f(g, N, N ). The fibres of f M,M are of dimension d M,M := dim(g d ) + dim O(M ) + dim O(M ) dim O(M M ). Using Equation (), an easy calculation yields d M,M = dim(g d ) + dim(g d ) + dim Hom Λ (M, M ) + dim Hom Λ (M, M ). Let M be in the image of f. We want to compute the dimension of the fibre f (M). Let T := {O(N ) O(N ) Z Z N N = M}. It follows from the Krull-Remak-Schmidt Theorem that T is a finite set. Thus the fibre f (M) is the disjoint union of the fibres f N,N (M), where O(N ) O(N ) runs through T. So we get dim(f (M)) = max{d N,N O(N ) O(N ) T }. Thus by upper semicontinuity there is a dense open subset V Z such that all fibres f (M) with M V have dimension d Z,Z := dim(g d ) + dim(g d ) + hom Λ (Z, Z ) + hom Λ (Z, Z ). By Chevalley s Theorem we have Using Lemma 5.8 we get dim(z) + d Z,Z = dim(g d ) + dim(z ) + dim(z ). dim(z) = dim(z ) + dim(z ) + (d d ) hom Λ (Z, Z ) hom Λ (Z, Z ). This finishes the proof. Lemma 5.0. For Z, Z, Z decirr(λ) with Z = Z Z we have c Λ (Z) = c Λ (Z ) + c Λ (Z ). Proof. For i =, let (d i, v i ) := dim(z i ), and let (d, v) := dim(z). We get c Λ (Z) = dim(z) dim(g d ) + end Λ (Z) = dim(z ) + dim(z ) dim(g d ) dim(g d ) + end Λ (Z ) + end Λ (Z ) = c Λ (Z ) + c Λ (Z ). The first equality follows directly from the definition of c Λ (Z). The second equality uses Lemma 5.7(i) and Lemma 5.9.

17 CALDERO-CHAPOTON ALGEBRAS 7 The following result is a version of Theorem 5. for strongly reduced components. Theorem 5.. For Z,..., Z t decirr(λ) the following are equivalent: (i) Z Z t is a strongly reduced irreducible component. (ii) Each Z i is strongly reduced and E Λ (Z i, Z j ) = 0 for all i j. Proof. Without loss of generality assume that t =. The general case follows by induction. Let Z decirr d,v (Λ) and Z decirr d,v (Λ). Assume that Z := Z Z is a strongly reduced component. Thus we have c Λ (Z) = E Λ (Z). Applying Lemma 5.0 and Lemma 5.7(ii) this implies c Λ (Z ) + c Λ (Z ) = E Λ (Z ) + E Λ (Z ) + E Λ (Z, Z ) + E Λ (Z, Z ). Since c Λ (Z i ) E Λ (Z i ) we get E Λ (Z, Z ) = E Λ (Z, Z ) = 0 and c Λ (Z i ) = E Λ (Z i ). Thus (i) implies (ii). To show the converse, assume that Z and Z are strongly reduced with E Λ (Z, Z ) = E Λ (Z, Z ) = 0. We claim that c Λ (Z) = c Λ (Z ) + c Λ (Z ) = E Λ (Z ) + E Λ (Z ) = E Λ (Z). For the first equality we use Lemma 5.0, the second equality is just our assumption that Z and Z are strongly reduced. Finally, the third equality follows from Lemma 5.7 together with our assumption that E Λ (Z, Z ) and E Λ (Z, Z ) are both zero. Thus Z is strongly reduced. Note that Theorems 5. and 5. imply that each Z decirr s.r. (Λ) is of the form Z = Z Z t with Z i decirr s.r. (Λ) and Z i indecomposable for all i. The next lemma follows directly from upper semicontinuity and Lemma 4.(i). Lemma 5.. For Z, Z, Z decirr(λ) with Z = Z Z we have g Λ (Z) = g Λ (Z ) + g Λ (Z ). Lemma 5.. For Z decirr s.r. d,v (Λ) we have d g Λ (Z) = dim(z) dim(g d ). Proof. It follows from the definitions that E Λ (Z) = end Λ (Z) + d g Λ (Z), and we have c Λ (Z) = dim(z) dim(g d ) + end Λ (Z). Now the claim follows, since c Λ (Z) = E Λ (Z). Corollary 5.4. Let Z decirr s.r. d,v (Λ) with d 0. If E Λ(Z) = 0, then d g Λ (Z) = end Λ (Z) < 0.

18 8 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER 5.. Parametrization of strongly reduced components. Let Λ = C Q /I be a finite-dimensional basic algebra. Plamondon [P] constructs a map P Λ : decirr(λ) Z n, which can be defined as follows: Let Z decirr(λ). Then there exist injective Λ-modules I0 Λ(Z) and IΛ (Z), which are uniquely determined up to isomorphism, and a dense open subset U πz such that for each representation M U we have I0 Λ(M) = IΛ 0 (Z) and I Λ(M) = IΛ (Z). For Z decirrs.r. d,v (Λ) define P Λ (Z) := dim(soc(i Λ 0 (Z))) + dim(soc(i Λ (Z))) + v. Let PΛ s.r. : decirr s.r. (Λ) Z n be the restriction of P Λ to decirr s.r. (Λ). For a representation M let add(m) be the category of all finite direct sums of direct summands of M. Plamondon [P] obtains the following striking result. Theorem 5.5 (Plamondon). For any finite-dimensional basic algebra Λ the following hold: (i) PΛ s.r. : decirr s.r. (Λ) Z n is bijective. (ii) For every Z decirr s.r. (Λ) we have add(i Λ 0 (Z)) add(i Λ (Z)) = 0. Note that Plamondon works with irreducible components, and not with decorated irreducible components. But his results translate easily from one concept to the other. We now generalize Theorem 5.5(i) to arbitrary basic algebras Λ. It turns out that decirr s.r. (Λ) is in general no longer parametrized by Z n but by a subset of Z n. Our proof is based on Plamondon s result and uses additionally truncations of basic algebras. For a basic algebra Λ let G Λ : decirr(λ) Z n be the map, which sends Z decirr(λ) to the generic g-vector g Λ (Z) of Z. For finitedimensional Λ, it follows immediately from Lemma. that G Λ = P Λ. Let be the restriction of G Λ to decirr s.r. (Λ). For a basic algebra Λ let G s.r. Λ : decirr s.r. (Λ) Z n decirr <p (Λ) be the set of irreducible components Z decirr(λ) such that (d, v) := dim(z) satisfies d < p. Define decirr s.r. <p(λ) := decirr <p (Λ) decirr s.r. (Λ). Lemma 5.6. For a basic algebra Λ the following hold: (i) For all p q we have decirr s.r. <p(λ p ) decirr s.r. <q (Λ q ) decirr s.r. (Λ).

19 CALDERO-CHAPOTON ALGEBRAS 9 (ii) We have decirr s.r. (Λ) = p>0 decirr s.r. <p(λ p ). Proof. Let Z decirr d,v (Λ), and let p > d. Thus we have Z decirr d,v (Λ p ) and Z decirr <p (Λ p ). Furthermore, we have c Λp (Z) = c Λ (Z) and E Λp (Z) = E Λ (Z). Thus Z decirr s.r. (Λ) if and only if Z decirr s.r. (Λ p ). This yields the result. Theorem 5.7. For a basic algebra Λ the following hold: (i) The map is injective. (ii) The following are equivalent: (a) G s.r. Λ is surjective. (b) Λ is finite-dimensional. G s.r. Λ : decirr s.r. (Λ) Z n Proof. Since Λ p is finite-dimensional for all p, we know from Plamondon s Theorem 5.5(i) that G s.r. Λ p : decirr s.r. (Λ p ) Z n is bijective. Now Lemma 5.6 yields that the map G s.r. Λ : decirr s.r. (Λ) Z n sends Z decirr s.r. <p(λ p ) to G s.r. Λ p (Z), and that GΛ s.r. is injective. This proves (i). Theorem 5.5(i) says that (b) implies (a). To show the converse, assume that Λ is infinite dimensional. Thus there exists some i n such that the injective envelope I i of the simple Λ-module S i is infinite dimensional. Let I i,p be the injective envelope of the simple Λ p -module S i. Using that I i is infinite dimensional, one can easily show that dim(i i,p ) p. Assume that e i is in the image of G s.r. Λ. (Here e i denotes the ith standard basis vector of Z n.) In other words, there is some Z decirr s.r. (Λ) such that G s.r. Λ (Z) = e i. By Lemma 5.6(ii) we know that Z decirr s.r. <p(λ p ) for some p. Since g Λ (Z) = e i, we have I Λp 0 (Z) = I i,p and I Λp (Z) = 0. This implies that Z is the closure of the orbit of the decorated representation (I i,p, 0). But dim(i i,p ) p and the dimension of all representations in Z is strictly smaller than p, a contradiction. The proof of Theorem 5.7(ii) yields the following result. Corollary 5.8. For a basic algebra Λ and i n the following are equivalent: (i) e i Im(G s.r. Λ ). (ii) I i is finite-dimensional. Let G Λ := Im(G s.r. Λ ) = {g Λ (Z) Z decirr s.r. (Λ)} be the set of generic g-vectors of the strongly reduced irreducible components.

20 0 G. CERULLI IRELLI, D. LABARDINI-FRAGOSO, AND J. SCHRÖER 6. Component graphs and CC-clusters 6.. The graph of strongly reduced components. Let Λ be a basic algebra. In [CBS] the component graph Γ(Irr(Λ)) of Λ is defined as follows: The vertices of Γ(Irr(Λ)) are the indecomposable irreducible components in Irr(Λ). There is an edge between (possibly equal) vertices Z and Z if ext Λ (Z, Z ) = ext Λ (Z, Z ) = 0. We want to define an analogue of Γ(Irr(Λ)) for strongly reduced components. The graph Γ(decIrr s.r. (Λ)) of strongly reduced components has as vertices the indecomposable components in decirr s.r. (Λ), and there is an edge between (possibly equal) vertices Z and Z if E Λ (Z, Z ) = E Λ (Z, Z ) = Component clusters. Let Γ be a graph, and let Γ 0 be the set of vertices of Γ. We allow only single edges, and we allow loops, i.e. edges from a vertex to itself. For a subset U Γ 0 let Γ U be the full subgraph, whose set of vertices is U. The subgraph Γ U is complete if for each i, j J with i j there is an edge between i and j. A complete subgraph Γ U is maximal if for each complete subgraph Γ U with U U we have U = U. We call a subgraph Γ U loop-complete if Γ U is complete and there is a loop at each vertex in U. The set of vertices of a maximal complete subgraph of Γ := Γ(decIrr s.r. (Λ)) is called a component cluster of Λ. A component cluster U of Λ is E-rigid provided E Λ (Z) = 0 for all Z U. (Recall that there is a loop at a vertex Z of Γ if and only if E Λ (Z, Z) = 0. If E Λ (Z) = 0, then E Λ (Z, Z) = 0, but the converse does not hold.) Proposition 6.. For each loop-complete subgraph Γ U of Γ := Γ(decIrr s.r. (Λ)) we have U n. Proof. Assume that Z,..., Z n+ are pairwise different vertices of a loop-complete subgraph Γ J of Γ(decIrr s.r. (Λ)). For i n + let g Λ (Z i ) be the generic g-vector of Z i. Since Γ J is loop-complete we know by Theorem 5. that Z a := Z a Za n+ n+ is again a strongly reduced component for each a = (a,..., a n+ ) N n+. additivity of g-vectors we get g Λ (Z a ) = a g Λ (Z ) + + a n+ g Λ (Z n+ ). By the Furthermore, we know from Theorem 5. that Z a = Z b if and only if a = b. Now one can essentially copy the proof of [GS, Theorem.] to show that there are a, b N n+ with g Λ (Z a ) = g Λ (Z b ) but a b. By Theorem 5.7 different strongly reduced components have different g-vectors. Thus we have a contradiction. Corollary 6.. Let Λ be a finite-dimensional basic algebra. Let M be a representation of Λ with Hom Λ (τ Λ (M), M) = 0. Then M has at most n isomorphism classes of indecomposable direct summands. The following conjecture might be a bit too optimistic. But it is true for Λ = C Q the path algebra of an acyclic quiver Q, see [DW, Corollary ] and Section 9.. Conjecture 6.. For any basic algebra Λ the following hold: (i) The component clusters of Λ have cardinality at most n. (ii) The E-rigid component clusters of Λ are exactly the component clusters of cardinality n.

21 CALDERO-CHAPOTON ALGEBRAS 6.. E-rigid representations. After most of this work was done, we learned that Iyama and Reiten [IR] obtained some beautiful results on socalled τ-rigid modules over finitedimensional algebras, which fit perfectly into the framework of Caldero-Chapoton algebras. Adapting their terminology to decorated representations of basic algebras, a decorated representation M of a basic algebra Λ is called E-rigid provided E Λ (M) = 0. The following theorem is just a reformulation of Iyama and Reiten s results on τ-rigid modules. Part (i) follows also directly from the more general statement in Proposition 6.. For M decrep(λ) let Σ(M) be the number of isomorphism classes of indecomposable direct summands of M. Theorem 6.4 ([IR]). Let Λ = C Q /I be a finite-dimensional basic algebra. For M decrep(λ) the following hold: (i) If M is E-rigid, then Σ(M) n. (ii) For each E-rigid M decrep(λ) there exists some N decrep(λ) such that M N is E-rigid and Σ(M N ) = n. (iii) For each E-rigid M decrep(λ) with Σ(M) = n there are exactly two nonisomorphic indecomposable decorated representations N, N decrep(λ) such that M N i is E-rigid and Σ(M N i ) = n for i =,. It is easy to find examples of infinite dimensional basic algebras Λ such that Theorem 6.4(iii) does not hold, see Section 9... A basic algebra Λ is representation-finite if there are only finitely many isomorphism classes of indecomposable representations in rep(λ). One easily checks that Λ is finitedimensional in this case. Corollary 6.5. Assume that Λ is a representation-finite basic algebra. Then the following hold: (i) Each component cluster of Λ is E-rigid. (ii) Each component cluster of Λ has cardinality n. (iii) There is bijection between the set of isomorphism classes of E-rigid representation of Λ to the set decirr s.r. (Λ) of strongly reduced components. Namely, one maps an E-rigid representation M to the closure of the orbit O(M). Proof. Since Λ is representation-finite, every irreducible component Z decirr(λ) is a union of finitely many orbits, and exactly one of these orbits has do be dense in Z. Thus we have c Λ (Z) = 0. This implies (i) and (iii). Now (ii) follows directly from Theorem 6.4(ii) Generic Caldero-Chapoton functions. For each (d, v) N n N n let C d,v : decrep d,v (Λ) Z[x ±,..., x± n ] be the function defined by M C Λ (M). The map C d,v is a constructible function. In particular, the image of C d,v is finite. Thus for an irreducible component Z decirr d,v (Λ) there exists a dense open subset U Z such that C d,v is constant on U. Define C Λ (Z) := C Λ (M) with M any representation in U. The element C Λ (Z) depends only on Z and not on the choice of U.