On sign coherence of $c$-vectors

Given a finite dimensional algebra $A$ over an algebraically closed field, we consider the $c$-vectors such as defined by Fu in \cite{Fu2017} and we give a new proof of its sign-coherence. Moreover, we characterise the modules whose dimension vectors are $c$-vectors as bricks respecting a functorially finiteness condition.


Introduction
Cluster algebras were introduced at the beginning of the century by Fomin and Zelevinsky in [15]. In the subsequent papers [16] and [17], they introduce two families of vectors with integer coefficients that index the cluster variables: the c-vectors and the g-vectors. They have been used to parametrise canonical bases of cluster algebras (see for instance [24,28]) and, in the last years, to build the scattering diagrams of cluster algebras by Gross, Hacking, Keel and Kontsevich in [19], a powerful tool that was used in order to solve several conjectures on cluster algebras at once and also relate cluster algebras with mirror symmetry.
From the moment Fomin and Zelevinsky defined the c-vectors, they noted that all non-zero entries of a given c-vector are either positive or negative and name this phenomenon as sign-coherence of c-vectors. They conjecture that c-vectors are always sign-coherent and this was proven true for quivers first by [14] and later by Nagao in [25]. The general case of sign-coherence for cluster was proven in [19].
Later, cluster algebras started to be categorified using the so-called cluster categories, first introduced by Caldero, Chapoton and Schiffler in [9] for the A n case, and, independently, by Buan, Marsh, Reineke, Reiten and Todorov in [8] for acyclic quivers. That lead people to give representation theoretic meaning to these families of vectors. First, Dehy and Keller introduced in [13] the set of g † -vectors for 2-Calabi-Yau categories, conjecturally equivalent to g-vectors in the corresponding cluster algebra. The equivalence follows from the work of Plamondon in [27]. On the other hand, Nagao proved in [25] that one can realise the c-vectors of an skewsymmetrizable cluster algebra as a subset of the dimension vectors of functorially finite bricks in the module category of the associated jacobian algebra. The reverse inclusion was proved by Nájera-Chavez in [10,26] for the acyclic and finite case, respectively.
In the recent years, Adachi, Iyama and Reiten introduced in [1] the τ -tilting theory. This theory succeeds to emulate the combinatorics of cluster algebras on the module category of an algebra, bypassing the construction of a cluster category. Therefore, problems that arise naturally in the cluster setting can be stated on this representation theoretic environment, even if the algebra considered does not have an associated cluster algebra.
For instance, of g-vectors in this context was given by Adachi, Iyama and Reiten in [1], generalising the approach of [13]. Afterwards, Fu introduced in [18] the cvectors for finite dimensional algebras as the columns of the inverse of the transpose of matrices of g-vectors (see section 2 for a rigorous definition). Then he showed that c-vectors are sign-coherent for every finite dimensional algebra A, by showing that c-vectors are dimension vectors of certain A-modules. Moreover, he gives an explicit description of the these modules when A is either quasitilted, representation directed or cluster-tilted of finite type.
In the present paper we take the definition of c-vectors given in [18] and we show that they are the dimension vectors of certain bricks, using the description of stability functions given in [7]. Our first theorem is the following. In particular the c-vectors of A are sign-coherent.
This result has two interesting consequences. The first one is that the arrows of the exchange graph of τ -tilting pairs of every algebra can be labeled with c-vectors as follows. Note that the previous labelling coincides with the brick labelling studied for instance in [12,5].
As we said already, c-vectors and g-vectors are closely related with the scattering diagrams of cluster algebras. In [6], Bridgeland proposed a scattering diagram for any finite dimensional algebra and showed that this scattering diagram coincide with the corresponding cluster scattering diagram if the algebra hereditary. In [7] the g-vectors were used to give an algebraic description of the wall and chamber structure of an algebra, which happens to be the support of the scattering diagram introduced by Bridgeland. As a consequence of our main theorem we prove that the walls surrounding chambers generated by g-vectors (this set includes all reachable chambers) are perpendicular to c-vectors. For more details on the wall and chamber structure of an algebra see section 4. The precise statement is the following. Now, it is known that the dimension vector of every brick in the module category of an arbitrary algebra is a c-vector. Our following result gives a classification of the bricks whose dimension vectors are positive c-vectors in terms of functorially torsion classes. . . , B k } be the set of bricks whose dimension correspond to the positive columns of the Cmatrix C (M,P ) of (M, P ). Then the following holds: • FacM is the minimal torsion class containing B + (M,P ) . Moreover, the reciprocal also holds. Namely, if is N is a set of bricks in modA such that Hom A (B, B ′ ) = 0 for all B, B ′ ∈ N and the minimal torsion class containing N is functorially finite. Then N = B + (M,P ) for some τ -tilting pair (M, P ). The structure of the article is the following. In section 2 we give the background material and we establish the setting for the rest of the paper. In section 3 we prove the main results of the paper. In section 4 we investigate how the results in the previous section improve the description of the wall and chamber structure an algebra given in [7]. In section 5, we finish the article with the complete analysis of a particular example.
It is important to remark that, independently, Speyer and Thomas (private communication), and Jørgensen and Yakimov in [22] proved similar results. While preparing this second version the author was informed that Asai also proved similar results in [2].
Acknowledgements The author thankfully acknowledges Peter Jørgensen for the exchanges that lead to the work in this paper. He also is grateful to Kiyoshi Igusa, Hongwei Niu and Sibylle Schroll for the useful discussions. He does not want to forget the valuable comments of Bernhard Keller and the anonymous referee on a previous version of the paper that lead to great improvement of this article. This project was supported by the EPSRC founded project EP/P016294/1.

Setting
In this paper A is a finite dimensional algebra over an algebraically closed field k. By modA we mean the category of finite generated right A-modules and τ represents the Auslander-Reiten translation in modA. If one considers A as a module over itself, then A can be written as The Grothendieck group of A is noted K 0 (A), where rk(K 0 (A)) = n. It is known that K 0 (A) is isomorphic to Z n . In this paper we consider the embedding Given an A-module M , we denote by |M | the number of non-isomorphic indecomposable direct summands of M .
When we write −, − we are referring to the canonical inner product in R n which is defined by We say that an A-module M is a brick if its endomorphism algebra End A (M ) is a division ring.

τ -tilting theory
The τ -tilting theory was introduced by Adachi, Iyama and Reiten in [1]. This theory provides a framework to study problems arising in cluster algebras in the module category of an arbitrary algebra. In the proofs of this paper the τ -rigid (τ -tilting) pairs play a central role. They are defined as follows. Moreover, we say that (M, P ) is τ -tilting (or almost τ -tilting) if |M | + |P | = n (or |M | + |P | = n − 1, respectively).
From now on, when we say that (M, P ) is a τ -tilting (τ -rigid) pair, we are assuming that M and P are basic and its decomposition as direct sum of indecomposable modules can be written as M = k i=1 M i and P = n j=k+1 P j (M = k i=1 M i and P = t j=k+1 P j , with t ≤ n, respectively). The τ -tilting is a generalisation of classical tilting theory which is capable of describe all the functorially finite torsion classes in the module category of an algebra in terms of τ -tilting pairs. Moreover, Φ is a bijection if we restrict it to τ -tilting pairs.
Recall that a torsion pair (T , F ) in modA is a pair T and F of full subcategories in modA such that Hom A (X, Y ) = 0 for every X ∈ T and Y ∈ F and which are maximal with respect to this property. Given a torsion pair (T , F ), we say that T is a torsion class and F is a torsion free class. Moreover, T is a full subcategory of modA closed under quotients and extensions, while F is a full subcategory of modA closed under submodules and extensions. It is well know that for every subcategory T closed under quotients and extensions there exists a torsion free class F such that (T , F ) is a torsion pair. In particular, if T = FacM for some A-module M , then F = M ⊥ = {N ∈ modA : Hom A (M, N ) = 0}.
In this paper, given a module M , we denote by T (M ) the minimal torsion class containing M .
Recall that for every torsion pair (T , F ) and every module M in modA there exists the a canonical short exact sequence where tM ∈ T and M/tM ∈ F , which is unique up to isomorphism.

Integer vectors associated to modules
In this paper we denote by K 0 (A) the Grothendieck group of A. By abuse of notation, for every M in modA we identify its equivalence class in K 0 (A) with its dimension vector, and we denote it by [M ].
For each cluster algebra, one can associate to it two different and complementary set of vectors: the c-vectors and g-vectors (see [17]). In this paper we study their representation theoretic versions. The definition of g-vectors is the following. Definition 2.3. [1, Section 5] Let A be an algebra, M be an A-module and There is important homological information that arises when g-vectors and dimension vectors interplay, as showed by Auslander and Reiten in [3].
In particular, if we restrict ourselves to τ -tilting pairs, g-vectors have other emerging features. See for instance the following result proved by Adachi, Iyama and Reiten in [1].
This important property of g-vectors allowed Fu to define c-vectors as follows.
Then the c-matrix of (M, P ) is defined as Also, each column of C (M,P ) is called a c-vector of A. Moreover, we denote by cv(A) the set of all c-vectors of A.
Remark 2.7. Note that, since modA is a Krull-Schmidt category, the definition of G (M,P ) for every τ -tilting pair is unique up to permutation of columns. Therefore, C (M,P ) is also uniquely defined up to permutation of columns.
In this paper we are interested in the property of sign-coherence of c-vectors. The formal definition is the following.
Finally, a set S of vectors is said to be sign-coherent if v is either positive or negative for every v ∈ S.
The sign-coherence of c-vectors was first formulated as a conjecture in the cluster setting by Fomin and Zelevinsky in [17]. This conjecture was proven to be true for quiver cluster algebras by Derksen, Weymann and Zelevinsky in [14]. Later, Fu introduced c-vectors for every finite dimensional algebra in [18] and prove the following. We denote the set of positive and negative c-vectors by cv + (A) and cv − (A), respectively.

Stability conditions
We will attack the problem of sign-coherence of c-vectors from the point of view of stability conditions. The definition of θ-semistables modules was introduced by King in [23] in terms of linear functionals. Given that we intend to have a geometrical realisation of these objects, we adapt the definition in terms of vectors as follows.
Given a vector θ ∈ R n , we denote by modA ss θ the category of θ-semistable modules. It is known that modA ss θ is an abelian category for every θ, where θ-stable modules correspond to the simple objects in modA ss θ . Note that a θ-stable module is necessarily a brick in modA by [29,Theorem 1] and [29,Proposition 3.4].
Sometimes, a more precise description of modA ss θ can be given. For instance, if θ can be defined using the g-vectors of a τ -rigid pair (M, P ), such a description was given in [7]. The result is the following.
Remark 2.12. The algebraÃ (M,P ) in the previous theorem correspond to the algebra arising from τ -tilting reduction developed by Jasso in [21]. The explicit construction of the algebra can be found in the original article. Also the main ideas are given in [7,Section 2]. Using the previous result, Brüstle, Smith and Treffinger constructed explicitly one θ α(M,P ) -stable module when (M, P ) is an almost τ -tilting pair.
Proposition 2.13. [7, Proposition 3.17] Let (M, P ) be an almost τ -tilting pair and θ α(M,P ) be as before. Then there is a θ α(M,P ) -semistable module N that is constructed as follows: Let (M 1 , P 1 ) and (M 2 , P 2 ) be the two τ -tilting pairs containing M and P as a direct factor. Order them such that FacM 1 ⊂ FacM 2 . Then N is the cokernel of the right addM -approximation of M 2 .

c-vectors as dimension vectors of bricks
This section is organised as follows. In the first subsection, using theorem 2.11, we find n bricks {B 1 , . . . , B n } which are naturally associated to a given τ -tilting pair (M, P ). Then we build a matrix X (M,P ) using of dimension vectors of those bricks, which will play a key role in the rest of the paper.
In the second we show that one can always build the matrix C (M,P ) from X (M,P ) . As a corollary it follows that every positive (negative) c-vectors is the (opposite of the) dimension vector of a brick in modA. Hence, the sign coherence of c-vectors follows at once.
In the final subsection we study the relation between c-vectors and torsion classes. More precisely, we show that the modules whose dimension vectors are positive the c-vectors of C (M,P ) are the minimal set of bricks in modA generating the torsion class FacM . We also show the reciprocal, that is, we show that the dimension vectors of every minimal set of bricks in modA generating a functorially finite torsion class are the positive columns of the C-matrix C (M,P ) for a certain τ -tilting pair (M, P ).

Bricks associated to a τ -tilting pair
Let (M, P ) be a τ -tilting pair. Then their decomposition as sum of indecomposable direct summands is the following.
We can suppose without loss of generality the that the indecomposable direct summands of M and P are pairwise non-isomorphic, i.e., M and P are basic. Now, given (M, P ), one can construct for every 1 ≤ r ≤ n an almost τ -tilting pair (M, P ) r as follows.
Then, define for every r the vector θ r in the following way.
Hence, theorem 2.11 implies that for every r there exists a brick B r which is θ rstable. Moreover, B r is unique up to isomorphism. Therefore, for every τ -tilting pair we have a set B (M,P ) of θ r -stable modules for every r between 1 and n. Formally: Now we are ready to define X (M,P ) as the square matrix having as r-th column the dimension vector [B r ] of the brick B r ∈ B (M,P ) .
If one takes the set of all g-vectors and the set of all bricks in the module category that can be obtained as we just did, there is no well defined function between these two sets. However, if we restrict ourselves to a particular τ -tilting pair the situation changes, since the previous construction gives a bijection between the g-vectors of the indecomposable direct summands of (M, P ) and the elements of B (M,P ) . Therefore, sometimes we may fall in an abuse of language saying that a the brick B r ∈ B (M,P ) is the brick associated or corresponding to the g-vector g Mr if 1 ≤ r ≤ k or −g Pr if k + 1 ≤ r ≤ n.  Remark that the fact that D is diagonal and invertible over Z implies that every in the diagonal is either 1 or −1. We start with the following proposition that will take care of the positive entries of D. Proof. Let N r the θ r -semistable module of proposition 2.13. It follows from N r is a generator of modA ss θ (M,P )r by [21,Theorem 3.15]. Moreover [7,Proposition 3.17] implies that N r belongs to FacM . Therefore any other θ r -semistable module N belong to FacM because FacM is a torsion class.
The following lemma is key to show that c-vectors are the dimension vectors of certain modules. Proof. Let B r ∈ B (M,P ) . As a first remark, one can see that theorem 2.11 implies that θ (M,P ) , [B r ] = 0 because (M, P ) is a τ -tilting module.
Suppose that k + 1 ≤ r ≤ n. Then the linearity of the inner product implies that By construction we have that B r is θ r -stable. This implies in particular that θ r , [B r ] = 0. Hence a contradiction with our hypothesis. Then we have that 1 ≤ r ≤ k. In that case Since B r is θ r -stable we have that B r ∈ FacM by proposition 3.2. Hence So, Therefore, is enough to show that dim k (Hom A (M r , B r )) = 1 to complete the proof. Let (M, P ) r be the almost τ -tilting pair as in subsection 3.1, the torsion pair associated to it and let be the canonical short exact sequence with respect to (T r , F r ).
We have that N r = M r /tM r by [7,Lemma 2.3]. Hence N r is an Ext-projective module in modA ss θ (M,P )r by [21,Theorem 3.15]. Therefore Theorem 2.11 implies that (G(Ã (M,P )r )) for some natural number t. Consider, for every 1 ≤ j ≤ t, the following commutative diagram where E j is the pullback of p and the canonical monomorphism ι j for every j.
Then E j ∈ FacM for every 1 ≤ j ≤ t because FacM is closed under extensions. Consequently, so does t j=1 E j . Consider the morphisms Then we construct the following commutative diagram.
Note that, by hypothesis, ι is an isomorphism and, by construction, id t is an epimorphism. Therefore, we can apply the snake lemma to show that f is an epimorphism. Since M r is an Ext-projective module for FacM and t j=1 E j ∈ FacM , we deduce that f must split. So, M r is an indecomposable direct factor of t j=1 E j ∈ FacM . Then, M r is a direct factor of some E j .
On the other hand, we know that dim k (Hom A (M r , B r )) = 0 and Moreover, B r is, by hypothesis, the image over the functor G : modÃ (M,P )r → modA of a representative of the unique isomorphism class of the simpleÃ (M,P )r -modules. Therefore dim k (Hom A (M r , B r )) ≤ dim k (HomÃ (M,P )r (Ã (M,P )r , S)) = 1 Then we can conclude that dim A (Hom A (M r , B r )) = 1 as claimed.
Now we need to consider the cases of bricks such that θ r , [B r ] < 0. However, in this case the inequality could come from the fact that either Hom A (B r , τ M r ) = 0 or Hom A (P r , M r ) = 0. Hence, using dual-like arguments is not enough. Instead, we will use some basic linear algebra.
Let consider a basis B = {v 1 , . . . , v n } of Z n and let B * = {v * 1 , . . . , v * n } be its dual basis, i.e., a set of linearly independent vectors in Z n such that In general the dual basis (B ′ ) * of B ′ can be quite different of the dual basis B * of B. However something can said, as shown in the following result. We include the proof for the convenience of the reader.
Proof. Given that B * is a basis of Z n , we have that (v ′ 1 ) can be written as a linear combination of the elements of B * . Moreover, given that B * is a dual basis of B, this linear combination has the following form.
Consider G T (M,P ) , the transpose of the g-matrix of the τ -tilting pair (M, P ), and multiply it to the right with the matrix X (M,P ) constructed in subsection 3.1.
Note that, by definition, All this argument shows that D is a diagonal matrix having only 1 or −1 in the diagonal. Moreover, is easy to see that D 2 is the identity matrix. Therefore, multiplying by D on the right we find the following equality. As an immediate consequence of the previous result we get the following corollaries.
Corollary 3.6. Let A be an algebra. Then the c-vectors of A are sign-coherent.
Proof. In theorem 3.5 is shown that every c-vector is either the dimension vector of a module or its opposite. Hence c-vectors are sign-coherent.  In previous work by Nájera-Chávez (see [26,Theorem 6] and [10,Theorem 11]) and Fu ([18, Theorem 3.1, 4.8, 4.13]) they have been showed that, for certain types of algebras, the set of positive c-vectors correspond to the dimension vectors of exceptional objects, that is, bricks without non-trivial self-extensions. However, in corollary 3.8 we only are able to prove that c-vectors correspond to dimension vectors of bricks. The following example shows that we can not do better. Hence the problem of classifying all the algebras whose positive c-vectors correspond to its exceptional objects arises naturally.

c-vectors and functorially finite torsion pairs
Let (M, P ) be a τ -tilting pair. In this subsection we study the relation between the set of bricks B (M,P ) associated to (M, P ) and the torsion pair (FacM, M ⊥ ) induced by it.
In order to state one of the main theorems of this paper, we need to introduce some terminology and notation.
Given two A-modules X and Y , we say that they are Hom-orthogonal if The following proposition is a direct consequence of the definition of theorem 3.5. Proof. The first part of the statement is a particular case proposition 3.2 and its dual. Now we prove the moreover part of the statement. By construction B s is a θ s -stable module. Then [7, Proposition 3.13] implies that Moreover, the fact that B s ∈ FacM yields an epimorphism p s : M s → B s . The same argument shows the existence of an epimorphism p t : M t → B t .
Now, every morphism f ∈ Hom A (B t , B s ) can be composed to the left with p t to get a morphism p t f : M t → B s . But M t is a direct summand of ⊕ i =s M i , so Hom A (M t , B s ) = 0. Hence f = 0 because p t is an epimorphism. The fact that Hom A (B s , B t ) = 0 is shown in the same fashion. This finishes the proof.
Consider a set N = {N 1 , . . . , N t } of A-modules and let N = t i=1 N i . Then we denote by T (N ) the full subcategory of modA having as objects that can be filtered by objects in FacN . More precisely, Is easy to see that T (N ) is closed under quotients and extensions, which implies that T (N ) is always a torsion pair. Moreover, as shown in [11,Proposition 3.3], T (N ) is the minimal torsion class containing N .
Let N be an A-module and N ′ be a self-extension of N . Then is easy to see that T (N ) = T (N ′ ). Therefore some redundancies needs to be avoided. In order to do that, from now on we restrict ourselves to the case where every B ∈ N is a brick.
Then, a natural question to ask is the following: Given a torsion class T in modA, what is minimal collection of bricks N such that T = T (N )?
This was answered by Barnard, Carrol and Zhu in [5] as follows.
Theorem 3.11. [5, Theorem 1.0.8] Let T be a torsion class in modA. Then the minimal set N T ⊂ T such that T (N ) = T is the set which is maximal for the following properties: Remark 3.12. Note that the maximality of N T implies that it is unique for every torsion class T . Now we are able to state and prove the main theorem of this paper, which gives a characterisation of all modules whose dimension vectors are positive c-vectors.

The wall and chamber structure of an algebra
If one looks at the Definition 2.10 of King's stability condition, one see that there are two directions to study them: either one fix a functional θ and study the category of θ-semistable modules, as is done in Theorem 2.11, or one fix a module M and study the functionals θ making M a θ-semistable module. In this subsection we continue the work of [7] considering the second option. Note that not every θ belongs to the stability space D(M ) for some nonzero module M . For instance, is easy to see that θ = (1, 1, . . . , 1) is an example of such a functional for every algebra A. This leads to the following definition.
Definition 4.2. Let A be an algebra such that rk(K 0 (A)) = n and be the maximal open set of all θ having no θ-semistable modules other that the zero object. Then a dimension n connected component C of R is called a chamber and this partition of R n is known as the wall and chamber structure of A.
One of the main objectives of [7] was to describe the wall and chamber structure of an algebra using τ -tilting theory. This was partially achieved, when they determined that every τ -tilting pairs induce a chamber and they described the walls surrounding that chamber. See [7,Corollary 3.18] In this context, one can use the results in this paper to go one step further, as shown in the following result. Proof. It was shown in [7, Corollary 3.18] that walls surrounding C (M,P ) are defined by the dimension vectors of θ r -(semi)stable modules, where 1 ≤ r ≤ n. Then the result follows directly from theorem 3.5.
It was already remarked in [7, Section 4] that the wall and chamber structure is dual to the exchange graph of τ -tilting pairs and that this duality induces a brick labelling of the exchange graph. Therefore, corollary 4.3 implies that we can label the arrows in the exchange lattice of τ -tilting pairs by c-vectors. Note that this labelling is coincides with the labelling studied in [12]. Proof. Suppose that (M, P ) and (M ′ , P ′ ) are two τ -tilting pairs such that one is a mutation of the other. Without loss of generality we can suppose that FacM ′ ⊂ FacM Then [7,Proposition 3.17] implies that the chambers C (M,P ) and C (M ′ ,P ′ ) inducing by them share a wall D(N ) for some module N . But, corollary 4.3 implies that N can be taken to be a brick, which is unique up to isomorphism. Therefore the dimension vector of N is a positive c-vector for the τ -tilting pair (M, P ) by proposition 3.2 and theorem 3.5. This finishes the proof.   Table 1: τ -tilting pairs with their corresponding positive c-vectors, bricks and torsion classes

A nice example
We finish illustrating theorem 3.5, theorem 3.13, corollary 4.3 and corollary 4.4 in the case of one particular algebra. Example 5.1. We consider the case of A, the path algebra of the quiver 1 α / / 2 β / / 3 modulo the ideal generated by the relation αβ. This algebra has 12 different τ -tilting pairs. We list them in Table 1 along with their corresponding G-matrices, C-matrices, positive c-vectors, B + (M,P ) and torsion classes, as proved in theorems 3.5 and theorem 3.13.
Also, one can see in Figure 1 the exchange graph of τ -tilting pairs in which every arrow is labeled with a positive c-vector, as showed in corollary 4.4.
Finally, in figure 2 one can see a representation of the wall and chamber structure of A in the style of [20,Example 4.0.2]. In this figure, each circle or arc correspond to the stereographic projection intersection of each wall with the unit sphere in R 3 . Note that in this projection the point at infinity correspond to the point (1, 1, 1) in R 3 . Moreover, the arcs and circles are labeled by the c-vectors which are perpendicular to them and each chamber is tagged by the τ -tilting pair that generates it.
Note that, given a τ -tilting pair (M, P ), the positive columns of the C-matrix C (M,P ) coincide with the non-convex arc surrounding C (M,P ) , while negative cvectors correspond to the convex ones.
Also, as we pointed out already, the exchange graph of τ -tilting pairs can be embedded into the wall and chamber structure of the algebra, where a mutation corresponds to crossing a wall. To illustrate this phenomenon, we have coloured the arrows in figure 1 with the colours of the walls that we are crossing at each mutation. Remark that arrows go from the chamber in which the given c-vector is positive to a chamber where it is negative.