March 09, 2025
\(F\)-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of \(F\)-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. \(E\)-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of \(F\)-invariant and \(E\)-invariant are totally different.
In this paper, we give a new proof of the mutation-invariance of \(F\)-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that \(F\)-invariant coincides with \(E\)-invariant on cluster monomials. We also give a proof of Reading’s conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of \(g\)-vectors.
Cluster algebras were introduced by Fomin and Zelevinsky [1] as a combinatorial approach to the upper global bases of quantum groups and to the theory of total positivity in algebraic groups. A cluster algebra of rank \(n\) (with trivial coefficients) is a \(\mathbb{Z}\)-subalgebra of a rational function field generated by a special set of generators called cluster variables, which are grouped into overlapping subsets of fixed size \(n\), called clusters. A seed \(t=({\boldsymbol{x}}_t,B_t)\) is a pair consisting of a cluster \({\boldsymbol{x}}_t=(x_{1;t},\ldots,x_{n;t})\) and a skew-symmetrizable integer matrix \(B_t=(b_{ij}^t)_{n\times n}\), called an exchange matrix. One can obtain new seeds from a given one by a procedure called mutation. The sets of cluster variables and clusters of a cluster algebra are determined by an initial seed and the iterative mutations. A cluster momomial is a monomial in variables from the same cluster. The remarkable Laurent phenomenon [1] says that the expansion of a cluster monomial \(u\) with repect to any cluster \({\boldsymbol{x}}_t\) is a Laurent polynomial.
The notions of \(g\)-vectors and \(F\)-polynomials associated to cluster monomials are introduced in [2]. To each cluster monomial \(u\) and any initial seed \(t=({\boldsymbol{x}}_t,B_t)\), one has the \(g\)-vector \(g_u^t\in\mathbb{Z}^n\) and \(F\)-polynomial \(F_u^t(y_1,\ldots,y_n)\in\mathbb{Z}[y_1,\ldots,y_n]\) with respect to seed \(t=({\boldsymbol{x}}_t,B_t)\). It is known from [3] that the \(F\)-polynomial \(F_u^t\) has non-negative coefficients and constant term \(1\). Fomin and Zelevinsky [2] proved that the Laurent expansion of \(u\) with respect to seed \(t\) can be written as: \[\begin{align} \label{eqn:u-expan} u={\boldsymbol{x}}_t^{{\boldsymbol{g}}_u^t}F_u^t(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}), \end{align}\tag{1}\] where \(\hat{y}_{k;t}={\boldsymbol{x}}_t^{B_t{\boldsymbol{e}}_k}\) and \({\boldsymbol{e}}_k\) is the \(k\)th column of \(I_n\). We call 1 the canonical expression of \(u\) with respect to seed \(t\).
The theory of quivers with potentials, introduced by Derksen, Weyman and Zelevinsky [4], [5], is a powerful tool to study skew-symmetric cluster algebras. The important notions in this theory include quivers with potentials and their mutations, decorated representations of quivers with potentials and their mutations. Various of the fundamental concepts in cluster algebras, such as seeds, mutations, cluster monomials, \(g\)-vectors and \(F\)-polynomials were categorified in the theory of quivers with potentials.
For any two decorated representations \(\mathcal{M}=(M,V)\) and \(\mathcal{N}=(N,W)\) of a quiver with potential \((Q,W)\), Derksen, Weyman and Zelevinsky [5] defined an integer \(E^{\rm inj}(\mathcal{M},\mathcal{N})\) by \[\begin{align} E^{\rm inj}(\mathcal{M},\mathcal{N}):=\dim \operatorname{\mathsf{Hom}}_{(Q,W)}(M,N)+\sum_{i=1}^nd_i(\mathcal{M})g_i(\mathcal{N}),\nonumber \end{align}\] where \(d_i(\mathcal{M})\) is the \(i\)th component of the dimension vector of \(M\) (the undecorated part of \(\mathcal{M}\)) and \(g_i(\mathcal{N})\) is the \(i\)th component of the \(g\)-vector of \(\mathcal{N}\) defined in [5]. The integer \(E^{\rm inj}(\mathcal{M},\mathcal{N})\) can be interpreted as the dimension of certain morphism space (see [5]), and thus \(E^{\rm inj}(\mathcal{M},\mathcal{N})\in\mathbb{Z}_{\geq 0}\).
The \(E\)-invariant \(E^{\rm sym}(\mathcal{M},\mathcal{N})\) associated to \(\mathcal{M}\) and \(\mathcal{N}\) is defined to be the symmetrized sum \[E^{\rm sym}(\mathcal{M},\mathcal{N})=E^{\rm inj}(\mathcal{M},\mathcal{N})+E^{\rm inj}(\mathcal{N},\mathcal{M})\geq 0.\] Derksen, Weyman and Zelevinsky [5] gave a recurrence relations for \(E^{\rm inj}(\mathcal{M},\mathcal{N})\) under mutations and thus they proved that the integer \(E^{\rm sym}(\mathcal{M},\mathcal{N})\) is mutation-invariant in the sense that \[E^{\rm sym}(\mathcal{M},\mathcal{N})=E^{\rm sym}(\mu_k(\mathcal{M}),\mu_k(\mathcal{N})),\] where \(\mu_k (\mathcal{M})\) and \(\mu_k(\mathcal{N})\) are decorated representations of the new quiver with potential \(\mu_k(Q,W)\) obtained by mutation in direction \(k\).
In a previous work [6], the author introduced two invariants (tropical invariant & \(F\)-invariant) for a pair of good elements (e.g. cluster monomials) in cluster algebras. The tropical invariant \(\langle u,u' \rangle\) for a pair of good elements \(u\) and \(u'\) is defined using tropicalization. More precisely, it is proved that the good element \(u'\) determines a semifield homomorphism \[\beta_{u'}:\mathbb{Q}_{\rm sf}(x_{1;t_0},\ldots,x_{m;t_0})\rightarrow \mathbb{Z}^{\max},\] where \(\mathbb{Q}_{\rm sf}(x_{1;t_0},\ldots,x_{m;t_0})\) is the universal semifield generated by the initial cluster variables (frozen and unfrozen) and \(\mathbb{Z}^{\max}=(\mathbb{Z},\;+,\;\max)\) is the tropical semifield (see Section 2.2 for the definition of semifield). The tropical invariant \(\langle u,u' \rangle\) is given by \[\langle u,u' \rangle:=\beta_{u'}(u),\] and it is a coordinate-free invariant [6]. The \(F\)-invariant \((u\mid\mid u')_F\) is defined to be the symmetrized sum of the tropical invariants: \[(u\mid\mid u')_F=\langle u,u' \rangle+\langle u',u \rangle.\] [6] says that for each seed \(t\), the \(F\)-invariant \((u\mid\mid u')_F\) can be written down explicitly in terms of the \(g\)-vectors and \(F\)-polynomials of \(u\) and \(u'\) with respect to seed \(t\), which is of the form 2 . In order to accurately present 2 , we need the notion of tropical polynomials.
Given a non-zero polynomial \(F=\sum_{{\boldsymbol{v}}\in\mathbb{N}^n}c_{{\boldsymbol{v}}}{\boldsymbol{y}}^{{\boldsymbol{v}}}\in\mathbb{Z}[y_1,\ldots,y_n]\) and a vector \({\boldsymbol{r}}\in \mathbb{Z}^n\), we denote by \[F[{\boldsymbol{r}}]:=\max\{{\boldsymbol{v}}^T{\boldsymbol{r}}\mid c_{{\boldsymbol{v}}}\neq 0\}\in\mathbb{Z}.\] We call the map \(F[-]:\mathbb{Z}^n\rightarrow\mathbb{Z}\) a tropical polynomial.
Obviously, if \(F\) has constant term \(1\), then \(F[{\boldsymbol{r}}]\in\mathbb{Z}_{\geq 0}\) for any \({\boldsymbol{r}}\in\mathbb{Z}^n\). For example, if we take \(F=1+y_1+y_1y_2\in\mathbb{Z}[y_1,y_2]\) and \({\boldsymbol{r}}=\begin{bmatrix} -2\\1 \end{bmatrix}\), then \[\begin{align} F[{\boldsymbol{r}}]=\max\{ \begin{bmatrix} 0,0 \end{bmatrix}\begin{bmatrix} -2\\1 \end{bmatrix}, \begin{bmatrix} 1,0 \end{bmatrix}\begin{bmatrix} -2\\1 \end{bmatrix}, \begin{bmatrix} 1,1 \end{bmatrix}\begin{bmatrix} -2\\1 \end{bmatrix} \}=\max\{0,-2,-1\}=0. \nonumber \end{align}\]
Definition 1 (\(F\)-invariant). Let \(\mathcal{A}\) be a cluster algebra and \(S=diag(s_1,\ldots,s_n)\) a skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). Let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\), and let \[u={\boldsymbol{x}}_t^{{\boldsymbol{g}}_u^t}F_u^t(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}) \;\;\;\;\text{and}\;\;\; u'={\boldsymbol{x}}_t^{{\boldsymbol{g}}_{u'}^t}F_{u'}^t(\hat{y}_{1;t},\ldots,\hat{y}_{n;t})\] be the canonical expressions of \(u\) and \(u'\) with repect to a seed \(t\). We call the integer \[\begin{align} \label{eqn:def-F-inv} (u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}]+F_{u'}^t[S{\boldsymbol{g}}_{u}^t] \end{align}\tag{2}\] the \(F\)-invariant between \(u\) and \(u'\), where \(t\) is any seed of \(\mathcal{A}\).
Remark 1. A priori, it is not obvious why the integer \((u\mid\mid u')_F\) is independent of the choice of the seed \(t\), i.e., why the \(F\)-invariant is invariant under the initial seed mutations. The viewpoint in [6] is that the \(F\)-invariant is the symmetrized sum of tropical invariants and the tropical invariants are invariant under the initial seed mutations [6]. In this paper, we will give a new proof of the mutation-invariance of \(F\)-invariant following Derksen-Weyman-Zelevinsky’s strategy [5] in proving the mutation-invariance of \(E\)-invariant.
Remark 2. For any two cluster monomials \(u\) and \(u'\), since the \(F\)-polynomials \(F_u^t\) and \(F_{u'}^t\) have constant term \(1\), we have \((u\mid\mid u')_F\in\mathbb{Z}_{\geq 0}\). It is proved in [6] that \((u\mid\mid u')_F=0\) if and only if the product \(uu'\) is still a cluster monomial.
Remark 3. The definition of \(F\)-invariant depends on a choice of a skew-symmetrizer \(S\). For this reason, we always fix a skew-symmetrizer in this paper. For skew-symmetric cluster algebras, we always fix \(S=I_n\). Of course, we could also choose \(S=2I_n\), then the obtained \(F\)-invariant is rescaled by \(2\). Different choices of skew-symmetrizer make no effect on the essential properties of \(F\)-invariant.
In this paper, we give a new proof the mutation-invariance of \(F\)-invariant following the strategy [5] used by Derksen, Weyman and Zelevinsky in proving the mutation-invariance of \(E\)-invariant (see Theorem 11). As a result, we prove that \(F\)-invariant and \(E\)-invariant are the same on cluster monomials (see Theorem 15). We also give a proof of Reading’s conjecture [7], which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of \(g\)-vectors (see Theorem 18).
Remark 4. The result that \(F\)-invariant and \(E\)-invariant are the same is well-known to some experts who know \(E\)-invariant very well. The reason is that both \(F\)-invariant and \(E\)-invariant are related to the components of \(f\)-vectors in cluster algebras. See Remark 12 and Remark 14. Nevertheless, the coincidence of \(F\)-invariant and \(E\)-invariant is not widely known for many people and this is a key reason for me to write this paper.
Remark 5. In the previous work [6], the author works on cluster algebras of full rank in which case one more data (i.e., compatible pair [8]) is available to us. This condition plays an important role in defining tropical invariant [6] for a pair of good elements. The full rank condition is important if one wants to have well-defined (extended) \(g\)-vectors and \(F\)-polynomials for good elements outside cluster monomials. Thanks to [6], we know that the \(F\)-invariant actually only depends on \(F\)-polynomials and \(g\)-vectors (namely, the principal part of the extended \(g\)-vectors). So when we work on cluster monomials, we can simply work on cluster algebras with trivial coefficients.
Fix a positive integer \(n\) and denote by \([1,n]:=\{1,2,\ldots,n\}\). An \(n\times n\) integer matrix \(B\) is said to be skew-symmetrizable, if there exists a diagonal integer matrix \(S=diag(s_1,\ldots,s_n)\) with \(s_i>0\) (\(i\in[1,n]\)) such that \(SB\) is skew-symmetric. Such a diagonal matrix \(S\) is called a skew-symmetrizer of \(B\).
Definition 2 (Matrix mutation). Let \(B=(b_{ij})_{n\times n}\) be a skew-symmetrizable matrix. The mutation of \(B\) in direction \(k\in[1,n]\) is defined to be the new integer matrix \(\mu_k(B)=B'=(b_{ij}')\) given by \[\begin{align} \label{eqn:b-mutation} b_{ij}^\prime&=&\begin{cases}-b_{ij}, & \text{if}\;i=k\;\text{or}\;j=k,\\ b_{ij}+[b_{ik}]_+[b_{kj}]_+-[-b_{ik}]_+[-b_{kj}]_+,&\text{otherwise},\end{cases} \end{align}\tag{3}\] where \([a]_+:=\max\{a,0\}\) for any \(a\in\mathbb{R}\).
It is easy to check that
The new matrix \(B'=\mu_k(B)\) is still skew-symmetrizable, and \(B, B'\) share the same skew-symmetrizers;
The mutation \(\mu_k\) is an involution, that is, \(\mu_k^2(B)=B\).
Let \(\mathbb{F}\) be a field of rational functions in \(n\) independent variables. A seed (with trivial coefficients) in \(\mathbb{F}\) is a pair \(({\boldsymbol{z}},B)\) such that
\({\boldsymbol{z}}=(z_1,\ldots,z_n)\) is an ordered set of free generators of \(\mathbb{F}\) over \(\mathbb{Q}\);
\(B=(b_{ij})\) is an \(n\times n\) skew-symmetrizable matrix.
Given a seed \(({\boldsymbol{z}},B)\) in \(\mathbb{F}\), we can perform two types of mutations: cluster seed mutations (or simply, mutations) and \(Y\)-seed mutations.
Definition 3 (Mutation and \(Y\)-seed mutation). (i) Let \(({\boldsymbol{x}},B)\) be a seed in \(\mathbb{F}\). The cluster seed mutation (or simply mutation) of \(({\boldsymbol{x}},B)\) in direction \(k\in[1,n]\) is the new seed \(({\boldsymbol{x}}', B')=\mu_k({\boldsymbol{x}}, B)\) given by \(B'=\mu_k(B)\) and \[\begin{align} \label{eqn:x-mutation} x_i^\prime=\begin{cases}x_i,& i\neq k,\\ x_k^{-1}\cdot (\prod_{j=1}^nx_j^{[b_{jk}]_+}+\prod_{j=1}^nx_j^{[-b_{jk}]_+}),&i= k.\end{cases} \end{align}\tag{4}\]
(ii) Let \(({\boldsymbol{y}}, B)\) be a seed in \(\mathbb{F}\). The \(Y\)-seed mutation of \(({\boldsymbol{y}}, B)\) in direction \(k\in[1,n]\) is the new seed \(({\boldsymbol{y}}', B'):=\hat{\mu}_k({\boldsymbol{y}}, B)\) given by \(B'=\mu_k(B)\) and \[\begin{align} \label{eqn:y-mutation} y_i^\prime&=&\begin{cases}y_k^{-1}, &i=k, \\ y_iy_k^{[b_{ki}]_+}(1+y_k)^{- b_{ki}},&i\neq k.\end{cases} \end{align}\tag{5}\]
It can be checked both \(\mu_k\) and \(\hat{\mu}_k\) are involutions. Let \(\mathbb{T}_n\) denote the \(n\)-regular tree. We label the edges of \(\mathbb{T}_n\) by \(1,\ldots, n\) such that the \(n\) different edges adjacent to the same vertex of \(\mathbb{T}_n\) receive different labels.
Definition 4 (Cluster pattern, \(Y\)-pattern and cluster ensemble). (i) A cluster pattern \[\mathcal{S}_X=\{({\boldsymbol{x}}_t, B_t)\mid t\in \mathbb{T}_n\}\] is an assignment of a seed \(({\boldsymbol{x}}_t, B_t)\) in \(\mathbb{F}\) to every vertex \(t\) of \(\mathbb{T}_n\) such that \(({\boldsymbol{x}}_{t'}, B_{t'})=\mu_k({\boldsymbol{x}}_t, B_t)\) whenever
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\).
(ii) A \(Y\)-pattern \(\mathcal{S}_Y=\{({\boldsymbol{y}}_t, B_t)\mid t\in \mathbb{T}_n\}\) is an assignment of a seed \(({\boldsymbol{y}}_t, B_t)\) in \(\mathbb{F}\) to every vertex \(t\) of \(\mathbb{T}_n\) such that \(({\boldsymbol{y}}_{t'}, B_{t'})=\hat{\mu}_k({\boldsymbol{y}}_t, B_t)\) whenever
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\).
(iii) Let \(\mathcal{S}_X=\{({\boldsymbol{x}}_t,B_t)\mid t\in\mathbb{T}_n\}\) be a cluster pattern, and let \(\mathcal{S}_Y=\{({\boldsymbol{y}}_t, \widehat B_t)\mid t\in\mathbb{T}_n\}\) be a \(Y\)-pattern. The pair \((\mathcal{S}_X, \mathcal{S}_Y)\) is called a cluster ensemble if \(B_t=\widehat B_t\) for any vertex \(t\in\mathbb{T}_n\).
We usually denote \({\boldsymbol{y}}_t=(y_{1;t},\ldots,y_{n;t}),\;{\boldsymbol{x}}_t=(x_{1;t},\ldots,x_{n;t})\) and \(B_t=(b_{ij}^t)\) and call them \(Y\)-cluster, cluster, exchange matrix at the vertex \(t\in\mathbb{T}_n\) respectively. Elements in \(Y\)-clusters are called \(y\)-variables and elements in clusters are called cluster variables.
The cluster algebra \(\mathcal{A}\) associated to a cluster pattern \(\mathcal{S}_X=\{({\boldsymbol{x}}_t, B_t)\mid t\in \mathbb{T}_n\}\) is the \(\mathbb{Z}\)-subalgebra of \(\mathbb{F}\) given by \[\mathcal{A}=\mathbb{Z}[x_{1;t},\ldots,x_{n;t}\mid t\in\mathbb{T}_n].\]
Theorem 6 ([1], Laurent phenomenon). Let \(({\boldsymbol{x}}_{t_0}, B_{t_0})\) be a seed of \(\mathcal{A}\). Then any cluster variable \(x_{k;t}\) can be written as a Laurent polynomial in \(\mathbb{Z}[x_{1;t_0}^{\pm 1},\ldots,x_{n;t_0}^{\pm 1}]\).
Recall that a semifield \((\mathbb{P},\cdot,\oplus)\) is an abelian multiplicative group \((\mathbb{P},\cdot)\) endowed with a binary operation \(\oplus\) of addition which is commutative, associative, and distributive with respect to the multiplication in \(\mathbb{P}\). For example, \[\mathbb{Z}^{\rm max}:=(\mathbb{Z},\;+,\; \max\{-,-\})\] is a semifield.
Let \(\mathbb{F}=\mathbb{Q}(y_1,\ldots,y_n)\) and denote by \(\mathbb{F}_{>0}:=\mathbb{Q}_{\rm sf}(y_1, \ldots, y_n)\) the set of all non-zero rational functions in \(y_1, \ldots, y_n\) that have subtraction free expressions. The set \(\mathbb{F}_{>0}\) is a semifield with respect to the usual operations of multiplication and addition. It is called an universal semifield.
Cluster algebras \(\mathcal{A}_{\rm prin}\) with principal coefficients at vertex \(t_0\in\mathbb{T}_n\) were introduced in [2]. For the precise definition, we refer to [2]. Cluster variables in \(\mathcal{A}_{\rm prin}\) are denoted by \(\overline{x}_{k;t}\), where \(k=1,\ldots, n\) and \(t\in\mathbb{T}_n\). By the Laurent phenomenon, \(\mathcal{A}_{\rm prin}\) is a subalgebra of \(\mathbb{Z}[y_1,\ldots,y_n][\overline{x}_{1;t_0}^{\pm 1},\ldots, \overline{x}_{n;t_0}^{\pm 1}]\).
Proposition 7. [2] Each cluster variable \(\overline{x}_{k;t}\) of \(\mathcal{A}_{\rm prin}\) is homogeneous with respect to \(\mathbb{Z}^n\)-grading in \(\mathbb{Z}[y_1,\ldots,y_n][\overline{x}_{1;t_0}^{\pm 1},\ldots, \overline{x}_{n;t_0}^{\pm 1}]\) given by \[\deg(\overline{x}_{i;t_0})={\boldsymbol{e}}_i\;\;\;\;\text{and}\;\;\;\deg(y_i)=-B_{t_0}{\boldsymbol{e}}_i,\] where \({\boldsymbol{e}}_i\) is the \(i\)th column of \(I_n\) and \(i=1,\ldots,n\).
Definition 5 (\(g\)-vector and \(F\)-polynomial). Let \(\mathcal{A}_{\rm prin}\) be a cluster algebra with principal coefficients at \(t_0\) and \(\overline{x}_{k;t}\) a cluster variable of \(\mathcal{A}_{\rm prin}\).
(i) The integer vector \({\boldsymbol{g}}_{k;t}^{B_{t_0};t_0}:=\deg(\overline{x}_{k;t})\) is called the \(g\)-vector of \({\overline{x}}_{k;t}\), which only depends on \((B_{t_0},t_0,k,t)\).
(ii) Writing \(\overline{x}_{k;t}\) as a Laurent polynomial in \(\mathbb{Z}[y_1,\ldots,y_n][x_{1;t_0}^{\pm1},\ldots,x_{n;t_0}^{\pm1}]\). The polynomial \[F_{k;t}^{B_{t_0};t_0}(y_1,\ldots,y_n):=\overline{x}_{k;t}\mid _{\overline{x}_{1;t_0}=\ldots=\overline{x}_{n;t_0}=1}\in\mathbb{Z}[y_1,\ldots,y_n],\] which only depends on \((B_{t_0},t_0,k,t)\), is called the \(F\)-polynomial of \(\overline{x}_{k;t}\).
Theorem 8. [3] The following statements hold.
(Row sign-coherence) For each \(i\in[1,n]\), the \(i\)th components of the \(g\)-vectors \({\boldsymbol{g}}_{1;t}^{B_{t_0};t_0},\ldots,{\boldsymbol{g}}_{n;t}^{B_{t_0};t_0}\) are simultaneously non-negative or simultaneously non-positive.
The \(F\)-polynomial \(F_{k;t}^{B_{t_0};t_0}(y_1,\ldots,y_n)\) is a polynomial in \(\mathbb{Z}_{\geq 0}[y_1,\ldots,y_n]\) with constant term \(1\).
Let \(\mathcal{A}_{\rm prin}\) be the cluster algebra with principal coefficients at vertex \(t_0\) and \(\mathcal{A}\) the cluster algebra with trivial coefficients. Suppose that \(\mathcal{A}_{\rm prin}\) and \(\mathcal{A}\) have the same initial exchange matrix at the vertex \(t_0\). Then by the separation formula [2], each cluster variable \(x_{k;t}\) in \(\mathcal{A}\) has an expression in terms of the \(g\)-vector and \(F\)-polynomial of \(\overline{x}_{k;t}\):
\[\begin{align} \label{eqn:x-gF} x_{k;t}={\boldsymbol{x}}_{t_0}^{{\boldsymbol{g}}_{k;t}^{B_{t_0};t_0}}F_{k;t}^{B_{t_0};t_0}(\hat{y}_{1;t_0},\ldots,\hat{y}_{n;t_0}), \end{align}\tag{6}\] where \(\hat{y}_{k;t_0}={\boldsymbol{x}}_{t_0}^{B_{t_0}{\boldsymbol{e}}_k}\) and \({\boldsymbol{e}}_k\) is the \(k\)th column of \(I_n\). In particular, this implies the correspondence \(\phi: \overline{x}_{k;t}\mapsto x_{k;t}\) is a well-defined map from the cluster variables of \(\mathcal{A}_{\rm prin}\) to those of \(\mathcal{A}\).
Proposition 9. [9] Keep the above setting. The map \(\phi: \overline{x}_{k;t}\mapsto x_{k;t}\) is a bijection.
Thanks to the above proposition, the correspondences \(x_{k;t}\mapsto {\boldsymbol{g}}_{k;t}^{B_{t_0};t_0}\) and \(x_{k;t}\mapsto F_{k;t}^{B_{t_0};t_0}\) are well-defined. Thus, we also call \({\boldsymbol{g}}_{k;t}^{B_{t_0};t_0}\) and \(F_{k;t}^{B_{t_0};t_0}\) the \(g\)-vector and \(F\)-polynomial of \(x_{k;t}\) with respect to vertex \(t_0\).
Recall that a cluster monomial in \(\mathcal{A}\) is a monomial in cluster variables from the same cluster. Let \(u\) be a cluster monomial, say \(u={\boldsymbol{x}}_t^{\boldsymbol{v}}\), where \({\boldsymbol{v}}=(v_1,\ldots,v_n)^T\in\mathbb{N}^n\). The integer vector and the polynomial below \[\sum_{k=1}^nv_k{\boldsymbol{g}}_{k;t}^{B_{t_0};t_0},\;\;\;\prod_{k=1}^n(F_{k;t}^{B_{t_0};t_0})^{v_i}\] are called the \(g\)-vector and \(F\)-polynomial of \(u\) with respect to vertex \(t_0\) and denoted by \({\boldsymbol{g}}_u^{t_0}\) and \(F_u^{t_0}\) respectively. By equality 6 , we obtain \[\begin{align} u={\boldsymbol{x}}_{t_0}^{{\boldsymbol{g}}_u^{t_0}}F_{u}^{t_0}(\hat{y}_{1;t_0},\ldots,\hat{y}_{n;t_0}).\nonumber \end{align}\]
By varying the rooted vertex \(t_0\in\mathbb{T}_n\), we have a family of integer vectors \(\{{\boldsymbol{g}}_u^w\mid w\in\mathbb{T}_n\}\) in \(\mathbb{Z}^n\) and a family of polynomials \(\{F_u^w\mid w\in\mathbb{T}_n\}\) in \(\mathbb{Z}[y_1,\ldots,y_n]\) for each cluster monomial \(u\) of \(\mathcal{A}\). Meanwhile, for each vertex \(w\in\mathbb{T}_n\), the Laurent expansion of \(u\) with respect to the seed at vertex \(w\) takes the form: \[\begin{align} \label{eqn:u-gF-t950} u={\boldsymbol{x}}_{w}^{{\boldsymbol{g}}_u^{w}}F_{u}^{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w}). \end{align}\tag{7}\] where \(\hat{y}_{k;w}={\boldsymbol{x}}_w^{B_w{\boldsymbol{e}}_k}\) and \({\boldsymbol{e}}_k\) is the \(k\)th column of \(I_n\).
In this subsection, we give a new proof of the mutation-invariance of \(F\)-invariant following the strategy used by Derksen, Weyman and Zelevinsky [5].
Let \(\mathcal{A}\) be a cluster algebra. Recall that we have a family of integer vectors \(\{{\boldsymbol{g}}_u^w\mid w\in\mathbb{T}_n\}\) in \(\mathbb{Z}^n\) and a family of polynomials \(\{F_u^w\mid w\in\mathbb{T}_n\}\) in \(\mathbb{Z}[y_1,\ldots,y_n]\) for each cluster monomial \(u\) of \(\mathcal{A}\). Now we summarize the recurrence relations under the initial seed mutations for the vectors in \(\{{\boldsymbol{g}}_u^w\mid w\in\mathbb{T}_n\}\) and for the polynomials in \(\{F_u^w\mid w\in\mathbb{T}_n\}\).
Lemma 1. [10] Let \(u\) be a cluster monomial of \(\mathcal{A}\) and
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
an edge in \(\mathbb{T}_n\). Then the \(g\)-vector \({\boldsymbol{g}}_{u}^{t'}=(g_{1;u}^{t'},\ldots,g_{n;u}^{t'})^T\) of \(u\) with respect to vertex \(t'\) is obtained from the \(g\)-vector \({\boldsymbol{g}}_{u}^{t}=(g_{1;u}^{t},\ldots,g_{n;u}^{t})^T\) of \(u\) with respect to vertex \(t\) by the following relations: \[\begin{align} \label{eqn:g-mutation} g_{i;u}^{t'}= \begin{cases}-g_{k;u}^t, &{i=k,}\\ g_{i;u}^t+[-b_{ik}^t]_+g_{k;u}^t+b_{ik}^t[g_{k;u}^t]_+,&{i\neq k}. \end{cases} \end{align}\tag{8}\]
Proof. We first consider the case that \(u\) is a cluster variable. Then by [10], the sign-coherence of \(c\)-vectors [GHKK18, Corollary 5.5] implies \[\begin{align} g_{i;u}^{t'}= \begin{cases}-g_{k;u}^t, &{i=k,}\\ g_{i;u}^t+[b_{ik}^t]_+g_{k;u}^t-b_{ik}^t{\rm min}\{g_{k;u}^t,0\},&{i\neq k},\nonumber \end{cases} \end{align}\] which can be rewritten as equality 8 using the relations \(\min\{g_{k;u}^t,0\}=-[-g_{k;u}^t]_+\) and \[[b_{ik}^t]_+g_{k;u}^t-[-b_{ik}^t]_+g_{k;u}^t=b_{ik}^tg_{k;u}^t=b_{ik}^t[g_{k;u}^t]_+-b_{ik}^t [-g_{k;u}^t]_+.\]
Now let us assume that \(u=\prod_{k=1}^n x_{k;w}^{v_i}\) is a cluster monomial in a seed \(({\boldsymbol{x}}_w,B_w)\). Then the required result follows from the case for cluster variables and Theorem 8 (i), which says that the \(j\)th components of the \(g\)-vectors \({\boldsymbol{g}}_{x_{1;w}}^t,\ldots, {\boldsymbol{g}}_{x_{n;w}}^t\) are simultaneously non-negative or simultaneously non-positive for each \(j\in[1,n]\). ◻
Corollary 1. Let \(S=diag(s_1,\ldots,s_n)\) be a skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). Let \(u\) be a cluster monomial of \(\mathcal{A}\) and \(\{{\boldsymbol{g}}_u^t\in\mathbb{Z}^n\mid t\in\mathbb{T}_n\}\) the collection of \(g\)-vectors of \(u\) with respect to vertices of \(\mathbb{T}_n\). Then the collection \(\{{\boldsymbol{q}}_u^t:=S{\boldsymbol{g}}_u^t\mid t\in\mathbb{T}_n\}\) of vectors satisfies the following recurrence relations: \[\begin{align} \label{eqn:q-mutation} q_{i;u}^{t'}= \begin{cases}-q_{k;u}^t, &{i=k,}\\ q_{i;u}^t+ [b_{ki}^t]_+q_{k;u}^t-b_{ki}^t[q_{k;u}^t]_+,&{i\neq k}. \end{cases} \end{align}\tag{9}\] for any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\).
Proof. By Lemma 1, we get \[\begin{align} q_{i;u}^{t'}=s_ig_{i;u}^{t'}= \begin{cases}-s_kg_{k;u}^t=-q_{k;u}^t, &{i=k,}\\ s_i(g_{i;u}^t+[-b_{ik}^t]_+g_{k;u}^t+b_{ik}^t[g_{k;u}^t]_+),&{i\neq k}.\nonumber \end{cases} \end{align}\] Since \(SB_t\) is skew-symmetric, we have \(s_ib_{ik}^t=-s_kb_{ki}^t\) and \[s_i[-b_{ik}^t]_+=[-s_ib_{ik}^t]_+=[s_kb_{ki}^t]_+=[b_{ki}^t]_+s_k.\] Thus \[\begin{align} s_i(g_{i;u}^t+[-b_{ik}^t]_+g_{k;u}^t+b_{ik}^t[g_{k;u}^t]_+)&=&s_ig_{i;u}^t+ [b_{ki}^t]_+(s_kg_{k;u}^t)-b_{ki}^t[s_kg_{k;u}^t]_+\nonumber\\ &=&q_{i;u}^t+ [b_{ki}^t]_+q_{k;u}^t-b_{ki}^t[q_{k;u}^t]_+.\nonumber \end{align}\] Then the result follows. ◻
Setting: Let \((\mathcal{S}_X, \mathcal{S}_Y)\) be a cluster ensemble, that is, \(\mathcal{S}_X\) and \(\mathcal{S}_Y\) have the same exchange matrix at each vertex \(t\in\mathbb{T}_n\). Let \(\mathcal{A}\) be the cluster algebra associated to \(\mathcal{S}_X\). Denote by \({\boldsymbol{y}}_t=(y_{1;t},\ldots,y_{n;t})\) the \(Y\)-cluster of \(\mathcal{S}_Y\) at \(t\in\mathbb{T}_n\). Denote by \(\mathbb{Q}_{\rm sf}(y_{1;t},\ldots,y_{n;t})\) the universal semifield generated by \(y\)-variables in \({\boldsymbol{y}}_t\). Since the mutation relations in 5 for \(Y\)-seeds are subtraction-free, we have \[\mathbb{Q}_{\rm sf}(y_{1;t},\ldots,y_{n;t})=\mathbb{Q}_{\rm sf}(y_{1;t'},\ldots,y_{n;t'})\] for any two vertices \(t,t'\in \mathbb{T}_n\). This common semifield is denoted by \[\mathbb{F}_{>0}^Y=(\mathbb{F}_{>0}^Y,\;\{{\boldsymbol{y}}_t\}_{t\in\mathbb{T}_n}),\] which admits a collection \(\{{\boldsymbol{y}}_t\}_{t\in\mathbb{T}_n}\) of ordered free generators determined by the \(Y\)-pattern \(\mathcal{S}_Y\).
Proposition 10. Keep the above setting. Let \(S=diag(s_1,\ldots,s_n)\) be a skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). Let \(u\) be a cluster monomial of \(\mathcal{A}\) and \(\{{\boldsymbol{g}}_u^t\in\mathbb{Z}^n\mid t\in\mathbb{T}_n\}\) the collection of \(g\)-vectors of \(u\) with respect to vertices of \(\mathbb{T}_n\). Then there exists a (unique) semifield homomorphism \[\rho_u: \mathbb{F}_{>0}^Y\rightarrow \mathbb{Z}^{\rm max}\] such that \(\rho_u({\boldsymbol{y}}_t)=(S{\boldsymbol{g}}_u^t)^T\) for any vertex \(t\in\mathbb{T}_n\).
Proof. Denote by \({\boldsymbol{q}}_u^t=S{\boldsymbol{g}}_u^t\) for \(t\in\mathbb{T}_n\). Let \(w\) be a fixed vertex of \(\mathbb{T}_n\). We define a semifield homomorphism \(\rho_u: \mathbb{F}_{>0}^Y\rightarrow \mathbb{Z}^{\rm max}\) by the condition \(\rho_u({\boldsymbol{y}}_{w})=({\boldsymbol{q}}_u^w)^T=(S{\boldsymbol{g}}_u^{w})^T\), i.e., \(\rho_u(y_{i;w})=q_{i;u}^w\) for \(i\in[1,n]\). It remains to check \[\rho_u({\boldsymbol{y}}_t)=({\boldsymbol{q}}_u^t)^T=(S{\boldsymbol{g}}_u^t)^T\] for any other vertices \(t\in\mathbb{T}_n\).
Since any \(Y\)-seed \(({\boldsymbol{y}}_t,B_t)\) can be obtained from the \(Y\)-seed \(({\boldsymbol{y}}_{w},B_{w})\) by a sequence of \(Y\)-seed mutations, it is enough to check that the required condition remains true under one step \(Y\)-seed mutation. Let us consider \(({\boldsymbol{y}}_{w'},B_{w'})=\hat{\mu}_k({\boldsymbol{y}}_{w},B_{w})\). Then by 5 , we have \[\begin{align} y_{i;w'}=\begin{cases}y_{k;w}^{-1}, &i=k, \\ y_{i;w}y_{k;w}^{[b_{ki}^{w}]_+}(1+y_{k;w})^{- b_{ki}^{w}},&i\neq k.\end{cases}\nonumber \end{align}\] By applying the semifield homomorphism \(\rho_u\), we have \[\begin{align} \label{eqn:y0-mutation} \rho_u(y_{i;w'})&=&\begin{cases}\rho_u(y_{k;w}^{-1}), &i=k, \\ \rho_u(y_{i;w})+[b_{ki}^{w}]_+ \rho_u(y_{k;w})-b_{ki}^{w}\cdot\max\{0, \;\rho_u(y_{k;w})\} ,&i\neq k.\end{cases}\nonumber\\ &=&\begin{cases}-q_{k;u}^w, &i=k, \\ q_{i;u}^w+[b_{ki}^{w}]_+ q_{k;u}^w-b_{ki}^{w}[q_{k;u}^w]_+,&i\neq k.\end{cases}\nonumber \end{align}\tag{10}\] By comparing with 9 , we have \(\rho_u(y_{i;w'})=q_{i;u}^{w'}\) and thus \(\rho_u({\boldsymbol{y}}_{w'})=({\boldsymbol{q}}_u^{w'})^T=(S{\boldsymbol{g}}_u^{w'})^T\). ◻
Lemma 2. [2], [5]Keep the setting as before. Let \(u\) be a cluster monomial of \(\mathcal{A}\). Let \[\{{\boldsymbol{g}}_u^t\in\mathbb{Z}^n\mid t\in\mathbb{T}_n\}\;\;\;\;\text{and}\;\;\;\{F_u^t\in\mathbb{Z}[y_1,\ldots,y_n]\mid t\in\mathbb{T}_n\}\] be the collections of \(g\)-vectors and \(F\)-polynomials of \(u\) with respect to vertices of \(\mathbb{T}_n\). Then for any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\), the \(F\)-polynomials \(F_u^{t'}\) and \(F_u^t\) are related as follows: \[\begin{align} \label{eqn:F-mutation} (1+y_{k;t'})^{-[-g_{k;u}^{t'}]_+}F_u^{t'}(y_{1;t'},\ldots,y_{n;t'})= (1+y_{k;t})^{-[-g_{k;u}^{t}]_+}F_u^{t}(y_{1;t},\ldots,y_{n;t}). \end{align}\tag{11}\]
Proof. We first consider the case that \(u\) is a cluster variable. Then by [5] (or [2]), we have \[(1+y_{k;t'})^{h_k'}F_u^{t'}(y_{1;t'},\ldots,y_{n;t'})= (1+y_{k;t})^{h_k}F_u^{t}(y_{1;t},\ldots,y_{n;t})\] where \(h_k\) and \(h_k'\) are defined in [2]. It is conjectured in [2] that \[h_k=-[-g_{k;u}^t]_+\;\;\;\;\text{and}\;\;\;h_k'=-[-g_{k;u}^{t'}]_+=-[g_{k;u}^t]_+.\] This conjecture is confirmed in [5] for skew-symmetric cluster algebras and in [11] for skew-symmetrizable cluster algebras. Thus 11 holds for the case that \(u\) is a cluster variable.
Now let us assume that \(u=\prod_{k=1}^n x_{k;w}^{v_i}\) is a cluster monomial in a seed \(({\boldsymbol{x}}_w,B_w)\). Then the required result follows from the case for cluster variables and Theorem 8 (i), which says that the \(j\)th components of the \(g\)-vectors \({\boldsymbol{g}}_{x_{1;w}}^t,\ldots, {\boldsymbol{g}}_{x_{n;w}}^t\) (resp. \({\boldsymbol{g}}_{x_{1;w}}^{t'},\ldots, {\boldsymbol{g}}_{x_{n;w}}^{t'}\)) are simultaneously non-negative or simultaneously non-positive for each \(j\in[1,n]\). ◻
Recall that given a non-zero polynomial \(F=\sum_{{\boldsymbol{v}}\in\mathbb{N}^n}c_{{\boldsymbol{v}}}{\boldsymbol{y}}^{{\boldsymbol{v}}}\in\mathbb{Z}[y_1,\ldots,y_n]\) and a vector \({\boldsymbol{r}}\in \mathbb{Z}^n\), we denote by \[F[{\boldsymbol{r}}]:=\max\{{\boldsymbol{v}}^T{\boldsymbol{r}}\mid c_{{\boldsymbol{v}}}\neq 0\}\in\mathbb{Z}.\]
Theorem 11. Let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\) and \(S=diag(s_1,\ldots,s_n)\) a skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). Then the following statements hold.
For any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\), we have \[\begin{align} \label{eqn:f-inv} F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]-F_u^t[S{\boldsymbol{g}}_{u'}^t]=s_k([-g_{k;u}^{t'}]_+[-g_{k;u'}^{t}]_+-[-g_{k;u}^t]_+[-g_{k;u'}^{t'}]_+). \end{align}\tag{12}\]
For any two vertices \(t,t'\in\mathbb{T}_n\), we have \[\begin{align} \label{eqn:F-inv-mut} F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]=F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]+F_{u'}^{t'}[S{\boldsymbol{g}}_u^{t'}]. \end{align}\tag{13}\] In particular, the \(F\)-invariant \((u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]\) only depends on \(u\) and \(u'\), not on the choice of vertex \(t\in\mathbb{T}_n\).
Proof. (i) Since \(u'\) is a cluster monomial and by Proposition 10, there exists a semifield homomorphism \(\rho_{u'}:\mathbb{F}_{>0}^Y\rightarrow\mathbb{Z}^{\max}\) such that \(\rho_{u'}({\boldsymbol{y}}_w)=(S{\boldsymbol{g}}_{u'}^w)^T\) for any vertex \(w\in\mathbb{T}_n\). Thus for a polynomial \(F(y_{1;w},\ldots,y_{n;w})=\sum_{{\boldsymbol{v}}\in\mathbb{N}^n}c_{\boldsymbol{v}}{\boldsymbol{y}}_w^{\boldsymbol{v}}\in\mathbb{F}_{>0}^Y\), we have \[\rho_{u'}(F(y_{1;w},\ldots,y_{n;w}))=\max\{{\boldsymbol{v}}^T S{\boldsymbol{g}}_{u'}^w\mid c_{\boldsymbol{v}}\neq 0\}=F[S{\boldsymbol{g}}_{u'}^w].\] Since 11 can be viewed as an equality in \(\mathbb{F}_{>0}^Y\), we apply \(\rho_{u'}\) to 11 and get \[-[-g_{k;u}^{t'}]_+\cdot \max\{0,\;\rho_{u'}(y_{k;t'})\}+F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]=-[-g_{k;u}^t]_+\cdot \max\{0,\;\rho_{u'}(y_{k;t})\}+F_u^t[S{\boldsymbol{g}}_{u'}^t].\] Since \(\rho_{u'}(y_{k;t'})=s_kg_{k;u'}^{t'}\) and \(\rho_{u'}(y_{k;t})=s_kg_{k;u'}^t\), we obtain \[-[-g_{k;u}^{t'}]_+[s_kg_{k;u'}^{t'}]_++F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]=-[-g_{k;u}^t]_+[s_kg_{k;u'}^t]_++F_u^t[S{\boldsymbol{g}}_{u'}^t].\] Thus \[\begin{align} \label{eqn:f0-inv} F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]-F_u^t[S{\boldsymbol{g}}_{u'}^t]=s_k([-g_{k;u}^{t'}]_+[g_{k;u'}^{t'}]_+-[-g_{k;u}^t]_+[g_{k;u'}^t]_+). \end{align}\tag{14}\] By Lemma 1, we have \(g_{k;u}^{t'}=-g_{k;u}^t\) and \(g_{k;u'}^{t'}=-g_{k;u'}^t\). So we can rewrite 14 as the form in 12 .
(ii) It suffices to prove that 13 holds for any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\). So let us assume \(({\boldsymbol{x}}_{t'},B_{t'})=\mu_k({\boldsymbol{x}}_t,B_t)\). By applying (i) to the pair \((u,u')\) of cluster monomials, we have \[F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]-F_u^t[S{\boldsymbol{g}}_{u'}^t]=s_k([-g_{k;u}^{t'}]_+[-g_{k;u'}^{t}]_+-[-g_{k;u}^t]_+[-g_{k;u'}^{t'}]_+).\] By applying (i) to the pair \((u',u)\) of cluster monomials, we have \[F_{u'}^{t'}[S{\boldsymbol{g}}_{u}^{t'}]-F_{u'}^t[S{\boldsymbol{g}}_{u}^t]=s_k([-g_{k;u'}^{t'}]_+[-g_{k;u}^{t}]_+-[-g_{k;u'}^t]_+[-g_{k;u}^{t'}]_+).\] Then by taking the sum of the two equalities, we obtain \[F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]=F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]+F_{u'}^{t'}[S{\boldsymbol{g}}_u^{t'}].\] Then by induction, we can see that \(F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]\) is invariant for \(t\in\mathbb{T}_n\). In particular, the \(F\)-invariant \((u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]\) only depends on \(u\) and \(u'\), not on the choice of vertex \(t\in\mathbb{T}_n\). ◻
The following remark aims to make a comparison with Remark 14.
Remark 12. (i) If \(u\) is a cluster monomial in seed \(({\boldsymbol{x}}_t,B_t)\), then \(F_u^t=1\) and thus \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=0\) for any cluster monomial \(u'\).
(ii) If \(u'=x_{k;t}\), then \({\boldsymbol{g}}_{u'}^t={\boldsymbol{e}}_k\) and thus \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=F_u^t[S{\boldsymbol{e}}_k]=s_kf_k\), where \(f_k\) is the maximal degree of \(y_k\) in \(F_u^t\in\mathbb{Z}[y_1,\ldots,y_n]\). In this case, \((u\mid\mid u')_F=s_kf_k+0=s_kf_k\). If we are in the skew-symmetric case and choose \(S=I_n\), then we have \((u\mid\mid u')_F=f_k\).
(iii) Since the \(F\)-polynomials \(F_u^t\) and \(F_{u'}^t\) have constant term \(1\), we have \(F_u^t[{\boldsymbol{r}}],\; F_{u'}^t[{\boldsymbol{r}}]\in\mathbb{Z}_{\geq 0}\) for any \({\boldsymbol{r}}\in\mathbb{Z}^n\). In particular, \((u\mid\mid u')_F\in\mathbb{Z}_{\geq 0}\).
Remark 13. (i) From 12 , we can see that in general \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) is not invariant for \(t\in\mathbb{T}_n\). The \(F\)-invariant \((u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_{u}^t]\) provides a way to complete \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) to a mutation invariant. Actually, the tropical invariant \(\langle u, u'\rangle\) in [6] is another way to complete \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) to a mutation invariant, but one should introduce frozen variables so that the extended exchange matrices have full rank in which case one more data (i.e., compatible pair [8]) is available to us.
(ii) From the author’s viewpoint, it is a more intrinsic way to define \(F\)-invariant \((u\mid\mid u')_F\) as the symmetrized sum of two mutation invariants \[(u\mid\mid u')_F=\langle u, u'\rangle+\langle u', u\rangle,\] rather than define it as \((u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_u^t]\), which is a symmetrized sum of two non mutation invariants. It also should mention that both (extended) \(g\)-vectors and \(F\)-polynomials are more natural notions in the full rank setting. In the full rank case, \(F\)-polynomials and (extended) \(g\)-vectors can be also defined for some good elements (beyond cluster monomials) in cluster algebras.
The theory of quivers with potentials, introduced by Derksen, Weyman and Zelevinsky [4], [5], is a powerful tool to study skew-symmetric cluster algebras. The important notions in this theory include quivers with potentials and their mutations, decorated representations of quivers with potentials and their mutations. Various of the fundamental concepts in cluster algebras, such as seeds, mutations, cluster monomials, \(g\)-vectors and \(F\)-polynomials were categorified in the theory of quivers with potentials. We refer to [4], [5] for the basic definitions in this theory.
For any two decorated representations \(\mathcal{M}=(M,V)\) and \(\mathcal{N}=(N,W)\) of a quiver with potential \((Q,W)\), Derksen, Weyman and Zelevinsky [5] defined an integer \(E^{\rm inj}(\mathcal{M},\mathcal{N})\) by \[\begin{align} \label{eqn:e-inj} E^{\rm inj}(\mathcal{M},\mathcal{N}):=\dim \operatorname{\mathsf{Hom}}_{(Q,W)}(M,N)+\sum_{i=1}^nd_i(\mathcal{M})g_i(\mathcal{N}), \end{align}\tag{15}\] where \(d_i(\mathcal{M})\) is the \(i\)th component of the dimension vector of \(M\) (the undecorated part of \(\mathcal{M}\)) and \(g_i(\mathcal{N})\) is the \(i\)th component of the \(g\)-vector of \(\mathcal{N}\) defined in [5].
The \(E\)-invariant \(E^{\rm sym}(\mathcal{M},\mathcal{N})\) associated to \(\mathcal{M}\) and \(\mathcal{N}\) is defined to be the symmetrized sum \[E^{\rm sym}(\mathcal{M},\mathcal{N})=E^{\rm inj}(\mathcal{M},\mathcal{N})+E^{\rm inj}(\mathcal{N},\mathcal{M}).\]
Remark 14. (i) If \(\mathcal{M}=(M,V)\) is a negative decorated representation, i.e., \(M=0\), then it is clear from 15 that \(E^{\rm inj}(\mathcal{M},\mathcal{N})=0\) for any decorated representation \(\mathcal{N}=(N,W)\).
(ii) If \(\mathcal{N}=(N,W)\) is a negative simple decorated representation, that is, \(N=0\) and \(W=S_k\) for some simple module at vertex \(k\in[1,n]\), then the \(g\)-vector \({\boldsymbol{g}}(\mathcal{N})\) of \(\mathcal{N}\) is the \(k\)th column of \(I_n\). It is easy to see that \(E^{\rm inj}(\mathcal{M},\mathcal{N})=d_k(\mathcal{M})\), which is the \(k\)th component of the dimension vector of \(M\). In this case, \[E^{\rm sym}(\mathcal{M},\mathcal{N})=d_k(\mathcal{M})+0=d_k(\mathcal{M})\geq 0.\] Under the categorification of cluster algebras, this integer corresponds to the maximal degree of \(y_k\) in the \(F\)-polynomial associated to \(\mathcal{M}\).
(iii) The integer \(E^{\rm inj}(\mathcal{M},\mathcal{N})\) can be interpreted as the dimension of certain morphism space (see [5]), and thus \(E^{\rm inj}(\mathcal{M},\mathcal{N})\in\mathbb{Z}_{\geq 0}\) and \(E^{\rm sym}(\mathcal{M},\mathcal{N})\in\mathbb{Z}_{\geq 0}\).
Now we refer to [5] to recall some important ideas in the study of cluster algebras using the theory of quivers with potentials. Let \(\mathcal{A}\) be a skew-symmetric cluster algebra. For each seed \(({\boldsymbol{x}}_t,B_t)\) of \(\mathcal{A}\), one can associate it with a quiver with potential \((Q_t,W_t)\) such that \((Q_{t'},W_{t'})=\mu_k(Q_t,W_t)\) whenever \(({\boldsymbol{x}}_{t'},B_{t'})=\mu_k({\boldsymbol{x}}_t,B_t)\). For each cluster monomial \(u\) of \(\mathcal{A}\), there is a family \(\{\mathcal{M}_u^t\mid t\in\mathbb{T}_n\}\) of “negative-reachable" 1 decorated representations, where \(\mathcal{M}_u^t=(M_u^t,V_u^t)\) is a decorated representation of \((Q_t,W_t)\). The decorated representation \(\mathcal{M}_u^t=(M_u^t,V_u^t)\) is used to encode the information of the \(g\)-vector \({\boldsymbol{g}}_u^t\) and \(F\)-polynomial \(F_u^t\) of \(u\) with respect to vertex \(t\in\mathbb{T}_n\).
Lemma 3. [5] Let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\), and let \(\{\mathcal{M}_{u}^t\mid t\in\mathbb{T}_n\}\) and \(\{\mathcal{M}_{u'}^t\mid t\in\mathbb{T}_n\}\) be the families of decorated representations corresponding to \(u\) and \(u'\). Then the following statements hold.
(i) For any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\), \[\begin{align} \label{eqn:e-inv} E^{\rm inj}(\mathcal{M}_u^{t'},\mathcal{M}_{u'}^{t'})-E^{\rm inj}(\mathcal{M}_u^{t},\mathcal{M}_{u'}^{t})=[-g_{k;u}^{t'}]_+[-g_{k;u'}^{t}]_+-[-g_{k;u}^t]_+[-g_{k;u'}^{t'}]_+. \end{align}\tag{16}\]
(ii) For any two vertices \(t,t'\in\mathbb{T}_n\), \[\begin{align} \label{eqn:E-inv} E^{\rm sym}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)=E^{\rm sym}(\mathcal{M}_u^{t'},\mathcal{M}_{u'}^{t'}). \end{align}\tag{17}\] In particular, the \(E\)-invariant is mutation invariant under the initial seed mutations.
Proof. (i) By [5], we have \[\begin{align} \label{eqn:e0-inv} E^{\rm inj}(\mathcal{M}_u^{t'},\mathcal{M}_{u'}^{t'})-E^{\rm inj}(\mathcal{M}_u^{t}, \mathcal{M}_{u'}^{t})=h_k(\mathcal{M}_u^{t'})h_k(\mathcal{M}_{u'}^t)-h_k(\mathcal{M}_u^{t})h_k(\mathcal{M}_{u'}^{t'}), \end{align}\tag{18}\] where \(h_k(\mathcal{M}_u^{w})=-[-g_{k;u}^{w}]_+\) and \(h_k(\mathcal{M}_{u'}^{w})=-[-g_{k;u'}^{w}]_+\) for any vertex \(w\in\mathbb{T}_n\), by [5]. Then the result follows.
(ii) This follows from (i) and the definition of \(E\)-invariant. ◻
Theorem 15. Let \(\mathcal{A}\) be a skew-symmetric cluster algebra and let \(S=I_n\) be the fixed skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). Let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\), and let \(\{\mathcal{M}_{u}^t\mid t\in\mathbb{T}_n\}\) and \(\{\mathcal{M}_{u'}^t\mid t\in\mathbb{T}_n\}\) be the families of decorated representations corresponding to \(u\) and \(u'\). Then we have \[E^{\rm inj}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)=F_u^t[{\boldsymbol{g}}_{u'}^t]\;\;\;\;\text{and}\;\;\;E^{\rm sym}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)=F_u^t[{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[{\boldsymbol{g}}_{u}^t]=(u\mid\mid u')_F,\] for any vertex \(t\in\mathbb{T}_n\).
Proof. Since \(u\) is a cluster monomial of \(\mathcal{A}\), we can take a vertex \(w\in\mathbb{T}_n\) such that \(u\) is a cluster monomial in \({\boldsymbol{x}}_w\). In this case, the decorated representation \(\mathcal{M}_u^w=(M_u^w,V_u^w)\) is a negative decorated representation, that is, \(M_u^w=0\). Then by Remark 14 (i), we have \(E^{\rm inj}(\mathcal{M}_u^w,\mathcal{M}_{u'}^w)=0\).
Since \(u\) is a cluster monomial in \({\boldsymbol{x}}_w\), we have \(F_u^w(y_1,\ldots,y_n)=1\). Thus \(F_u^w[{\boldsymbol{r}}]=0\) for any \({\boldsymbol{r}}\in\mathbb{Z}^n\). In particular, we have \[F_u^w[{\boldsymbol{g}}_{u'}^w]=0=E^{\rm inj}(\mathcal{M}_u^w,\mathcal{M}_{u'}^w).\]
By Theorem 11 (i) and Lemma 3 (i), we know that \(E^{\rm inj}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)\) and \(F_u^t[{\boldsymbol{g}}_{u'}^t]\) satisfy the same recurrence relations for \(t\in\mathbb{T}_n\). Then by \(E^{\rm inj}(\mathcal{M}_u^w,\mathcal{M}_{u'}^w)=F_u^w[{\boldsymbol{g}}_{u'}^w]\), we obtain that \[\begin{align} \label{eqn:e-inj61f-inv} E^{\rm inj}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)=F_u^t[{\boldsymbol{g}}_{u'}^t] \end{align}\tag{19}\] for any vertex \(t\in\mathbb{T}_n\). Similarly, we have \(E^{\rm inj}(\mathcal{M}_{u'}^t,\mathcal{M}_{u}^t)=F_{u'}^t[{\boldsymbol{g}}_{u}^t]\) for any vertex \(t\in\mathbb{T}_n\). Thus \[E^{\rm sym}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)= E^{\rm inj}(\mathcal{M}_u^t,\mathcal{M}_{u'}^t)+ E^{\rm inj}(\mathcal{M}_{u'}^t,\mathcal{M}_{u}^t) =F_u^t[{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[{\boldsymbol{g}}_{u}^t]=(u\mid\mid u')_F\] holds for any vertex \(t\in\mathbb{T}_n\). ◻
Remark 16. (i) By the theorem above, we know that the \(E\)-invariant and \(F\)-invariant are the same for cluster monomials of skew-symmetric cluster algebras. One of the advantages of \(F\)-invariant is that it can be also defined for skew-symmetrizable cluster algebras. In this case, \(E\)-invariant is not defined.
(ii) We remark that 19 is first proved in [12] by Jiarui Fei with a different presentation in his study of tropical \(F\)-polynomials of representations for finite-dimensional algebras.
Let \(\mathcal{A}\) be a cluster algebra and \(S\) a fixed skew-symmetrizer for the exchange matrices of \(\mathcal{A}\).
Definition 6. Let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\). Let \(\{{\boldsymbol{g}}_{u}^t\mid t\in\mathbb{T}_n\}\) and \(\{{\boldsymbol{g}}_{u'}^t\mid t\in\mathbb{T}_n\}\) be the families of \(g\)-vectors of \(u\) and \(u'\) with respect to vertices of \(\mathbb{T}_n\). We say that \(u\) and \(u'\) are sign-coherent, if for any vertex \(t\in\mathbb{T}_n\) and any \(k\in[1,n]\), we have \(g_{k;u}^tg_{k;u'}^t\geq 0\).
In the study universal geometric cluster algebras, Reading [7] conjectured a separation property for cluster variables of \(\mathcal{A}\), which can be reformulated as follows.
Conjecture 17. [7] Let \(u=x_{i;t}\) and \(u'=x_{j;t'}\) be two cluster variables of \(\mathcal{A}\). Then \(u\) and \(u'\) are contained in the same cluster if and only if \(u\) and \(u'\) are sign-coherent.
This conjecture implies that if two cluster variables are not contained in any common cluster, then they can be separated by the sign-coherence of \(g\)-vectors.
Theorem 18. Let \(\mathcal{A}\) be a cluster algebra and let \(u\) and \(u'\) be two cluster monomials of \(\mathcal{A}\). Then the product \(uu'\) is still a cluster monomial if and only if \(u\) and \(u'\) are sign-coherent. In particular, Reading’s conjecture is true.
Proof. “\(\Rightarrow\)": Suppose that the product \(uu'\) is still a cluster monomial of \(\mathcal{A}\). Then there exists a vertex \(w\in\mathbb{T}_n\) such that \(u, u'\) and \(uu'\) are cluster monomials in \({\boldsymbol{x}}_w\). We can assume that \(u=\prod_{i=1}^n x_{i;w}^{v_i}\) and \(u'=\prod_{i=1}^n x_{i;w}^{v_i'}\). Let \(t\in\mathbb{T}_n\). By Theorem 8 (i), we know that the \(k\)th components of the \(g\)-vectors \({\boldsymbol{g}}_{x_{1;w}}^t,\ldots, {\boldsymbol{g}}_{x_{n;w}}^t\) are simultaneously non-negative or simultaneously non-positive for each \(k\in[1,n]\). This implies \[g_{k;u}^tg_{k;u'}^t=(\sum_{i=1}^nv_ig_{k;x_{i;w}}^t)(\sum_{i=1}^nv_i'g_{k;x_{i;w}}^t)\geq 0,\] where \(t\in\mathbb{T}_n\) and \(k\in[1,n]\). So \(u\) and \(u'\) are sign-coherent.
“\(\Leftarrow\)": Suppose \(u\) and \(u'\) are sign-coherent. Let \(S\) be a skew-symmetrizer for the exchange matrices of \(\mathcal{A}\). We have the following claims.
Claim (a): \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) and \(F_{u'}^t[S{\boldsymbol{g}}_{u}^t]\) are invariant for \(t\in\mathbb{T}_n\).
Claim (b): \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=0=F_{u'}^t[S{\boldsymbol{g}}_{u}^t]\) for any vertex \(t\in\mathbb{T}_n\). In particular, \((u\mid\mid u')_F=0\).
Proof of claim (a): We first show that \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) is invariant for \(t\in\mathbb{T}_n\). It suffices to show that \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]\) for any edge
(0,1)*+t="A",(10,1)*+t’="B",^k"A";"B"
in \(\mathbb{T}_n\). In this case, we have \(g_{k;u}^{t'}=-g_{k;u}^t\) and \(g_{k;u'}^{t'}=-g_{k;u'}^t\), by Lemma 1. Since \(u\) and \(u'\) are sign-coherent, we have \(g_{k;u}^tg_{k;u'}^t\geq 0\). Thus \[(-g_{k;u}^{t'})(-g_{k;u'}^t)=-g_{k;u}^tg_{k;u'}^t\leq 0\;\;\;\text{and}\;\;\;(-g_{k;u}^t)(-g_{k;u'}^{t'})=-g_{k;u}^tg_{k;u}^t\leq 0.\] So we have \[[-g_{k;u}^{t'}]_+[-g_{k;u'}^t]_+=0=[-g_{k;u}^t]_+[-g_{k;u'}^{t'}]_+.\] Then by Theorem 11 (i), we get \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=F_u^{t'}[S{\boldsymbol{g}}_{u'}^{t'}]\). By induction, we see that \(F_u^t[S{\boldsymbol{g}}_{u'}^t]\) is invariant for \(t\in\mathbb{T}_n\). By the same argements, we can show that \(F_{u'}^t[S{\boldsymbol{g}}_{u}^t]\) is invariant for \(t\in\mathbb{T}_n\).
Proof of claim (b): Since \(u\) is a cluster monomial of \(\mathcal{A}\), we can assume that \(u\) is a cluster monomial in \({\boldsymbol{x}}_w\) for some \(w\in\mathbb{T}_n\). In this case, \(F_u^w=1\). Thus \(F_u^w[{\boldsymbol{r}}]=0\) for any \({\boldsymbol{r}}\in \mathbb{Z}^n\). In particular, we have \(F_u^w[S{\boldsymbol{g}}_{u'}^w]=0\). For any vertex \(t\in\mathbb{T}_n\), by claim (a), we have \(F_u^t[S{\boldsymbol{g}}_{u'}^t]=F_u^w[S{\boldsymbol{g}}_{u'}^w]=0\). By the same arguments, we have \(F_{u'}^t[S{\boldsymbol{g}}_{u}^t]=0\) for any vertex \(t\in\mathbb{T}_n\). Thus \[(u\mid\mid u')_F=F_u^t[S{\boldsymbol{g}}_{u'}^t]+F_{u'}^t[S{\boldsymbol{g}}_{u}^t]=0.\] Since \((u\mid\mid u')_F=0\) and by [6], we know that the product \(uu'\) is still a cluster monomial.
Therefore, the product \(uu'\) is still a cluster monomial if and only if \(u\) and \(u'\) are sign-coherent. In the case that \(u\) and \(u'\) are cluster variables, this corresponds to the statement in Reading’s conjecture. ◻
The term “negative-reachable" means that there is a vertex \(w\in\mathbb{T}_n\) such that \(\mathcal{M}_u^w=(M_u^w,V_u^w)\) is a negative decorated representation, i.e., \(M_u^w=0\).↩︎