The valuation pairing on an upper cluster algebra. (English) Zbl 1540.13051
In this article, the authors introduce a valuation pairing on upper cluster algebras, and apply the valuation pairing to obtain many results concerning factoriality, \(d\)-vectors, \(F\)-polynomials, and cluster Poisson variables.
Let \(\mathbb{A}\) be the set of cluster variables in an upper cluster algebra \(\mathcal{U}\). The valuation pairing is a map \begin{align*} (-||-)_v:\mathbb{A}\times \mathcal{U}& \longrightarrow\mathbb{N}\cup\{\infty\}\\ (A,M)&\longmapsto \max\{s\in \mathbb{N}\mid M/A^s\in \mathcal{U}\}. \end{align*} This valuation pairing satisfies the following properties (Proposition 3.3):
The authors also introduce the following local factorization in an upper cluster algebra \(\mathcal{U}\): for any non-zero element \(M\in \mathcal{U}\) and any cluster \(t=\{A_{1;t},\dots, A_{n;t}\}\), we can factor \(M=NL\) with \(N\) being a cluster monomial in \(t\) and \((A_{k;t}||L)_v=0\) for all \(k=1,\dots, n\).
They then prove in Theorem 3.7 that for a full rank upper cluster algebra \(\mathcal{U}\), the local factorization of any non-zero element \(M\) with respect to any cluster \(t\) is unique.
One important question in commutative algebra is to address the factoriality of an algebra. The authors prove in Theorem 4.9 that \(\mathcal{U}\) is factorial (UFD) if and only if the cluster variables in one cluster are prime (i.e., generating a prime principal ideal).
Recall that the exchange matrix \(B\) of a seed is an \(n\times m\) matrix where \(n\) is the number of cluster variables in the seed and \(m\) is the number of mutable cluster variables in the seed. The exchange matrix \(B\) is said to be primitive if each column vector is primitive (in the sense that the gcd of the entries is \(1\)). We say \(\mathcal{U}\) is primitive if it admits a primitive exchange matrix. The authors then prove in Theorem 4.13 that if \(\mathcal{U}\) is primitive and full-rank, then \(\mathcal{U}\) is factorial (UFD), and the numerator polynomial of any cluster variable in the Laurent expansion with respect to any given seed is irreducible. Since any upper cluster algebra with principal coefficients is primitive and full rank, this theorem holds for any upper cluster algebra with prinicipal coefficients. Moreover, the authors prove in Corollary 4.21 that for a primitive and full rank upper cluster algebra \(\mathcal{U}\), its cluster algebra \(\mathcal{A}\) is factorial (UFD) if and only if \(\mathcal{A}=\mathcal{U}\).
In cluster algebra III, Berenstein, Fomin, and Zelevinsky proved the star fish lemma, stating that if an upper cluster algebra \(\mathcal{U}\) is full rank, then it is equal to the intersection of the Laurent polynomial rings associated with an initial seed together with all its once-mutated neighboring seeds (\(m+1\) in total where \(m\) is the number of mutable vertices). In this article, the authors prove the ray fish lemma (Theorem 4.23), stating that if \(\mathcal{U}\) is primitive and full rank, and \(t_0\) and \(t\) are two seeds that share no common mutable cluster variables, then \(\mathcal{U}=\mathcal{L}_{t_0}\cap \mathcal{L}_t\) where \(\mathcal{L}_{t_0}\) and \(\mathcal{L}_t\) are the Laurent polynomial rings associated with the seeds \(t_0\) and \(t\).
By the Laurent phenomenon, we can fix an initial cluster \(t_0=\{A_1,\dots, A_n\}\) and expand any non-zero element \(M\) as \(P_M(A_1,\dots, A_n)/\prod_{i=1}^n A_i^{d_i}\). The polynomial \(P_M\) is called the numerator polynomial of \(M\) with respect to the initial cluster and the vector \(d_M:=(d_1,\dots, d_n)\) is called the \(d\)-vector of \(M\) with respect to the initial cluster. The authors prove in Theorem 5.1 that if \(\mathcal{U}\) is full rank, then \(d_k=(A_k||P_M)_v-(A_k||M)_v\). As a corollary, if \((A_k||M)_v=0\), then \(d_M\) is uniquely determined by \(P_M\). Note that \((A_k||M)_v=0\) holds whenever \(M\) is a cluster monomial that does not contain the initial cluster variables, and thus the above corollary is in particular applicable to such a cluster monomial.
For a fixed initial cluster \(t_0=\{A_1,\dots, A_n\}\), one can define for each mutable index \(i\) a Laurent monomial \(\hat{X}_i=\prod_j A_j^{b_{ij}}\). Then any cluster monomial \(M\) can be factored as \(\left(\prod_i A_i^{g_i}\right)F_M(\hat{X}_1,\dots, \hat{X}_m)\); the vector \(g_M:=(g_1,\dots, g_n)\) is called the \(g\)-vector of \(M\) with respect to the initial cluster and the polynomial \(F_M\) (in variables \(X_1,\dots, X_m\)) is called the \(F\)-polynomial with respect to the initial cluster. In Theorem 6.1, the authors prove that in an (upper) cluster algebra with principal coefficients, cluster monomials in non-initial cluster variables are uniquely determined by their \(F\)-polynomials. In theorem 6.2, they further prove that the \(F\)-polynomial of a non-initial cluster variable is irreducible (as a polynomial in \(X_1,\dots, X_m\)).
The variables \(X_1,\dots, X_n\) are also known as the cluster Poisson variables of Fock and Goncharov, and they obey their own set of cluster mutation rules. Moreover, one can construct an upper cluster algebra with universal coefficients by merging the cluster mutation rules for the \(A\)-variables and the \(X\)-variables together: \[ \mu_k(A_i)=\left\{\begin{array}{ll} \frac{X_{k;t}}{1+X_{k;t}}\frac{\prod_{b_{jk;t}>0}A_{j;t}^{b_{jk;t}}}{A_{k;t}}+\frac{1}{1+X_{k;t}}\frac{\prod_{b_{jk;t}<0}A_{j;t}^{-b_{jk;t}}}{A_{k;t}} & \text{if \(i=k\),}\\ A_i & \text{if \(i\neq k\).} \end{array}\right. \] Note that this is not Reading’s cluster algebra with universal coefficients for finite type cluster algebras; rather it is what Fock and Goncharov called the symplectic double. Let \(t_0\) and \(t\) be two cluster seeds of an upper cluster algebra with universal coefficients. Suppose the mutation of \(t_0\) at \(j\) produces a new cluster variable \(\frac{X_{j;t_0}}{1+X_{j;t_0}}N_1+\frac{1}{1+X_{j;t_0}}N_2\) and the mutation of \(t\) at \(k\) produces a new cluster variable \(\frac{X_{k;t}}{1+X_{k;t}}M_1+\frac{1}{1+X_{k;t}}M_2\). The authors prove in Theorem 7.5 that the cluster Poisson variables \(X_{j;t_0}=X_{k;t}\) if and only if \((A_{j;t_0},N_1,N_2)=(A_{k;t},M_1,M_2)\). As a corollary, they prove that the cluster Poisson seeds that contain a particular cluster Poisson variable form a connected subgraph of the exchange graph.
Let \(\mathbb{A}\) be the set of cluster variables in an upper cluster algebra \(\mathcal{U}\). The valuation pairing is a map \begin{align*} (-||-)_v:\mathbb{A}\times \mathcal{U}& \longrightarrow\mathbb{N}\cup\{\infty\}\\ (A,M)&\longmapsto \max\{s\in \mathbb{N}\mid M/A^s\in \mathcal{U}\}. \end{align*} This valuation pairing satisfies the following properties (Proposition 3.3):
- ●
- \((A||M)_v=\infty\) if and only if \(M=0\).
- ●
- \((A||M+L)_v\geq \min\{(A||M)_v,(A||L)_v\}\).
- ●
- For any \(s\geq 0\), \((A||A^sM)_v=s+(A||M)_v\).
The authors also introduce the following local factorization in an upper cluster algebra \(\mathcal{U}\): for any non-zero element \(M\in \mathcal{U}\) and any cluster \(t=\{A_{1;t},\dots, A_{n;t}\}\), we can factor \(M=NL\) with \(N\) being a cluster monomial in \(t\) and \((A_{k;t}||L)_v=0\) for all \(k=1,\dots, n\).
They then prove in Theorem 3.7 that for a full rank upper cluster algebra \(\mathcal{U}\), the local factorization of any non-zero element \(M\) with respect to any cluster \(t\) is unique.
One important question in commutative algebra is to address the factoriality of an algebra. The authors prove in Theorem 4.9 that \(\mathcal{U}\) is factorial (UFD) if and only if the cluster variables in one cluster are prime (i.e., generating a prime principal ideal).
Recall that the exchange matrix \(B\) of a seed is an \(n\times m\) matrix where \(n\) is the number of cluster variables in the seed and \(m\) is the number of mutable cluster variables in the seed. The exchange matrix \(B\) is said to be primitive if each column vector is primitive (in the sense that the gcd of the entries is \(1\)). We say \(\mathcal{U}\) is primitive if it admits a primitive exchange matrix. The authors then prove in Theorem 4.13 that if \(\mathcal{U}\) is primitive and full-rank, then \(\mathcal{U}\) is factorial (UFD), and the numerator polynomial of any cluster variable in the Laurent expansion with respect to any given seed is irreducible. Since any upper cluster algebra with principal coefficients is primitive and full rank, this theorem holds for any upper cluster algebra with prinicipal coefficients. Moreover, the authors prove in Corollary 4.21 that for a primitive and full rank upper cluster algebra \(\mathcal{U}\), its cluster algebra \(\mathcal{A}\) is factorial (UFD) if and only if \(\mathcal{A}=\mathcal{U}\).
In cluster algebra III, Berenstein, Fomin, and Zelevinsky proved the star fish lemma, stating that if an upper cluster algebra \(\mathcal{U}\) is full rank, then it is equal to the intersection of the Laurent polynomial rings associated with an initial seed together with all its once-mutated neighboring seeds (\(m+1\) in total where \(m\) is the number of mutable vertices). In this article, the authors prove the ray fish lemma (Theorem 4.23), stating that if \(\mathcal{U}\) is primitive and full rank, and \(t_0\) and \(t\) are two seeds that share no common mutable cluster variables, then \(\mathcal{U}=\mathcal{L}_{t_0}\cap \mathcal{L}_t\) where \(\mathcal{L}_{t_0}\) and \(\mathcal{L}_t\) are the Laurent polynomial rings associated with the seeds \(t_0\) and \(t\).
By the Laurent phenomenon, we can fix an initial cluster \(t_0=\{A_1,\dots, A_n\}\) and expand any non-zero element \(M\) as \(P_M(A_1,\dots, A_n)/\prod_{i=1}^n A_i^{d_i}\). The polynomial \(P_M\) is called the numerator polynomial of \(M\) with respect to the initial cluster and the vector \(d_M:=(d_1,\dots, d_n)\) is called the \(d\)-vector of \(M\) with respect to the initial cluster. The authors prove in Theorem 5.1 that if \(\mathcal{U}\) is full rank, then \(d_k=(A_k||P_M)_v-(A_k||M)_v\). As a corollary, if \((A_k||M)_v=0\), then \(d_M\) is uniquely determined by \(P_M\). Note that \((A_k||M)_v=0\) holds whenever \(M\) is a cluster monomial that does not contain the initial cluster variables, and thus the above corollary is in particular applicable to such a cluster monomial.
For a fixed initial cluster \(t_0=\{A_1,\dots, A_n\}\), one can define for each mutable index \(i\) a Laurent monomial \(\hat{X}_i=\prod_j A_j^{b_{ij}}\). Then any cluster monomial \(M\) can be factored as \(\left(\prod_i A_i^{g_i}\right)F_M(\hat{X}_1,\dots, \hat{X}_m)\); the vector \(g_M:=(g_1,\dots, g_n)\) is called the \(g\)-vector of \(M\) with respect to the initial cluster and the polynomial \(F_M\) (in variables \(X_1,\dots, X_m\)) is called the \(F\)-polynomial with respect to the initial cluster. In Theorem 6.1, the authors prove that in an (upper) cluster algebra with principal coefficients, cluster monomials in non-initial cluster variables are uniquely determined by their \(F\)-polynomials. In theorem 6.2, they further prove that the \(F\)-polynomial of a non-initial cluster variable is irreducible (as a polynomial in \(X_1,\dots, X_m\)).
The variables \(X_1,\dots, X_n\) are also known as the cluster Poisson variables of Fock and Goncharov, and they obey their own set of cluster mutation rules. Moreover, one can construct an upper cluster algebra with universal coefficients by merging the cluster mutation rules for the \(A\)-variables and the \(X\)-variables together: \[ \mu_k(A_i)=\left\{\begin{array}{ll} \frac{X_{k;t}}{1+X_{k;t}}\frac{\prod_{b_{jk;t}>0}A_{j;t}^{b_{jk;t}}}{A_{k;t}}+\frac{1}{1+X_{k;t}}\frac{\prod_{b_{jk;t}<0}A_{j;t}^{-b_{jk;t}}}{A_{k;t}} & \text{if \(i=k\),}\\ A_i & \text{if \(i\neq k\).} \end{array}\right. \] Note that this is not Reading’s cluster algebra with universal coefficients for finite type cluster algebras; rather it is what Fock and Goncharov called the symplectic double. Let \(t_0\) and \(t\) be two cluster seeds of an upper cluster algebra with universal coefficients. Suppose the mutation of \(t_0\) at \(j\) produces a new cluster variable \(\frac{X_{j;t_0}}{1+X_{j;t_0}}N_1+\frac{1}{1+X_{j;t_0}}N_2\) and the mutation of \(t\) at \(k\) produces a new cluster variable \(\frac{X_{k;t}}{1+X_{k;t}}M_1+\frac{1}{1+X_{k;t}}M_2\). The authors prove in Theorem 7.5 that the cluster Poisson variables \(X_{j;t_0}=X_{k;t}\) if and only if \((A_{j;t_0},N_1,N_2)=(A_{k;t},M_1,M_2)\). As a corollary, they prove that the cluster Poisson seeds that contain a particular cluster Poisson variable form a connected subgraph of the exchange graph.
Reviewer: Daping Weng (Davis)
MSC:
| 13F60 | Cluster algebras |
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