Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

Cluster algebras are first constructed by Fomin and Zelevinsky in [FZ1]. Generally speaking, it is a commutative algebra with so-called exchange relations given by an extra combinatorial structure. Later, researchers found many relationships from the theory of cluster algebras to other topics, such as Lie theory, quantum groups, representation theory, Riemann surfaces with triangulation, number theory, tropical geometry and Grassmanian theory as well as many interesting properties. The most significant properties among them are the Laurent phenomenon and positivity of varieties which claim that each cluster variables can be expressed as a Laurent polynomial in any cluster over ${\mathbb{N}}{\mathbb{P}}$. However, the calculation of the Laurent expression of a cluster variable in a given cluster is in general difficult. One of our aims in this paper is to provide recurrence formulas as a program to make the above calculation easier.

Two bases related to an upper cluster algebra $\mathcal{U}({\mathcal{A}})$, called the greedy basis and the theta basis respectively, are constructed in [LLZ] and [GHKK], which both contain coefficient free cluster monomials. It is known that each element in the above two bases satisfies the Laurent phenomenon and positivity, that is, its expression in every cluster is a Laurent polynomial over ${\mathbb{N}}{\mathbb{P}}$. So in some sense such element can be seen as a generalization of cluster monomials. Moreover, the constant coefficients of the Laurent expression in the initial cluster are related to counting of some combinatorial objects. However, the greedy basis is only constructed for rank $2$ case, while the theta basis relies on the cluster scattering diagram. Another goal of this paper is to directly construct a basis of $\mathcal{U}({\mathcal{A}})$ consisting of some universally indecomposable Laurent polynomials as a generalization of cluster monomials in general case. In order to achieve it, one useful tool we will apply is the Nowton polytopes of $F$-polynomials associated to cluster variables.

In [F], Jiarui Fei defined the Newton polytope of an $F$-polynomial associated to representations of a finite-dimensional basic algebra, as well as showed some interesting combinatorial properties of such Newton polytopes. On the other hand, the authors of [LLZ] and [LLS] focused on Newton polytopes of cluster variables in cluster algebras of rank $2$ and rank $3$ respectively. By definitions, up to a translation, the Newton polytope of a cluster variable can be obtained from that of the related $F$-polynomial by a transformation induced by its exchange matrix $B$ since $\hat{y}_{j;t}=\prod\limits_{i=1}^{m}x_{i;t}^{b_{ji}}$ in the case of geometric type. However, on the other hand, when $B$ is not invertible, it is not apparent to obtain the latter from the former. In this a aspect, it seems that the Nowton polytopes of $F$-polynomials keeps more information. So in this paper, we mainly focus on the Nowton polytopes of $F$-polynomials in initial $Y$-variables associated to cluster variables.

Based on the study of the Newton polytope $N_{l;t}$ of $F_{l;t}$, we introduce the polytope $N_{h}$ associated to a vector $h\in{\mathbb{Z}}^{n}$ and the polytope functions $\rho_{h}$ as a generalization of $N_{l;t}$ and $x_{l;t}$ respectively. Then the properties of $N_{h}$ and $\rho_{h}$ naturally induce those of $N_{l;t}$ and $x_{l;t}$. Moreover, it will also be proved that the polytope functions compose a strongly positive basis of $\mathcal{U}({\mathcal{A}})$ for a cluster algebras with principal coefficients as well as certain cluster algebra over arbitrary semifield. As applications, several conjectures are confirmed for TSSS cluster algebras, including the positivity conjectures of cluster variables and of $d$-vectors respectively.

In this section, we recall some preliminaries of cluster algebras, $F$-polynomials and $d$-vectors mainly based on [FZ4].

We would like to introduce the following notations for convenience: for any $n\in{\mathbb{N}}$, $x\in{\mathbb{Z}}$,

And for a vector $\alpha=(\alpha_{1},\cdots,\alpha_{r})\in{\mathbb{Z}}^{r}$, $[\alpha]_{+}=([\alpha_{1}]_{+},\cdots,[\alpha_{r}]_{+})$. We always represent the elements in ${\mathbb{R}}^{n}$ as row vectors unless otherwise specified.

An $n\times n$ integer matrix $B=(b_{ij})$ is called sign-skew-symmetric if either $b_{ij}=b_{ji}=0$ or $b_{ij}b_{ji}<0$ for any $i,j\in[1,n]$. A skew-symmetric matrix is a sign-skew-symmetric matrix with $b_{ij}=-b_{ji}$ for any $i,j\in[1,n]$. Moreover, a skew-symmetrizable matrix is a sign-skew-symmetric matrix such that there is a positive diagonal integer matrix $D$ satisfying that $DB$ is skew-symmetric.

For a sign-skew-symmetric matrix $B$, we define another $n\times n$ matrix $B^{\prime}=(b_{ij}^{\prime})$ satisfying that for any $k,i,j\in[1,n]$,

We call the formula (1) the exchange relation for sign-skew-symmetric matrices. Denote by $B^{\prime}=\mu_{k}(B)$ the mutation of $B$ in direction $k$.

For $k_{1},k_{2}\in[1,n]$, if $B^{\prime}=\mu_{k_{1}}(B)$ is also sign-skew-symmetric, then we can mutate $B^{\prime}$ in direction $k_{2}$ to obtain $B^{\prime\prime}=\mu_{k_{2}}\mu_{k_{1}}(B)$.

The notion of totally sign-skew-symmetric matrices was introduced in [BFZ]. It is well-known that skew-symmetric and skew-symmetrizable matrices are totally sign-skew-symmetric matrices. An example of a $3\times 3$ sign-skew-symmetric matrix which is not skew-symmetrizable was given in [BFZ]. In that paper, Berenstein etc. conjectured that any acyclic sign-skew-symmetric matrices are total. In [HL], Ming Huang and Fang Li proved this conjecture.

Hence, on sign-skew-symmetric matrices, one of the most important remaining problems is the condition under which sign-skew-symmetric matrices are total. In this paper, we always assume the involved sign-skew-symmetric matrices are totally sign-skew-symmetric.

For convenience, we will denote a totally sign-skew-symmetric matrix (respectively, cluster algebra defined subsequently) briefly as a TSSS matrix (respectively, TSSS cluster algebra).

Let $({\mathbb{P}},\oplus,\cdot)$ be a semifield, i.e., a free abelian multiplicative group endowed with a binary operation of (auxiliary) addition $\oplus$ which is commutative, associative and distributive with respect to the multiplication in ${\mathbb{P}}$. And ${\mathcal{F}}$ is the field of rational functions in $n$ independent variables with coefficients in ${\mathbb{Q}}{\mathbb{P}}$.

$X$ defined above is called a cluster with cluster variables $x_{i}$, $y_{i}$ is called a $Y$-variable and $B$ is called an exchange matrix.

It can be easily checked that $\Sigma{{}^{\prime}}$ is a seed and the seed mutation $\mu_{k}$ ia an involution.

In this paper, the seed assigned to a vertex $t$ is denoted by $\Sigma_{t}=(X_{t},Y_{t},B_{t})$ with

where $B_{t}$ is totally sign-skew-symmetric.

Now we are ready to introduce the definition of cluster algebras.

If there is a skew-symmetrizable (respectively, skew-symmetric) exchange matrix in a cluster algebra ${\mathcal{A}}$, then all exchange matrices of ${\mathcal{A}}$ are skew-symmetrizable (respectively, skew-symmetric). So, in this case we call ${\mathcal{A}}$ a skew-symmetrizable (respectively, skew-symmetric) cluster algebra.

In this paper, when saying a cluster algebra, we always mean a TSSS cluster algebra. And we always assume ${\mathcal{A}}$ is a cluster algebra with cluster variables $x_{l;t}$ for any $l\in[1,n],t\in{\mathbb{T}}_{n}$.

It can be seen from the definition that the cluster algebra ${\mathcal{A}}$ is related to the choice of semifield ${\mathbb{P}}$. There are two special semifields which play important roles.

In particular, we say a cluster algebra ${\mathcal{A}}$ is of geometry type if ${\mathbb{P}}$ is a tropical semifield. In this case, we can also denote ${\mathbb{P}}$ as $Trop(x_{n+1},x_{n+2},\cdots,x_{m})$. Then according to the definition, $y_{j;t}$ is a Laurent monomial of $x_{n+1},x_{n+2},\cdots,x_{m}$ for any $j\in[1,n],t\in{\mathbb{T}}_{n}$. Hence we can define $b_{ij}^{t}$ for $i\in[n,m],j\in[1,n]$ as

Let $\tilde{B}_{t}$ be the $m\times n$ matrix $\tilde{B}_{t}=(b_{ij}^{t})_{i\in[1,m],j\in[1,n]}$ and $\tilde{X}_{t}=(x_{1;t},\cdots,x_{n;t},x_{n+1},\cdots,x_{m})$. Then the seed assigned to $t$ can be represented as $(\tilde{X}_{t},\tilde{B}_{t})$. The mutation formulas are the same for $\tilde{B}$ while those of $\tilde{X}$ at direction $k$ become

Hence a cluster algebra having principal coefficients at some vertex is of geometric type. Then if we use $(\tilde{X},\tilde{B})$ to represent a seed, the definition is equivalent to that there is a seed $(\tilde{X}_{t_{0}},\tilde{B}_{t_{0}})$ at vertex $t_{0}$ satisfying $\tilde{B}_{t_{0}}=\left(\begin{array}[]{c}B_{t_{0}}\\ I\end{array}\right),$ where $I$ is a $n\times n$ identity matrix.

Given a cluster algebra ${\mathcal{A}}$ with initial seed $\Sigma_{t_{0}}=(X_{t_{0}},Y_{t_{0}},B_{t_{0}})$, we denote by ${\mathcal{A}}_{prin}$ the cluster algebra with principal coefficients associated to $B_{t_{0}}$, which is called the principal coefficients cluster algebra corresponding to ${\mathcal{A}}$ since it is unique up to cluster isomorphisms.

The Laurent phenomenon, given in [FZ1, FZ3], is the most fundamental result in cluster theory, which says that for a cluster algebra ${\mathcal{A}}$ and its fixed seed $(X,Y,B)$, every cluster variable of ${\mathcal{A}}$ is a Laurent polynomial over ${\mathbb{Z}}{\mathbb{P}}$ in cluster variables in $X$.

Thus for a seed $(X_{t_{0}},Y_{t_{0}},B_{t_{0}})$ and any cluster variable $x_{l;t}$ in ${\mathcal{A}}$, we can express it as a Laurent polynomial in cluster $X_{t_{0}}$:

such that $P_{l;t}^{t_{0}}$ is a (non-Laurent) polynomial with no non-trivial monomial factor. Here and in the following, we call $P_{l;t}^{t_{0}}$ the absolute numerator of $x_{l;t}$ with respect to $X_{t_{0}}$. The denominator vector $d_{l;t}^{t_{0}}=(d_{1}^{t_{0}}(x_{l;t}),d_{2}^{t_{0}}(x_{l;t}),\cdots,d_{n}^{t_{0}}(x_{l;t}))$ is called the $d$-vector of $x_{l;t}$ with respect to the cluster $X_{t_{0}}$. Moreover, if ${\mathcal{A}}$ has principal coefficients at $t_{0}$, then $P_{l;t}^{t_{0}}$ belongs to ${\mathbb{Z}}[x_{1,t_{0}},\cdots,x_{n,t_{0}};y_{1,t_{0}},\cdots,y_{n,t_{0}}]$. $F_{l;t}^{t_{0}}=P_{l;t}^{t_{0}}|_{x_{i;t_{0}}\rightarrow 1,\forall i\in[1,n]}$ is a polynomial in $y_{1,t_{0}},\cdots,y_{n,t_{0}}$ called the $F$-polynomial of $x_{l;t}$ with respect to $X_{t_{0}}$. Under the canonical ${\mathbb{Z}}^{n}$-grading given by $deg(x_{i})=e_{i},deg(y_{i})=-(b_{i}^{t_{0}})^{\top}$ for any $i\in[1,n]$ where $e_{1},\cdots,e_{n}$ are standard basis in ${\mathbb{Z}}^{n}$, and $b_{i}^{t_{0}}$ is the $i$-th column of $B_{t_{0}}$, the Laurent expression of $x_{l;t}$ in $X_{t_{0}}$ is homogeneous with degree $g_{l;t}^{t_{0}}$, which is called the $g$-vector^aaaUsually, $d$-vectors and $g$-vectors are written as column vectors. But in this paper, because we will write the coordinates specifically in some discussion, it is more convenient for us to use row vectors. So for the sake of consistency we always write vectors in ${\mathbb{R}}^{n}$, including $d$-vectors and $g$-vectors, as row vectors. of $x_{l;t}$ corresponding to $X_{t_{0}}$. Or generally $g$-vectors can also be defined recurrently as follows: $g_{j;t_{0}}^{t_{0}}=e_{j}$, and

where $t$ and $t{{}^{\prime}}$ are connected by an edge labeled $k$ in ${\mathbb{T}}_{n}$.

Next Theorem shows the importance of principal coefficients case in the study of cluster algebras.

We denote $\hat{Y}_{t}=\{\hat{y}_{1;t},\cdots,\hat{y}_{n;t}\}$ for any $t\in{\mathbb{T}}_{n}$.

When ${\mathcal{A}}$ is a cluster algebra of geometric type with initial seed $\Sigma_{t_{0}}=(\tilde{X}_{t_{0}},\tilde{B}_{t_{0}})$, we also denote its seed as $\Sigma_{t}=(X_{t},X^{fr}_{t},\tilde{B}_{t})$ for any $t\in{\mathbb{T}}_{n}$, where $X_{t}=\{x_{1;t},\cdots,x_{n;t}\}$ and $X^{fr}_{t}=\{x_{n+1},\cdots,x_{m}\}$ to distinguish two kinds of variables.

In this paper, we will construct $\rho_{h}\in{\mathbb{N}}[[\hat{Y}]]X^{h}$ for each $h\in{\mathbb{Z}}^{n}$, that is, $\rho_{h}$ is of the form $\sum\limits_{p\in{\mathbb{N}}^{n}}a_{p}\hat{Y}^{p}X^{h}$ and $\rho_{h}|_{x_{i}\rightarrow 1,\forall i\in[1,n]}$ is a power series of $Y$, where $a_{p}\in{\mathbb{N}}$. In order to emphasis that we regard $X$ as variables while $Y$ as coefficients, we will slightly abuse the notation to denote $\rho_{h}\in{\mathbb{N}}[Y][[X^{\pm 1}]]$ and call it a formal Laurent polynomial in $X$ with coefficients in ${\mathbb{N}}[Y]$.

For any $t,t{{}^{\prime}}\in{\mathbb{T}}_{n}$ connected by an edge labeled $k$ and any homogeneous Laurent polynomial

with grading $h$, we naturally have $f=F|_{{\mathcal{F}}}(\hat{Y}_{t{{}^{\prime}}})X_{t^{\prime}}^{h}$, where $F$ is obtained from the Laurent expression of $f$ in $X_{t{{}^{\prime}}}$ by specilizing $x_{i;t{{}^{\prime}}}$ to 1 for any $i\in[1,n]$. We modify it into the Laurent polynomial

and

with coefficients in $Y_{t}$ belonging to ${\mathbb{Z}}Trop(Y_{t})$. The motivation of such definition is as follows.

Applying $L^{t}$ on $f$ changes the semifield from $Trop(Y_{t{{}^{\prime}}})$ to $Trop(Y_{t})$, and compensating for it with dividing $y_{k;t{{}^{\prime}}}^{[h_{k}]_{+}}$ as in (5), we will show in Remark LABEL:remark_after_the_theorem that $L^{t}(\rho_{h}^{t{{}^{\prime}}})$ equals the Laurent expression of $\frac{F_{h}^{t{{}^{\prime}}}|_{{\mathcal{F}}}(\hat{Y}_{t{{}^{\prime}}})}{F_{h}^{t{{}^{\prime}}}|_{Trop(Y_{t})}(Y_{t{{}^{\prime}}})}X_{t{{}^{\prime}}}^{h}$ in $X_{t}$ with coefficients in $Y_{t}$ belonging to ${\mathbb{Z}}Trop(Y_{t})$, which realizes $\rho_{h}^{t{{}^{\prime}}}$ and $L^{t}$ as a generalization of a coefficient free cluster monomial and its mutation respectively in some aspect and thus justifies the compensation. On the other hand, later we will use $L^{t}$ to introduce a strong restriction on the support of homogeneous Laurent polynomial (see $\mathcal{U}^{+}(\Sigma_{t})$ or $\widehat{\mathcal{U}}^{+}(\Sigma_{t})$) so as to make $\rho_{h}$ which we are going to construct a special element under this restriction.

Then we can define $L^{t;\gamma}(f)=L^{t}\circ L^{t^{(1)}}\circ\cdots\circ L^{t^{(s)}}(f)$ for any path $\gamma=t-t^{(1)}-\cdots-t^{(s)}-t{{}^{\prime}}$ in ${\mathbb{T}}_{n}$ if $L^{t}\circ L^{t^{(1)}}\circ\cdots\circ L^{t^{(j)}}(f)\in{\mathbb{Z}}[Y_{t^{(j)}}^{\pm 1}][X_{t^{(j)}}^{\pm 1}]$ for $j\in[0,s]$.

Later we will show that $L^{t;\gamma}(f)$ only depends on the endpoints $t$ and $t{{}^{\prime}}$ in Remark LABEL:remark_after_the_theorem, so we usually omit the path in the superscript.

For any $t\in{\mathbb{T}}_{n}$ denote

and

where $t_{i}\in{\mathbb{T}}_{n}$ is the vertex connected to $t$ by an edge labeled $i$. We say an element in $\mathcal{U}^{+}_{\geqslant 0}(\Sigma_{t})$ to be indecomposable if it can not be written as a sum of two nonzero elements in $\mathcal{U}^{+}_{\geqslant 0}(\Sigma_{t})$.

And we add the hat “ $\widehat{\quad}\;$” to represent their completion. That is, denote

and

where $t_{i}\in{\mathbb{T}}_{n}$ is the vertex connected to $t$ by an edge labeled $i$.

In the sequel, for a cluster algebra ${\mathcal{A}}$, we will always denote by $t_{0}$ the vertex of the initial seed unless otherwise specified. And when a vertex is not written explicitly, we always mean the initial vertex $t_{0}$. For example, we use $X$, $x_{i}$, $P_{l;t}$ to denote $X_{t_{0}}$, $x_{i;t_{0}}$, $P_{l;t}^{t_{0}}$ respectively. For any cluster $X_{t}$ and any vector $\alpha=(\alpha_{1},\cdots,\alpha_{n})\in{\mathbb{Z}}^{n}$, we denote $X_{t}^{\alpha}=\prod\limits_{i=1}^{n}x_{i;t}^{\alpha_{i}}$.

Let ${\mathcal{A}}$ be a cluster algebra over ${\mathbb{P}}$ with initial seed $(X,Y,B)$.

For any $k\in[1,n]$, $t\in{\mathbb{T}}_{n}$ and Laurent polynomial $P\in{\mathbb{Z}}{\mathbb{P}}[X_{t}^{\pm 1}]$, we will always denote by

the exchange binomial in direction $k$ at $t$ and by $deg_{x_{k;t}}(P)$ the $x_{k;t}$-degree of $P$. Trivially, $M_{k;t}$ is a polynomial in ${\mathbb{Z}}{\mathbb{P}}[x_{1;t},\cdots,x_{k-1;t},x_{k+1;t},\cdots,x_{n;t}]$.

A cluster monomial in ${\mathcal{A}}$ is a monomial in a cluster $X_{t}$ for some $t\in{\mathbb{T}}_{n}$. In the following, when we mention a cluster monomial, it is of the form $aY_{t}^{p}X_{t}^{q}$ with $a\in{\mathbb{Z}},p,q\in{\mathbb{Z}}^{n}$ and $t\in{\mathbb{T}}_{n}$. For such a cluster monomial $f=aY^{p}X^{q}$, we call $a$ (respectively, $aY^{p}$) the constant coefficient (respectively, coefficient) of $f$ and say $f$ is constant coefficient free (respectively, coefficient free) if $a=1$ (respectively,$aY^{p}=1$). Similarly, we define a cluster polynomial to be a polynomial in a cluster $X_{t}$.

For a variable $x$, we say a polynomial $P$ is $x$-homogeneous if $deg_{x}(p)$ are the same for all monomial summands $p$ of $P$.

According to the mutation formula (2), $\widetilde{deg}_{k}^{t}(P)$ is the maximal integer $a$ such that $\frac{P}{x_{k;t}^{a}}$ can be expressed as a Laurent polynomial in $X_{t_{k}}$, where $t_{k}\in{\mathbb{T}}_{n}$ is the vertex connected to $t$ by an edge labeled $k$.

Following the definitions in [LLZ], a Laurent polynomial $p$ in $X$ is called universally positive if $p\in{\mathbb{N}}{\mathbb{P}}[X_{t}^{\pm 1}]$ for any $t\in{\mathbb{T}}_{n}$. And a universally positive Laurent polynomial is said to be universally indecomposable if it cannot be expressed as a sum of two nonzero universally positive Laurent polynomials. Universal indecomposability can be regarded as the “minimalism” in the set of universally positive Laurent polynomials. Since the above two definitions are given for all $t\in T_{n}$, they are naturally mutation invariants.

A ${\mathbb{Z}}{\mathbb{P}}$-basis $\{\alpha_{s}\}_{s\in I}$ of $\mathcal{U(A)}$ is called strongly positive if for any $i,j\in I$, $\alpha_{i}\alpha_{j}=\sum\limits_{s\in I}a_{ij}^{s}\alpha_{s}$, where $a_{ij}^{s}\in{\mathbb{N}}{\mathbb{P}}$ for any $s\in I$.

Next we briefly introduce some concepts and notations about polytopes mainly from [Z], which will be used in this paper with slight modification.

In this paper, unless otherwise specified, we always fix the following notations and notions:
(i) Denote by $z_{1},\cdots,z_{n}$ the coordinates of ${\mathbb{R}}^{n}$;
(ii) Points imply lattice points in ${\mathbb{Z}}^{n}$;
(iii) Polytopes imply those whose vertices are lattice points;
(iv) The partial order “$\leqslant$” in ${\mathbb{Z}}^{n}$ is defined as $a\leqslant b$ for any $a=(a_{1},\cdots,a_{n}),b=(b_{1},\cdots,b_{n})\in{\mathbb{Z}}^{n}$ if $a_{i}\leqslant b_{i}$ for all $i\in[1,n]$.

Here we modify the original definition of polytopes by associating weight to each lattice point in it.