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This page largely follows Keller’s Cluster algebras and derived categories survey.

Skew-symmetric cluster algebras with trivial coefficients

In what follows, we let $Q = (Q_0, Q_1, s, t)$ be a (finite) quiver without oriented $2$-cycles or loops. The skew-symmetric matrix $B = B_Q$ of $Q$ has entries $b_{ij}$ where

In fact, skew-symmetric matrices define quivers of this form.

[proof] Skew-symmetry implies the diagonal consists only of zeros. In addition, we define $Q$ to have $|b_{ij}|$ arrows from $i$ to $j$ (or $j$ to $i$ — whichever is the correct orientation given by $\sgn(b_{ij})$). By construction, there can’t be any $2$-cycles. [/proof]

If $B$ is the skew-symmetric matrix associated with $Q$, and $B’$ the one associated with $\mu_k(Q)$, then

The first case corresponds to flipping the orientation of arrows incident to $k$ in quiver mutation. The second case is the steps where we add arrows and cancel out $2$-cycles. The $\max(0, b_{ik}b_{kj})$ part just ensures that we don’t accidentally look at paths $i - k - j$ where the orientations of $i-k$ and $k-j$ aren’t consistent.

Now let $n \ge 1$ be an integer and $\mathcal{F}$ be the field $\QQ(x_1, \ldots, x_n)$ where $x_1, \ldots, x_n$ are all algebraically independent. A seed is a pair $(R, u)$ where $R$ is a quiver and $u$ is a sequence $u_1, \ldots, u_n$ of $\mathcal{F}$ which freely generate the field $\mathcal{F}$. If $(R, u)$ is a seed and $k$ a vertex of $R$, the mutation $\mu_k(R, u)$ is the seed $(R’, u’)$ where $R’ = \mu_k(R)$ and $u’$ is obtained from $u$ by replacing $u_k$ by $u_k’$ defined by the exchange relation

where both products range over all (relevant) arrows of $R$. This also has a matrix-variant given by

[definition] Fix a quiver $Q$. The initial seed of $Q$ is $(Q, \{x_1, \ldots x_n\})$. A cluster associated with $Q$ is a sequence $u$ which appears in a seed $(R, u)$ obtained from the initial seed by iterated mutation. The cluster variables are the elements of the clusters. The (skew-symmetric) cluster algebra $\mathcal{A}_Q$ is the $\QQ$-subalgebra of $\mathcal{F}$ generated by the cluster variables. The rank of $\mathcal{A}_Q$ is $n$. [/definition]

Some quick properties:

[theorem=(Laurent Phenomenon)] Each cluster variable of $\mathcal{A}_Q$ is a Laurent polynomial with integer coeffients in the cluster variables from any cluster. [/theorem]

The Laurent Phenomenon was proven by Fomin and Zelevinsky (see Theorem 3.1 of Cluster algebras I: Foundations).

[theorem=(Finite-Type Classification)] The cluster algebra $\mathcal{A}_Q$ is finite-type (has finitely many cluster variables) if and only if $Q$ is mutation equivalent to a simply-laced Dynkin quiver of type $\Delta$. In this case, we say that the cluster type of $Q$ is $\Delta$. [/theorem]

The finite-type classification was proven by Fomin and Zelevinsky (see Theorem 1.4 of Cluster algebras II: Finite type classification).

[theorem] If $Q$ is a simply-laced Dynkin quiver of type $\Delta$, then the non-initial cluster variables of $\mathcal{A}_Q$ are in bijection with the positive roots of the root system of $\Delta$. In particular, if $\alpha_1, \ldots, \alpha_n$ are the simple roots, then for each positive root

there is a unique non-initial cluster variable $X_\alpha$ whose denominator is

[/theorem]

This parameterization result was proved by Fomin and Zelevinsky (see Theorem 1.9 of Cluster algebras II: Finite type classification).

A cluster monomial is a product of nonnegative powers of cluster variables belonging to the same cluster.

[theorem] The cluster monomials are linearly independent over the field $\QQ$. [/theorem]

In Keller’s survey, this is stated as a conjecture. This is now a known result due to Irelli-Keller-Labardini-Fragoso-Plamondon.

[theorem=(Positivity)] The cluster variables are Laurent polynomials with nonnegative integer coefficients in the variables of each cluster. [/theorem]

This was proved for all skew-symmetric cluster algebras by Lee-Schiffler.

Skew-symmetrizable cluster algebras with trivial coefficients(?)

The main ingredient of these cluster algebras is the notion of a valued quiver.

[definition] A valued quiver is a quiver $Q$ equipped with a valuation map $v:Q_1 \to \ZZ^2$ such that

a) There are no loops in $Q$.

b) There is at most one arrow between any two vertices of $Q$.

c) There is a function $d:Q_0 \to \NN$ such that $d(i) > 0$ for all vertices $i$ and, for each arrow $i\xrightarrow{\alpha}j$, we have

where $v(\alpha) = (v(\alpha)_1, v(\alpha)_2)$. [/definition]

Let $Q$ be a valued quiver with vertex set $I$. The integer matrix $B = (b_{ij})_{i,j\in I}$ associated with $Q$ is given by

If $D$ is the diagonal $I\times I$-matrix with entries $d_{ii} = d(i)$ for $i \in I$, then the matrix $DB$ is skew-symmetric. That is, $B$ is skew-symmetrizable.

We define mutation of valued quivers by using the matrix-mutation formula above. Similarly, we extend the notion of seeds by using the exchange relation written for matrices. Given a valued quiver $Q$, clusters, cluster variables, cluster monomials, and cluster algebras are all defined analogously.

Given a valued quiver $(Q, v)$, it’s associated Cartan matrix $C$ is a $Q_0 \times Q_0$ matrix whose entries are given by

[theorem=(Finite-Type Classification)] The cluster algebra $\mathcal{A}_{(Q,v)}$ is finite type if and only if $Q$ is mutation-equivalent to a valued quiver whose associated Cartan matrix corresponds to a finite root system. [/theorem]

The finite-type classification was proven by Fomin and Zelevinsky (see Theorem 1.4 of Cluster algebras II: Finite type classification).

According to Keller, the independence conjecture is mostly open as is the positivity conjecture. Dupont showed the positivity conjecture to be true in rank two.

Skew-symmetrizable cluster algebras of geometric type

Let $1 \le n \le m$. Let $\widetilde{Q}$ be an ice quiver of type (n, m) (a quiver with $m$ vertices). The principal part of $\widetilde{Q}$ is the full subquiver $Q$ whose vertices are $[n]$. The vertices $[m] \setminus [n]$ are called the frozen vertices. The cluster algebra $\mathcal{A}_{\widetilde{Q}}$ is defined in the way as above but:

• Only mutations at non-frozen vertices are allowed.
• No arrows between frozen vertices are added in mutation (note: this can actually be disregarded because such arrows have no contribution to the exchange relations).
• The variables $x_{n+1}, \ldots x_m$ are called coefficients.
• The cluster type of $\widetilde{Q}$ is that of its principle part $Q$ (if it is defined).

As usual, we associate an integer matrix to our quiver. Let $\widetilde{B}$ be the integer $m\times n$-matrix whose coefficient $b_{ij}$ is given by

over $i \in [m]$ and $j \in [n]$. The top $n\times n$ part $B$ of $\widetilde{B}$ is called its principal part.

Of course, one can also define the cluster algebra associated with a valued ice quiver with an integer skew-symmetrizable $m\times n$-matrix.

As proven in Cluster algebras II: Finite type classification, there is a stronger version of the Laurent phenomenon.

[theorem=(Laurent Phenomenon)] Each cluster variable in $\mathcal{A}_{\widetilde Q}$ is a Laurent polynomial in the initial variables $x_1, \ldots, x_n$ with coefficients in $\ZZ[x_{n+1}, \ldots, x_m]$. [/theorem]

There is a localization perspective of $\mathcal{A}_\widetilde{Q}$ given by inverting some of the coefficients. Let $K/\QQ$ be a field extension and $A$ a commutative $K$-algebra that is an integral domain. A cluster structure of type $\widetilde Q$ on $A$ is given by an isomorphism $\phi:\mathcal{A}_\widetilde{Q} \otimes_\QQ K \to A$. This isomorphism is determined by the immages of the coefficients and of the initial cluster variables. Under this terminology, we call the datum of $\widetilde Q$ and of the $\phi(x_i)$ an initial seed for $A$.

[proposition=(4.2 of Keller’s)] Let $X$ be a rational quasi-affine irreducible algebraic variety over $\CC$. Let $\widetilde{Q}$ be an ice quiver of type $(n,m)$. Assume that we are given a regular function $\phi_c$ on $X$ for each coefficient $c = x_i$ (for all $n < i \le m$), and a regular function $\phi_x$ on $X$ for each cluster variable $x$ of $\mathcal{A}_{\widetilde Q}$ such that

a) The dimension of $X$ equals $m$.

b) The functions $\phi_x$ and $\phi_c$ generate the coordinate algebra $\CC[X]$

c) The correspondence $x \mapsto \phi_x$, $c \mapsto \phi_c$ takes each exchange relation of $\mathcal{A}_{\widetilde Q}$ to an equality in $\CC[x]$.

Then the correspondence $x \mapsto \phi_x$, $c \mapsto \phi_c$ extends to an algebra isomorphism $\phi:\mathcal{A}_{\widetilde Q} \otimes_\QQ \CC \xrightarrow{\simeq} \CC[X]$ so that $\CC[X]$ carries a cluster algebra structure of type $\widetilde{Q}$ with initial seed $\phi_x$, $1 \le i \le m$. [/proposition]

We should note that this is really Proposition 11.1 of Cluster algebras II: Finite type classification due to Fomin and Zelevinsky. In any case, this explains why there is a natural cluster algebra structure of type $A_n$ on the Grassmannian $\Gr(2, n + 3)$.

[example=(Cluster structure of $\Gr(2,n+3)$)] Let $n \ge 1$ be an integer and $A = \CC[\Gr(2,n+3)]$. This algebra is generated by the Plucker coordinates subject to the Plucker relations. The cluster algebra structure is given by taking a regular $n+3$-gon $P$ whose vertices are labeled by $[n+3]$ clockwise. Then:

• The coefficients are the variables $x_{ij}$ associated with the sides of $P$.
• The cluster variables are the variables $x_{ij}$ associated with the diagonals of $P$.
• The clusters are the $n$-tuples of cluster variables corresponding to diagonals which form a triangulation of $P$. [/example]

General cluster algebras

$n$-regular tree parametrization

There is a parametrization of the seeds in the mutation class of a given inital seed. Let $1 \le n \le m$ be integers and $\widetilde Q$ be a valued ice quiver of type $(n, m)$. Let $X = \{x_1, \ldots, x_m\}$ be the initial cluster and $(\widetilde Q, X)$ the initial seed. Let $\mathbb T_n$ be the $n$-regular tree:

• Edges are labeled by $[n]$ so that the $n$ edges incident to a given vertex are all distinctly labeled.
• Fix a vertex $t_0$ of $\mathbb T_n$. To each vertex $t$ of $\mathbb T_n$, we associate a seed $(\widetilde Q(t), X(t))$ such that:
  • If $t = t_0$, we have the initial seed.
  • Otherwise, whenever $t$ adjacent to $t’$ by an edge labeled by $k$, the seeds associated with $t$ and $t’$ are related by mutation at $k$.

We will write $x_i(t)$ ($1 \le i \le n$) for the cluster variables in the seed $X(t)$ and $\widetilde B(t)$ for the matrix associated with $\widetilde Q(t)$.

Principal coefficients

Let $n \ge 1$ be an integer and $Q$ a valued quiver with $n$ vertices. Let $B = B_Q$ be the associated $n\times n$-matrix.

Principal coefficients: $c$-vectors

Let $Q_{pr}$ be the principal extension of $Q$. That is, it is the valued quiver obtained from $Q$ by adding new vertices $n + 1, \ldots, 2n$ and new arrows $i + n \to i$ (for all $1 \le i \le n$) for each vertex $i$ of $Q$. The cluster algeebra with principal coefficients associated with $Q$ is the cluster algebra associated with $Q_{pr}$ and we write $B_{pr}$ for the associated $2n \times n$-matrix. Clearly, $B_pr$ is obtained from $B$ by

For a vertex $t$ of the $n$-regular tree, the matrix of $c$-vectors $C(t)$ is the $n\times n$-matrix appearing in the bottom part of

The columns of $C(t)$ are the $c$-vectors at $t$.

[orangebox=Conjecture] Each $c$-vector associated with a valued quiver is nonzero and has either all components nonnegative or all components nonpositive. [/orangebox]

Principal coefficients: $F$-polynomials and $g$-vectors

For each vertex $t$ of the $n$-regular tree, the cluster variable $x_j(t)$ belongs to the ring

by the Laurent phenomenon. The $F$-polynomial

is by definition the specialization of $x_{j}(t)$ by setting

To define $g$-vectors, we grade the ring

with the $\ZZ^n$-grading defined by

For each vertex $t$ of the $n$-regular tree, the cluster variables $x_j(t)$ of $\mathcal{A}(Q_{pr})$ are homogeneous under this grading and its degree is the $g$-vector $g_j(t)$. The matrix of $g$-vectors $G(t)$ has as its columns $g_j(t)$.

[theorem] Let $p_{ij}$ be the number of paths from $i$ to $j$ and $\alpha_1, \ldots, \alpha_n$ the simple roots of the root system corresponding to the underlying graph of $Q$. The linear map taking $e_j$ to $\sum_{i=1}^n p_{ij}\alpha_i$ is a bijection from the set of $g$-vectors of $Q$ to the union of the set of real positive Schur roots with the set of negative simple roots. [/theorem]

This result was due to Caldero-Keller.

Cluster algebras with coefficients in a semifield

A semifield is an abelian group $\mathbb{P}$ endowed with a binary operation $\oplus:\mathbb{P} \times \mathbb{P} \to \mathbb{P}$ which is commutative, associative, and distributive with respect to the group law of $\mathbb{P}$.

The tropical semifield $\operatorname{Trop}(u_1, \ldots, u_n)$ is the free (multiplicative) abelian group generated by the indeterminates $u_i$ endowed with the operation $\oplus$ defined by

Thus, this semifield is isomorphic to $\ZZ_{\mathrm{trop}}^n$ where $\ZZ_{\mathrm{trop}}$ is the abelian group $\ZZ$ endowed with the operation $\oplus$ defined by $x \oplus y = \min(x, y)$.

In Lemma 2.1.6 of Parametrizations of Canonical Bases and Totally Positive Matrices by Berenstein-Fomin-Zelevinsky, it was shown that the universal semifield $\QQ_{sf}(x_1, \ldots, x_n)$ on given indeterminates $x_1, \ldots, x_n$ is the closure, in $\QQ(x_1, \ldots, x_n)$, of the set $\{x_1, \ldots, x_n\}$ under multiplication, division, and addition. An important remark is that this closure contains polynomials with coefficients that may not be positive. For instance,

Note that abelian groups underlying a semifield $\mathbb{P}$ must be torsion-free since if $x \in \mathbb P$ satisfies $x^m = 1$, then

by writing $x^m = 1$ then using commutativity of $\oplus$. Thus, the group ring $\ZZ \mathbb{P}$ is an integral domain.

[definition=($Y$-seeds)] Fix a semifield $\mathbb{P}$ and an integer $n \ge 1$. A $Y$-seed of rank $n$ with values in $\mathbb{P}$ is a pair $(Q, Y)$ formed by a valued quiver $Q$ with $n$ vertices and by a sequence $Y = (y_1, \ldots, y_n)$ of elements of $\mathbb{P}$. Let $B$ be the skew-symmetrizable matrix corresponding to $Q$. If $(Q, Y)$ is a $Y$-seed and $k$ a vertex of $Q$, the mutated $Y$-seed $\mu_k(Q, Y)$ is the $Y$-seed $(Q’, Y’)$ where $Q’ = \mu_k(Q)$ and, for $1 \le j \le n$, we have

[/definition]

[definition=(Seeds with coefficients)] Let $\mathbb{QP}$ be the fraction field of the group ring $\mathbb{ZP}$ and $\mathcal{F}$ be any field obtained from $\mathbb{QP}$ by adjoining $n$ indeterminates. A seed with coefficients in $\mathbb P$ is a triple $(Q, Y, X)$ where $(Q, Y)$ is a $Y$-seed of rank $n$ with values in $\mathbb{P}$ and $X$ is a sequence of $\mathcal{F}$ which freely generate the field $\mathcal{F}$. If $(Q, Y, X)$ is a seed and $k$ a vertex of $Q$, the mutation $\mu_k(Q, Y, X)$ is the seed formed by the mutation $\mu_k(Q, Y)$ and the sequence $X’$ with $x_j’ = x_j$ for $j \neq k$ and $x_k’$ defined by the exchange relation

[/definition]

[definition=(Seed pattern)] A seed pattern is the datum for each vertex $t$ of the $n$-regular tree, of a seed $(Q(t), Y(t), X(t))$ such that if $t$ and $t’$ are adjacent via an edge labeled $k$, then the seeds corresponding to $t$ and $t’$ are linked by a mutation at $k$. [/definition]

[definition=(Cluster algebra)] The cluster algebra is the $\mathbb{ZP}$-subalgebra of the field $\mathcal{F}$ generated by the cluster variables. [/definition]

To recover the cluster algebras of geometric type, we define $\mathbb{P}$ to be the semifield $\operatorname{Trop}(x_{n+1}, \ldots, x_m)$ and the initial $Y$-variables to be

The separation formulas

The following formula was proved by Fomin and Zelevinsky.

[theorem] For each vertex $t$ of the $n$-regular tree and each $1 \le j \le n$, we have

where $\hat y_\ell = y_j \prod_i x_i^{b_{i\ell}}$. [/theorem]