Generalized Cluster Algebra of Geometric Type

• Generalized cluster algebras of geometric type are algebraic structures that extend classical versions using arbitrary fixed polynomials and tropical coefficient systems.
• They bridge combinatorial and geometric phenomena by incorporating polynomial exchange rules, leading to well-defined invariants like c-vectors and g-vectors.
• Their structure enables companion ordinary cluster algebras, explicit standard monomial bases, and categorical realizations through quotient and mutation frameworks.

A generalized cluster algebra of geometric type is an algebraic structure in which exchange relations—central to the mutation dynamics—are generalized from binomials (as in classical cluster algebras) to arbitrary fixed polynomials with combinatorial and geometric constraints. This broadens the cluster algebra paradigm introduced by Fomin and Zelevinsky, capturing an enlarged array of algebraic and combinatorial phenomena while retaining essential features such as the Laurent phenomenon, recursive invariants, and intricate links to quiver representations, geometry, and combinatorics. The geometric type refers to the use of a coefficient semifield with tropical addition, imparting geometric and polyhedral structure to the theory.

1. Formal Definition and Mutation Structure

Let $n, m$ be integers with $m\geq n$. A generalized cluster algebra of geometric type is defined using:

• An extended cluster $\mathbf{x} = (x_1, \dots, x_m)$ of algebraically independent variables;
• An $m \times n$ extended exchange matrix $B$ whose principal $n \times n$ part is skew-symmetrizable;
• For each mutable index $i = 1, \dots, n$, a positive integer $d_i>0$ dividing all entries in column $i$ of the principal part, and a string of polynomials (or monomials) $\rho_i = (p_{i,0},\ldots,p_{i,d_i})$ in the frozen variables, with $p_{i,0}=p_{i,d_i}=1$;
• The data $(\mathbf{x}, \rho, B)$ constitute a generalized seed.

Mutation in direction $k$ yields a new seed $(\mathbf{x}', \rho', B')$, by:

$$x_k x_k' = \sum_{r=0}^{d_k} p_{k, r} \prod_{j=1}^m x_j^{r[\beta_{j k}]_+ + (d_k - r)[-\beta_{j k}]_+},$$

where $\beta_{j k} = b_{j k}/d_k$, and $[y]_+ = \max(y,0)$. The coefficient strings satisfy $\rho'_{k, r} = \rho_{k, d_k - r}$, while $\rho'_i = \rho_i$ for $i\neq k$, and the exchange matrix is mutated by the Fomin–Zelevinsky formula (Bai et al., 2020).

When all $d_i = 1$ and $\rho_i = (1,1)$, the definition recovers the classical (binomial) cluster algebra.

2. Fundamental Properties: Laurent Phenomenon and Bounds

Generalized cluster algebras of geometric type retain the Laurent phenomenon: every cluster variable is a Laurent polynomial in the variables of any one seed, with coefficients in the group ring $\mathbb{Z}\mathbb{P}$ of the frozen variables (Bai et al., 2020, Huang et al., 21 Jan 2024).

For acyclic and coprime seeds, the algebra admits further structural theorems:

• The lower bound algebra (generated by $\{x_i, x'_i\}$) and the upper bound algebra (the intersection of Laurent rings over all mutations in all directions) coincide with the cluster algebra:

$\mathcal{A} = \mathcal{L} = \mathcal{U},$

and possess a standard monomial basis in $\{x_i, x'_i\}$ excluding $x_i x'_i$ as factors (Bai et al., 2020, Huang et al., 21 Jan 2024).

• In the case of acyclicity (quiver with no oriented cycles) but lacking coprimality, the cluster algebra equals the generalized upper cluster algebra, the intersection of Laurent rings over all possible seeds [(Bai et al., 2020) Appendix A].

3. Invariants and Recursion: $c$-Vectors, $g$-Vectors, $F$-Polynomials

Generalized cluster algebras of geometric type are equipped with key combinatorial invariants analogous to those of Fomin–Zelevinsky:

• $c$-vectors describe the exponents of $y$-variables in the tropicalization of coefficient mutation sequences.
• $g$-vectors encode the grading (degrees of cluster variables) under a natural multi-grading, reflecting the mutation path.
• $F$-polynomials are obtained by evaluating cluster variables at $x_i = 1$, capturing the combinatorial content of variable transformations.

The recursion relations for these invariants extend the classical case, introducing dependence on the exchange degrees $d_i$:

• $c$-vector mutation:

$$c_{ij;t'} = \begin{cases} -c_{ik;t} & j=k,\ c_{ij;t} + c_{ik;t} [d_k b_{kj;t}]_+ + [ -c_{ik;t} ]_+ d_k b_{kj;t} & j\neq k. \end{cases}$$

• $g$-vector mutation and $F$-polynomial recursion possess analogous, degree-dependent formulas (Nakanishi et al., 2015).

These invariants are central to the polyhedral geometry (mutation fans, universal coefficients (Reading, 2012)) and provide a bridge to companion (ordinary) cluster algebras via specialization.

4. Companion Cluster Algebras and Specialization Formulas

Given a generalized cluster algebra of geometric type with exchange degrees $(d_1, \ldots, d_n)$, two companion ordinary cluster algebras are naturally associated:

• Left-companion algebra: formed on the ambient field $\mathbb{Q}\mathbb{P}(x_i^{1/d_i})$ with exchange matrix $D B$, and variables $x_i^{1/d_i}, y_i$.
• Right-companion algebra: formed with the same exchange matrix, but variables $x_i, y_i^{d_i}$ and exchange matrix $B D$.

The $c$-, $g$-, and $F$-invariants of the generalized algebra relate to those of the companions via explicit scaling and specialization formulas, with the generalized polynomial recurrences for invariants specializing to binomial recursions under suitable substitutions:

• For $F$-polynomials, setting $z_{i,s} = \binom{d_i}{s}$ or $z_{i,s} = 0$ specializes the generalized algebra to the left or right companion, respectively (details in (Nakanishi et al., 2015)).

The generalized cluster algebra thus "controls" two ordinary cluster algebras and allows its variables and coefficients to be recovered from those of the companions.

5. Representation as Quotients and Structural Isomorphism

Recent advances demonstrate that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of an ordinary cluster algebra of higher rank:

• The canonical enlargement of the exchange matrix is constructed by forming block-diagonals according to the mutation degrees;
• Ordinary cluster algebra seeds of rank $\sum_i d_i$ are associated with these enlarged matrices;
• The generalized algebra is realized as an explicitly described quotient, with composite variables ($X_{i;t} = \prod_{l=1}^{d_i} x_{i,l;t}$) constrained by factorization identities aligned with the exchange polynomials (Akagi et al., 24 Dec 2025).

The key isomorphism theorem confirms that for any coefficient semifield, the generalized cluster algebra of geometric type is recovered as a quotient of a subalgebra of an ordinary cluster algebra, with the correspondence of all main invariants: $C$-, $G$-matrices, and $F$-polynomials (Akagi et al., 24 Dec 2025).

6. Standard Monomial and PBW-Like Bases; Acyclicity and Coprimality

Acyclicity (absence of oriented cycles in the quiver) and coprimality (coprimality of exchange polynomials) guarantee a strong basis theorem:

• The set of standard monomials in $\{x_i, x'_i\}$, i.e., monomials not divisible by any $x_i x'_i$, forms a $\mathbb{Z}\mathbb{P}$-basis of the cluster algebra;
• When the projective cluster variables are adjoined, this extends to a dual PBW basis, indexed by $\mathbb{Z}^n$, with ordering by lexicographic rules on degrees (Huang et al., 21 Jan 2024).

Furthermore, if the standard monomials are linearly independent, the quiver cannot contain oriented 3-cycles—a restriction on the algebra's combinatorial type.

7. Internal Structure, Subalgebras, and Geometric Models

The internal structure of generalized cluster algebras of geometric type has been analyzed via partial seed endomorphisms and Green's relations. These tools classify subalgebras through semigroup theory, providing a bijection between isomorphism classes of sub-rooted cluster algebras and regular $\mathcal{D}$-classes in the semigroup of partial seed endomorphisms (Huang et al., 2016, Huang et al., 2015).

For cluster algebras associated to surfaces, the topological model of "paunched surfaces" (cutting along arcs, adding boundary components) parameterizes subalgebras geometrically, which correspond to sub-seeds and their algebraic invariants.

In categorical frameworks, every rooted cluster subalgebra admits an associated monoidal categorification, reflecting structural inheritance through subcategories of monoidal categories (Huang et al., 2015).

---

References:

• "On the Generalized Cluster Algebras of Geometric Type" (Bai et al., 2020)
• "Some properties of generalized cluster algebras of geometric types" (Huang et al., 21 Jan 2024)
• "Companion cluster algebras to a generalized cluster algebra" (Nakanishi et al., 2015)
• "Relation between generalized and ordinary cluster algebras" (Akagi et al., 24 Dec 2025)
• "Universal geometric cluster algebras" (Reading, 2012)
• "On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces" (Huang et al., 2016)
• "On Structure of cluster algebras of geometric type I: In view of sub-seeds and seed homomorphisms" (Huang et al., 2015)