Generalized Cluster Algebra with y-Variables

• Generalized cluster algebras with y-variables are algebraic structures that extend ordinary cluster algebras by incorporating arbitrary exchange polynomials and enriched semifield coefficients.
• They enable systematic treatment of mutation dynamics, impacting the study of integrable systems, projective geometry, and supersymmetric gauge theory through explicit coefficient transformations.
• Their composite embedding into ordinary cluster frameworks provides new structural invariants and computational techniques, unifying diverse algebraic and geometric phenomena.

A generalized cluster algebra with $y$-variables is an algebraic structure that extends the framework of ordinary cluster algebras by allowing exchange relations governed by arbitrary polynomials and by enriching the notion of coefficients to arbitrary semifields. The inclusion of $y$-variables enables a systematic treatment of transformations (mutations) of coefficients under iteration, with significant implications for the structure and applications of cluster algebras in both algebraic and geometric contexts. This concept forms the technical foundation of several developments in algebra, geometry, and mathematical physics, including the study of integrable systems, projective geometry (notably the generalized pentagram map), and supersymmetric gauge theory correspondences.

1. Semifield Structure, Seeds, and Generalized Exchange Relations

Let $(\mathbb P, \oplus, \cdot)$ denote a semifield in the sense of Fomin-Zelevinsky, i.e., an abelian group $(\mathbb P, \cdot)$ together with a commutative semigroup $(\mathbb P, \oplus)$ satisfying distributivity: $p\cdot(q_1 \oplus q_2) = p q_1 \oplus p q_2$ for all $p, q_1, q_2 \in \mathbb P$. The algebraic structure above (including the split $\mathbb P_0$ and group ring $\mathbb Z \mathbb P$) allows the definition of generalized exchange polynomials with coefficients in $\mathbb P$.

A generalized seed is a quadruple $\Sigma = (\mathbf x, \mathbf y, \mathbf Z, B)$, where:

• $\mathbf x = (x_1, \dots, x_n)$ are generators of an ambient field $\mathcal F = \mathbb Q \mathbb P(x_1, \dots, x_n)$,
• $\mathbf y \in \mathbb P^n$ (the $y$-variables, or coefficient tuple),
• $\mathbf Z = (Z_1(u), \dots, Z_n(u))$ are exchange polynomials $Z_i(u) = \sum_{\ell=0}^{r_i} \hat z_{i\ell} u^\ell$ with coefficients $\hat z_{i\ell} \in \mathbb{NP}$, $\hat z_{i0} = \hat z_{i r_i} = 1$,
• $B = (b_{ij}) \in \operatorname{Mat}_n(\mathbb Z)$ is a skew-symmetrizable integer matrix.

The associated mutation rules $\mu_k^{\mathbf r}$ for $k \in \{1, \ldots, n\}$ act by: $$\begin{aligned} x'_k &= x_k^{-1} \left( \prod_{j=1}^n x_j^{[-b_{jk}]_+} \right)^{r_k} \frac{Z_k(\hat y_k)}{Z_k|_{\mathbb P_0}(y_k)}, \quad x'_i = x_i \ \text{for } i\neq k \ y'_k &= y_k^{-1}, \qquad y'_i = y_i (y_k^{[b_{ki}]_+})^{r_k} (Z_k|_{\mathbb P_0}(y_k))^{-b_{ki}}, \ \text{for } i\neq k \ Z'_k(u) &= u^{r_k} Z_k(u^{-1}), \quad Z'_i(u) = Z_i(u) \ (i\neq k), \ b'_{ij} &= \begin{cases} -b_{ij} & \text{if } i = k \text{ or } j=k \ b_{ij} + r_k (b_{ik} [b_{kj}]_+ + [-b_{ik}]_+ b_{kj}) & \text{otherwise} \end{cases} \end{aligned}$$ where $\hat y_k = y_k \prod_{j=1}^n x_j^{b_{jk}}$. These rules generalize the ordinary cluster exchange relations by allowing higher degree and more complex polynomial dependence in the mutations of $x$ and $y$ variables (Akagi et al., 24 Dec 2025).

2. Embedding into Ordinary Cluster Algebras via Composite Patterns

Every generalized cluster algebra with $y$-variables and arbitrary exchange polynomials can be realized as a subquotient of an ordinary cluster algebra. The construction proceeds by enlarging the rank, associating a "composite" pattern to each generalized pattern, and tracking variables via auxiliary parameters.

Let $n$ be the rank and $\mathbf r = (r_1, \dots, r_n)$ the degrees of exchange polynomials, with total "pseudo-rank" $\mathcal N = \sum_i r_i$. The enlarged exchange matrix $\mathcal B$ is a block matrix whose indices correspond to $(i,l)$ with $i=1, \ldots, n$, $l=1, \ldots, r_i$, each block filled with entries $b_{ij}$. For each $k$, the "composite mutation" $\mu_k^{\mathrm{c}}$ is the composition of mutations at all $r_k$ copies corresponding to $k$, yielding a new seed with the rotated exchange matrix.

Within this framework, the generalized $y$-pattern is embedded into the ordinary $y$-pattern via identification: $$y_{i;t}^{\mathrm g} \longmapsto \left[ \prod_{l=1}^{r_i} y_{i l; t}^{\mathrm c} \right]^{1/r_i}$$ in an appropriate quotient of a larger semifield. The ambient algebras satisfy

$$\mathcal A^{\mathrm g} \cong \mathcal A^{\mathrm c}/\mathcal I \subset \mathcal A,$$

where $\mathcal I$ is the ideal encoding the specialization from symmetric polynomials to exchange polynomial coefficients in the original pattern (Akagi et al., 24 Dec 2025).

3. Mutation and $Y$-Pattern Dynamics

The mutation formulas for $y$-variables are fundamental to the cluster algebra structure, and take the following form for both ordinary and generalized cluster algebras: $$y'_k = y_k^{-1}; \quad y'_j = y_j\, y_k^{[b_{kj}]_+}\, (1+y_k)^{-b_{kj}}, \quad j \neq k$$ in the principal coefficient setting (Yamazaki, 2016). For generalized cluster algebras, these formulas are enriched to incorporate the evaluation of exchange polynomials at the $y$-variables: $$y'_i = y_i\ (y_k^{[b_{ki}]_+})^{r_k}\ (Z_k|_{\mathbb P_0}(y_k))^{-b_{ki}}.$$

The cluster $y$-variables hence govern the evolution of coefficients in the algebra under sequences of mutations, forming a $Y$-pattern. The Laurent phenomenon and positivity for these variables hold under suitable conditions, mirroring the foundational results for ordinary cluster algebras.

4. Geometric and Physical Realizations: Y-Meshes, Pentagram Maps, and Gauge Theories

Cluster algebras with $y$-variables have deep geometric interpretations. In the context of projective geometry, "Y-meshes" and "Y-pins" provide a configuration-theoretic setting for generalized pentagram maps. A Y-mesh of type $S=\{a,b,c,d\}$ assigns points $P_r \in \mathbb{RP}^D$ to each $r \in \mathbb{Z}^2$, subject to collinearity and distinctness conditions that enable the following projective cross-ratio variable: $y_r = -[P_{r+a}, P_{r+c}, P_{r+b}, P_{r+d}],$ which satisfies octahedral recurrence relations that precisely correspond to $y$-pattern mutations: $$y_{r+a+b} y_{r+c+d} = \frac{(1 + y_{r+a+c})(1 + y_{r+b+d})}{(1 + y_{r+a+d})(1 + y_{r+b+c})}$$ for all $r \in \mathbb{Z}^2$ (Glick et al., 2015).

In mathematical physics, $y$-variables can be interpreted as exponentials of complexified FI parameters in 2d $\mathcal{N}=(2,2)$ supersymmetric gauge theories. In this context, the exact $S^2$ partition function R-matrix $\mathcal R(u; y_v)$ exhibits dependence on $y$-variables that mutate under Seiberg duality in precise accordance with cluster algebra mutation rules. This gives rise to a "cluster-enriched" Yang-Baxter equation, providing a dynamical and physically-motivated generalization of the classical setting (Yamazaki, 2016).

5. Structural Invariants: $C$-matrices, $G$-matrices, and $F$-polynomials

The $C$-matrices, $G$-matrices, and $F$-polynomials encode deep combinatorial and algebraic information about the cluster algebra structure. In the composite embedding, these invariants for the generalized pattern can be recovered as blockwise combinations and specializations of those from the ordinary (composite) pattern. Specifically:

• $C$-matrices: $$c_{ij;t}^{\mathrm g} = \sum_{l=1}^{r_i} c_{il,jm;t}^{\mathrm c} = r_i \tilde c_{ij;t}^{\mathrm c} + \sigma_{j;t} \delta_{ij}$$
• $G$-matrices: $$g_{ij;t}^{\mathrm g} = \sum_{m=1}^{r_j} g_{il,jm;t}^{\mathrm c} = r_j \tilde g_{ij;t}^{\mathrm c} + \sigma_{j;t} \delta_{ij}$$
• $F$-polynomials: $$F_{i;t}^{\mathrm g}(\mathbf y, \mathbf z) = \left. \prod_{l=1}^{r_i} F_{il;t}^{\mathrm c}(s_{11} y_1, \dots, s_{n r_n} y_n) \right|_{\{e_{jm}=z_{jm}\}}$$

where the $\tilde c_{il,jm;t}^{\mathrm c}$ and $\tilde g_{il,jm;t}^{\mathrm c}$ are appropriately shifted composite entries, and the product and substitution reflect the passage to the generalized setting (Akagi et al., 24 Dec 2025).

6. New Phenomena and Applications

Generalized cluster algebras with $y$-variables support new structural phenomena not present in the ordinary theory:

• Quivers associated to these algebras can be embedded on higher genus surfaces (e.g., tori), leading to "cluster integrable systems."
• The geometric dynamics (e.g., generalized pentagram maps) require nontrivial coplanarity and span-intersection consistency in higher dimensions, realized via the Y-mesh formalism.
• The reduction of recurrence order and the emergence of maps with fractal substructure ("gopher," "rabbit," etc.) highlight additional combinatorial richness.
• The occurrence of $y$-variables associated to cross-ratios, triple ratios, and generalized multi-ratios indicates the presence of algebraic invariants beyond the classical cross-ratio.

In physical settings, the cluster-enriched Yang-Baxter equation and quantizations thereof raise new possibilities for integrable systems and the interplay between cluster algebraic and physical dualities (Glick et al., 2015, Yamazaki, 2016).

7. Interrelations and Universality

The universality of generalized cluster algebras with $y$-variables is established via their realization as subquotients (via composite patterns and suitable identifications) of ordinary cluster algebras. This embedding not only unifies the algebraic structures but also allows the transfer of structural results, computational techniques, and invariants between ordinary and generalized settings. This connection also facilitates explicit computations, structural analysis, and the study of representation-theoretic and geometric applications by leveraging the ordinary cluster algebra machinery (Akagi et al., 24 Dec 2025).