Growth of infinite frieze patterns of affine type

We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze patterns given by both the homogeneous and non-homogeneous stable tubes all have the same growth coefficients. We also derive and verify an explicit formula for the $k$-th growth coefficient, expressed directly in terms of data from homogeneous tubes, or, alternatively, from appropriate elements of the corresponding cluster algebra.

💡 Research Summary

This paper investigates the growth coefficients of infinite frieze patterns that arise from cluster algebras of affine type. The authors combine three main perspectives—cluster modular groups, surface triangulations, and representation theory of quivers—to obtain a unified description of these coefficients.

First, the authors recall the classical definition of a frieze pattern as an infinite strip of integers satisfying the diamond relation, and they formalize a frieze as a homomorphism from a cluster algebra to the integers. The “quiddity row” (the first non‑trivial row) determines the whole pattern, and the difference between entries in successive rows defines the k‑th growth coefficient sₖ(F). Proposition 2.1 shows that these differences are constant along each row and satisfy the linear recurrence sₖ₊₁ = s₁·sₖ – sₖ₋₁. Consequently, all growth coefficients are given by the normalized Chebyshev polynomials of the first kind: sₖ = Tₖ(s₁).

In Section 3 the authors interpret s₁ geometrically via the cluster modular group. For affine type A they consider an annulus Aₘ,ₙ with a triangulation T that yields an infinite periodic frieze. The entry x_{i,i+n−1} corresponds to a self‑intersecting arc γ_{i,i+n−1}. By applying skein relations to smooth the self‑intersection, they obtain the relation x_{i,i+n−1}=x_{i+1,i+n}+Θ−1, where Θ is the value of the frieze on the unique simple closed curve σ winding once around the annulus. Hence s₁ = Θ. The same argument works for higher k: the k‑th growth coefficient equals the frieze value on the k‑fold bracelet (σ traversed k times). An analogous discussion for affine type D uses a twice‑punctured disc; the simple closed curve enclosing both punctures again yields the common growth coefficient for all three infinite friezes associated with that surface.

Section 4 turns to representation theory. For an affine quiver Q the associated cluster category C_Q decomposes into a transjective component and infinitely many regular components, each a tube. Tubes are either homogeneous (all objects have the same quasi‑length) or non‑homogeneous (rank n tubes). Using the Caldero–Chapoton map, the authors send the τ‑orbit of the quasi‑simple objects in a non‑homogeneous tube to cluster variables, specialize the initial cluster to 1, and obtain an integer sequence that serves as the quiddity row. This construction produces the same infinite periodic friezes described in Section 3.

The central representation‑theoretic results are Theorem 4.5 and Proposition 4.8. Theorem 4.5 proves that any two tubes belonging to the same mutation class of Q give rise to friezes with identical growth coefficients. The proof exploits the linear relations between dimension vectors of objects in homogeneous tubes and the Caldero–Chapoton images of objects in non‑homogeneous tubes. Proposition 4.8 provides an explicit formula for sₖ in terms of data from a homogeneous tube: if X_R denotes the Caldero–Chapoton image of a quasi‑simple object R in a homogeneous tube, then sₖ = Tₖ(X_R).

Section 5 supplies concrete examples. For affine A with (m,n) = (3,5) the authors compute the λ‑length Θ of the core curve σ to be 7, yielding s₁ = 7, s₂ = 2·7²−2 = 96, s₃ = 2·7·96−7 = 1335, which match the differences observed in the generated frieze. For affine D they consider a twice‑punctured disc with five boundary points; the closed curve around the punctures gives Θ = 5, and all three associated friezes share this growth coefficient.

The paper also discusses recent independent work by Plamondon and Stella (PS26), who defined infinite friezes for all affine types using root systems and theta functions, proving that friezes from the same root system share growth coefficients. The present results agree with theirs but are obtained via cluster modular groups, skein relations, and AR‑quiver analysis, offering a complementary viewpoint.

In conclusion, the authors demonstrate that for any fixed affine cluster category, every infinite frieze pattern arising from either homogeneous or non‑homogeneous tubes possesses the same sequence of growth coefficients, and these coefficients are precisely the Chebyshev polynomials evaluated at the frieze value of the fundamental closed curve. This unifies geometric, combinatorial, and categorical approaches to the study of infinite friezes in affine type and provides explicit, computable formulas for all growth coefficients.