Infinite string bricks and Sturmian words over some gentle algebras

TL;DR

This work classifies infinite string bricks over the double-Kronecker gentle algebra by translating string data into Sturmian words. The authors develop a rich combinatorial bridge between representation theory and binary word theory: strings and envelopes in gentle algebras correspond to cutting sequences and Sturmian/Christoffel words, allowing an explicit Sturmian-based description of brick modules. They first achieve a complete classification for the double-Kronecker case, showing bricks arise exactly from characteristic Sturmian words in appropriate Str-sets, and then extend the approach to a broader family of gentle algebras with similar kissing configurations via an envelope/covering framework. The results illuminate deep connections between brick theory, string modules, and low-complexity binary words, with potential applications to generic bricks and broader gentle-algebra families.

Abstract

We study infinite string modules that are bricks over some gentle algebras. In particular, we first give a complete classification of these modules over the double-Kronecker gentle algebra and prove that each family is in bijection with a family of Sturmian (binary) words. We then generalize some of our results to a larger family of gentle algebras.

Figures (6)

• Figure 1: Geometric presentation of lower cutting word $r_{m,0}^{D}$ for $D = (0, \infty)$ and $m = \frac{2}{1 + \sqrt{5}}$.
• Figure 2: Geometric presentation of Christoffel words
• Figure 3: A bound quiver $(Q,I)$, where $A = kQ/I$ is gentle. The dashed arcs depict the generators of $I$, given by $\alpha\beta$ and $\delta\gamma$.
• Figure 4: General configuration of a quotient envelope of a string
• Figure 5: $n$-fold Kronecker gentle algebra.

Theorems & Definitions (64)

• Definition 2.1
• Example 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.2: Theorem 1.3.13 in Lo
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Example 2.6
• Remark 2.7