The Brick Word

A brick in representation theory is an indecomposable module whose endomorphism ring is a division ring — the algebraic analogue of having no internal symmetries to exploit. Determining whether a module is a brick typically requires algebraic computation in the module category.

Kuber and Sengupta show that for string algebras, bricks are characterized by a property of words. They build a multi-entry inverse automaton that processes pointed binary words, and string modules are bricks if and only if their corresponding pointed words satisfy specific combinatorial conditions over this automaton.

The translation is exact. An algebraic indecomposability property — no nontrivial endomorphisms — maps to a pattern-matching condition on binary strings. The automaton accepts precisely the words whose modules are bricks, converting a representation-theoretic question into a formal language question.

This generalizes a known connection between infinite string bricks in gentle algebras and Sturmian words — those binary words with minimal combinatorial complexity. The surprise in the earlier result was that algebraic simplicity (being a brick) corresponded to combinatorial simplicity (being Sturmian). The generalization preserves the surprise while extending it: the correspondence between algebraic structure and word structure is not an accident of gentle algebras but a feature of string algebras broadly.

Two fields separated by their entire vocabulary — automata theory and module categories — turn out to be asking the same question in different notation. The automaton does not approximate the algebraic test. It is the algebraic test, expressed in a language where the answer is computable by a finite machine.