Introduction to Cluster Algebras

This is a Lean formalization of Chapters 1–7 of Fomin, Williams and Zelevinsky's Introduction to Cluster Algebras.

Main Results

• Mutating a quiver and then taking its extended exchange matrix agrees with mutating the matrix.
• Matrix mutation preserves the determinant when the mutated diagonal entry vanishes.
• Matrix mutation preserves the rank when the mutated diagonal entry vanishes.
• Matrix mutation preserves the gcd of the entries, assuming mutation in that direction is an involution.
• A skew-symmetrizing vector of a matrix with connected diagram is unique up to a scalar.
• The Markov quiver is isomorphic to each of its mutations.
• The arrows between the chambers of a wiring diagram form a quiver.
• A braid move of a wiring diagram mutates its quiver.
• A double wiring diagram on n + 2 strands has (n + 2)² chambers, (n + 1)² of them bounded.
• Orientations of non-isomorphic finite trees are not mutation equivalent, assuming the Caldero–Keller theorem.

See §Formal Challenge for a formal certificate.

Dependencies

This depends on Mathlib.

Formal Challenge

A formal challenge file certifying that this repository does formalize the results claimed above is located at Challenge/Basic.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.

This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:

lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.