[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"H1XuiB1CuG":3},"[![](logo.svg)](https://axiommath.ai/)\n\n# Introduction to Cluster Algebras\n\nThis is a Lean formalization of Chapters 1–7 of Fomin, Williams and Zelevinsky's *Introduction to Cluster Algebras*.\n\n## Main Results\n\n* Mutating a quiver and then taking its extended exchange matrix agrees with mutating the matrix.\n* Matrix mutation preserves the determinant when the mutated diagonal entry vanishes.\n* Matrix mutation preserves the rank when the mutated diagonal entry vanishes.\n* Matrix mutation preserves the gcd of the entries, assuming mutation in that direction is an involution.\n* A skew-symmetrizing vector of a matrix with connected diagram is unique up to a scalar.\n* The Markov quiver is isomorphic to each of its mutations.\n* The arrows between the chambers of a wiring diagram form a quiver.\n* A braid move of a wiring diagram mutates its quiver.\n* A double wiring diagram on `n + 2` strands has `(n + 2)²` chambers, `(n + 1)²` of them bounded.\n* Orientations of non-isomorphic finite trees are not mutation equivalent, assuming the Caldero–Keller theorem.\n\nSee [§Formal Challenge](#formal-challenge) for a formal certificate.\n\n## Dependencies\n\nThis depends on [Mathlib](https://github.com/leanprover-community/mathlib4).\n\n## Formal Challenge\n\nA formal challenge file certifying that this repository does formalize the results claimed above is located at [Challenge/Basic.lean](Challenge/Basic.lean). This file only depends on Mathlib. It contains formal statements of [§Main Results](#main-results) with `sorry` as proof.\n\nThis 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:\n```\nlake env comparator Comparator/comparator.json\n```\n\nThis repository has been locally verified with the comparator.\n",1793473260734]