Pythagorean triples and cluster algebras

This post is coincidentally a week after my previous one [I originally wrote the new year post on 01/04 and this a week later before being persuaded to convert to WordPress] however I doubt that such regularity will last once the semester starts. I’ve been having some interesting conversations with my good friend Alex – who has the same new year’s resolution as me: his blog is coming soon! – over the Christmas break about a most classical of subjects: Pythagorean triples. It was shown in the 1930s that primitive Pythagorean triples fall into the structure of a ternary tree linked by certain linear transformations that are guaranteed to produce new Pythagorean triples from old. This immediately reminded me of the Markov equation. See my recent talk notes for some details of how this seemingly innocuous equation often crops up in geometry. The Markov equation has a cluster algebra underpinning it in the sense that the transformations linking its solutions can be realised as quiver mutations, and that the seeds of the corresponding cluster algebra biject with solutions. For some winter entertainment, I decided to try to tell a similar story with Pythagoras’ equation a^2+b^2=c^2 as protagonist. There turned out to be some positivity issues that obstructed such a concise and complete tale in this case, however I made some decent progress! You can read my account of it here.

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