# Clusters and the semicanonical basis

In the final instalment of this trilogy comparing Lusztig’s semicanonical basis on $\mathbb{C}[N]$ with Fomin-Zelevinsky’s cluster basis, I will discuss Geiss-Leclerc-Schroer’s construction of the cluster structure on $\mathbb{C}[N]$ in dual terms to Lusztig’s Lagrangian construction of $U\mathfrak{n}$, which then makes the comparison transparent.

Their construction also clarifies why the Dynkin types $\Gamma$ of the preprojective algebras $\text{PP}(\Gamma)$ with only finitely many indecomposable modules coincide with the types for which the cluster structure on $\mathbb{C}[N]$ is finite-type, where $N$ is a maximal unipotent subgroup inside a simple Lie group of type $\Gamma$.

As a bonus, I included some notes – adapted from a recent talk by Paul Hacking here – that act as a prologue for my Lie groups class paper; they elaborate on one of the entrances cluster algebras make into mirror symmetry, and how this fosters a geometrical expectation of some sort of ‘semicanonical’ basis in situations where they arise.