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.

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