Self-injective Jacobian algebras from Postnikov diagrams

Abstract

We study a finite-dimensional algebra $\Lambda$ constructed from a Postnikov diagram $D$ in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus $\Lambda$ is isomorphic to the stable endomorphism algebra of the cluster tilting module $T\in \underline{\mathrm{CM}}(B)$ introduced by Jensen-King-Su in order to categorify the cluster algebra structure of $\mathbb{C}[{\mathrm{Gr}}_k({\mathbb{C}}^n)]$. We show that $\Lambda$ is self-injective if and only if $D$ has a certain rotational symmetry. In this case, $\Lambda$ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.

Fixed a mistake in the proof of Theorem 5.6, updated references, minor changes and fixes. Final version, to appear in Algebr. Represent. Theory. 32 pages, 31 figures