Self-injective Jacobian algebras from Postnikov diagrams

Skew group algebras of Jacobian algebras

“…It turns out that in all known examples of self-injective planar QPs a Nakayama automorphism acts by a rotation, hence they fit nicely in our setting. Recently it has been shown that Postnikov diagrams have connections with planar self-injective QPs: in [Pas17] it is proved that the QP coming from an (a, n)-Postnikov diagram on a disk (as in [BKM16]) is self-injective if and only if the diagram is rotation invariant. Thus, our construction produces many examples of symmetric Jacobian algebras, one for every such Postnikov diagram.…”

“…There is a way of producing strongly planar QPs with a group acting by rotations by means of socalled Postnikov diagrams (see [Pos06], [BKM16], [Pas17]). A Postnikov diagram is a collection of oriented curves in a disk subject to some axioms depending on two integer parameters a, n ≥ 1, and it naturally gives rise to a planar QP.…”

Existence of symmetric maximal noncrossing collections of k-element sets

“…One can also look at the stable category CM(B) of CM(B), which is triangulated and in fact 2-Calabi-Yau Remark 7.1. In [BKM16] and [Pas17], the focus is on Postnikov diagrams (or alternating strand diagrams). By [OPS15, Theorem 11.1], there is a bijection between Postnikov diagrams and maximal noncrossing collections, so the two concepts are interchangeable.…”

“…, n} symmetric if it is invariant under adding k (mod n) to all indices. In a recent paper [Pas17] Pasquali showed that symmetric maximal noncrossing collections naturally give rise to self-injective Jacobian algebras. More precisely he showed that any maximal symmetric noncrossing collection for a pair (k, n) gives rise to a self-injective Jacobian algebra whose quiver with potential can be obtained from an embedding of the collection into the plane.…”

Quiddity sequences for $\mathrm{SL}_3$-frieze patterns

“…We denote the set of clusters of Plücker coordinates in the cluster structure of CrGrpk, nqs by S k,n . The dimer algebras associated to each such cluster are interesting in their own right, and recent studies include [2], [8], [11], [13], [18], [20]; see also [4] and [12] for their connections to other areas of research.…”