Rowmotion in slow motion. (English) Zbl 1459.06010
Summary: Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal \(x\) to the order ideal generated by the minimal elements not in \(x\). It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if \(A\) is a representation finite algebra and mod \(A\) has no cycles, then the torsion classes of \(A\) ordered by inclusion form a trim lattice.
MSC:
| 06D75 | Other generalizations of distributive lattices |
| 06B10 | Lattice ideals, congruence relations |
| 05E16 | Combinatorial aspects of groups and algebras |
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