Rowmotion in slow motion

Abstract

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 modA has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

Tamari Lattices for Parabolic Quotients of the Symmetric Group

“…(We can always order some of the irreducible elements of a lattice in such a way, the extremality of L guarantees that this is an ordering of all irreducibles.) With the help of this ordering we may follow [21, Definition 2(b)] and define the Galois graph of L (see also [42,Section 2.3]). This is the directed graph G(L) with vertex set [s], where i → k if and only if i = k, and j i ≤ m k .…”

“…In [42,Theorem 5.5] it was shown that the complement of the undirected Galois graph of an extremal semidistributive lattice L is precisely the 1-skeleton of the so-called canonical join complex of L. This is the simplicial complex whose faces are canonical join representations of L. By Proposition 4.10 the canonical join representations in T J n are precisely the J-noncrossing partitions. We thus have the following corollary (which may also be verified directly).…”

Independence posets

“…Suppose top(G) is a lattice; since it only has |G| join and |G| meet irreducible elements, and since it has a chain of length |G| by Lemma 3.4, it is extremal. In [TW17], we represented extremal lattices in the following way, following a construction of Markowsky [Mar92]. Any acyclic directed graph G gives rise to an extremal lattice L(G), as follows: for X, Y ⊆ G with X ∩ Y = ∅, we say (X, Y ) is an orthogonal pair if there is no edge from any i ∈ X to any k ∈ Y , and we say it is a maximal orthogonal pair if X and Y are maximal with that property.…”

“…Conversely, we can associate an acyclic directed grath G(L) to any extremal lattice called its Galois graph with the property that L(G(L)) ≃ L. We refer to [TW17] for further details, including Markowsky's generalization of Birkhoff's fundamental theorem of distributive lattices to extremal lattices.…”

The CDE property for skew vexillary permutations

“…Following the work of Reading [36] in the case of weak order on permutations, Barnard [2] defined a canonical way to label the edges of the Hasse diagram (i.e., the cover relations) of any semidistributive lattice with join irreducible elements. As Thomas and Williams [54] further emphasized, this edge labeling gives a natural way to extend the notion of toggling to a semidistributive lattice. This extended notion of toggling also allows us to generalize the notions of toggle-symmetric distributions, and tCDE posets, to the semidistributive setting.…”