An analogue of distributivity for ungraded lattices

In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Other than distributive lattices, the main examples of trim lattices are the Tamari lattices and various generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim, and we conjecture that all Cambrian lattices are trim.Comment: 19 pages, 4 figures. Version 2 includes small improvements to exposition, corrections of typos, and a new section showing that if a group G acts on a trim lattice by lattice automorphisms, then the sublattice of L consisting of elements fixed by G is tri

Trimness of Closed Intervals in Cambrian Semilattices

In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. This means that if such an interval has length $k$, then there exists a maximal chain of length $k$ consisting of left-modular elements, and there are precisely $k$ join- and $k$ meet-irreducible elements in this interval. Consequently every graded interval in $\mathcal{C}_{\gamma}$ is distributive. This problem was open for any Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure

Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity

The $\gamma$-Cambrian semilattices $\mathcal{C}_{\gamma}$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, and they are lattices if and only if $W$ is finite. In the case where $W$ is the symmetric group $\mathfrak{S}_{n}$ and $\gamma$ is the long cycle $(1\;2\;\ldots\;n)$ the corresponding $\gamma$-Cambrian lattice is isomorphic to the well-known Tamari lattice $\mathcal{T}_{n}$. Recently, Kallipoliti and the author have investigated $\mathcal{C}_{\gamma}$ from a topological viewpoint, and showed that many properties of the Tamari lattices can be generalized nicely. In the present article this investigation is continued on a structural level using the observation of Reading and Speyer that $\mathcal{C}_{\gamma}$ is semidistributive. First we prove that every closed interval of $\mathcal{C}_{\gamma}$ is a bounded-homomorphic image of a free lattice (in fact it is a so-called $\mathcal{H\!H}$-lattice). Subsequently we prove that each closed interval of $\mathcal{C}_{\gamma}$ is trim, we determine its breadth, and we characterize the closed intervals that are dismantlable.Comment: This paper has been withdrawn by the author due to a gap in the proof of Theorem 1.1(i). The results in Theorems 1.1(ii)-(iv) and 1.2, and those needed for their proofs remain true, and will be addressed in separate articles. I suspect that the claim of Theorem 1.1(i) is still true. In fact, I suspect that quotients of HH-lattices are HH-lattices again. Comments are very welcom

Tamari lattices and noncrossing partitions in type B

AbstractThe usual, or type An, Tamari lattice is a partial order on TnA, the triangulations of an (n+3)-gon. We define a partial order on TnB, the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be considered the Bn Tamari lattice. We also define a bijection between TnB and the noncrossing partitions of type Bn defined by Reiner

Towards m-Cambrian Lattices

For positive integers $m$ and $k$, we introduce a family of lattices $\mathcal{C}_{k}^{(m)}$ associated to the Cambrian lattice $\mathcal{C}_{k}$ of the dihedral group $I_{2}(k)$. We show that $\mathcal{C}_{k}^{(m)}$ satisfies some basic properties of a Fuss-Catalan generalization of $\mathcal{C}_{k}$, namely that $\mathcal{C}_{k}^{(1)}=\mathcal{C}_{k}$ and \bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr). Subsequently, we prove some structural and topological properties of these lattices---namely that they are trim and EL-shellable---which were known for $\mathcal{C}_{k}$ before. Remarkably, our construction coincides in the case $k=3$ with the $m$-Tamari lattice of parameter 3 due to Bergeron and Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the context of other Coxeter groups, in particular we conjecture that the lattice completion of the analogous construction for the symmetric group $\mathfrak{S}_{n}$ and the long cycle $(1\;2\;\ldots\;n)$ is isomorphic to the $m$-Tamari lattice of parameter $n$.Comment: 20 pages, 13 figures. The results of this paper are subsumed by arXiv:1312.2520, and it will therefore not be publishe

Positivity results on ribbon Schur function differences

There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur-positivity is essentially a product of two chains.Comment: 20 pages, 5 figures. Minor expository changes. Final version, to appear in the European J. Combin

Rowmotion Markov Chains

Rowmotion is a certain well-studied bijective operator on the distributive lattice $J(P)$ of order ideals of a finite poset $P$. We introduce the rowmotion Markov chain ${\bf M}_{J(P)}$ by assigning a probability $p_x$ to each $x\in P$ and using these probabilities to insert randomness into the original definition of rowmotion. More generally, we introduce a very broad family of toggle Markov chains inspired by Striker's notion of generalized toggling. We characterize when toggle Markov chains are irreducible, and we show that each toggle Markov chain has a remarkably simple stationary distribution. We also provide a second generalization of rowmotion Markov chains to the context of semidistrim lattices. Given a semidistrim lattice $L$, we assign a probability $p_j$ to each join-irreducible element $j$ of $L$ and use these probabilities to construct a rowmotion Markov chain ${\bf M}_L$. Under the assumption that each probability $p_j$ is strictly between $0$ and $1$, we prove that ${\bf M}_{L}$ is irreducible. We also compute the stationary distribution of the rowmotion Markov chain of a lattice obtained by adding a minimal element and a maximal element to a disjoint union of two chains. We bound the mixing time of ${\bf M}_{L}$ for an arbitrary semidistrim lattice $L$. In the special case when $L$ is a Boolean lattice, we use spectral methods to obtain much stronger estimates on the mixing time, showing that rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.Comment: 20 pages, 4 figures. The new version introduces and studies toggle Markov chains and proves cutoff for rowmotion Markov chains of Boolean lattice