We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the Cambrian lattice is the weak order on Coxeter-sortable elements. These results exhibit W-Catalan combinatorics arising in the context of the lattice theory of the weak order on W.
In this article, we give a short algebraic proof that all closed intervals in a $γ$-Cambrian semilattice $\mathcal{C}_γ$ are trim for any Coxeter group $W$ and any Coxeter element $γ\in W$. 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 $\mathcal{C}_γ$ is distributive. This problem was open for any Coxeter group that is not a Weyl group.
In this article, we investigate the set of $γ$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $γ\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_γ$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_γ$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $γ\in W$ the lattice $\mathcal…
In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahor…
For each complete meet-semilattice $M$, we define an operator $\mathsf{Pop}_M:M\to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot x\}\cup\{x\}).\] When $M$ is the right weak order on a symmetric group, $\mathsf{Pop}_M$ is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of $\mathsf{Pop}_{\text{Tam}(ν)}$, where $\text{Tam}(ν)$ is the $ν$-Tamari lattice. We determine the maximum size of a forward orbit of $\mathsf{Pop}_{\te…
The $γ$-Cambrian semilattices $\mathcal{C}_γ$ defined by Reading and Speyer are a family of meet-semilattices associated with a Coxeter group $W$ and a Coxeter element $γ\in W$, and they are lattices if and only if $W$ is finite. In the case where $W$ is the symmetric group $\mathfrak{S}_{n}$ and $γ$ is the long cycle $(1\;2\;\ldots\;n)$ the corresponding $γ$-Cambrian lattice is isomorphic to the well-known Tamari lattice $\mathcal{T}_{n}$. Recently, Kallipoliti and the author have investigated $\mathcal{C}_γ$ from a topological viewpoint, and showed that many properties of the Tamari lattices…
Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$ in the right weak order. When $W$ is the symmetric group $S_n$, $\mathsf{Pop}$ coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where $h$ is the Coxeter number of $W$ (with $h=\infty$ if $W$ is infinite) and $O_f(w)$ denotes…
Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator $\mathsf{Pop}_M : M \to M$ for each complete meet-semilattice $M$ by $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}).$$ This paper concerns the dynamics of $\mathsf{Pop}_{\mathrm{Tam}_n}$, where $\mathrm{Tam}_n$ is the $n$-th Tamari lattice. We say an element $x\in \mathrm{Tam}_n$ is $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}_M^t (x)$ is the minimal element and we let $h_t(n)$ denote the number of $t$-$\mathsf{Pop}$-sortable elements in $\mathrm{Tam}_n$. We find an explicit formula for…
In this expository note, I showcase the relevance of Coxeter groups to quiver representations. I discuss (1) real and imaginary roots, (2) reflection functors, and (3) torsion free classes and c-sortable elements. The first two topics are classical, while the third is a more recent development. I show that torsion free classes in rep Q containing finitely many indecomposables correspond bijectively to c-sortable elements in the corresponding Weyl group. This was first established in Dynkin type by Ingalls and Thomas; it was shown in general by Amiot, Iyama, Reiten, and Todorov. The proof in th…
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 generaliza…
Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the con…
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we…
Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this polytope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construction agrees with…
Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $Δ=\{\a_s | s\in S\}$ of the same length and fundamental weights $Δ^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}κ_s v_s$, $κ_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained …
Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf S}_\equiv:W\to W$ by ${\bf S}_\equiv(w)=w\left(π_\downarrow^\equiv(w)\right)^{-1}$, where $π_\downarrow^\equiv(w)$ is the unique minimal element of the congruence class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's stack-sorting map. When $\equiv$ is the descent congruence on $S_n$, the operator ${\bf S}_\equiv$ is the pop-stack-sorting map. We establish …
