The Affine Tamari Lattice

TL;DR

The paper develops two finite lattices, the cyclic Tamari lattice $\mathsf{CTam}_{312}$ and the affine Tamari lattice $\mathsf{ATam}_{312}$, as lattice quotients of Dyer-type orders on translation-invariant total orders, for fixed $n\ge2$. It provides a versatile network of descriptions—via TITOs, TIBITs, arc diagrams, torsion classes, type $D_n$ triangulations, and $g$-vectors—establishing self-duality, semidistributivity, and Catalan-type enumerations $|\mathsf{CTam}|=\binom{2n}{n}$ and $|\mathsf{ATam}|=\frac{3n-2}{n}\binom{2n-2}{n-1}$; it also analyzes rowmotion through cyclic sieving phenomena and derives maximal green sequence length intervals $[2n-1,\binom{n+1}{2}]$ for CTam and $[2n-2,\binom{n+1}{2}-1]$ for ATam. The work connects these lattices to quiver representations (via cycle and $D$-type algebras), type $D_n$ triangulations, and stability/g-vector fans, highlighting deep links to cluster algebras and representation theory. It closes with several future directions, including associahedra, generalized Tamari frameworks, shard orders, and operator dynamics such as pop-stack mutations. Overall, the paper provides a rich, multi-interpretation framework for affine and cyclic Tamari lattices and inaugurates a broad program of future study in combinatorics and representation theory.

Abstract

Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which is the infinite lattice of translation-invariant total orders under containment of inversion sets. The affine Tamari lattice is a quotient of the Dyer lattice, which in turn is a quotient of the cyclic Dyer lattice and is isomorphic to the collection of biclosed sets of the root system of type $\widetilde{A}_{n-1}$ under inclusion. We provide numerous combinatorial and algebraic descriptions of these lattices using translation-invariant total orders, translation-invariant binary in-ordered trees, noncrossing arc diagrams, torsion classes, triangulations, and translation-invariant noncrossing partitions. The cardinalities of the cyclic and affine Tamari lattices are the Catalan numbers of types $B_n$ and $D_n$, respectively. We show that these lattices are self-dual and semidistributive, and we describe their decompositions coming from the Fundamental Theorem of Finite Semidistributive Lattices. We also show that the rowmotion operators on these lattices have well-behaved orbit structures, which we describe via the cyclic sieving phenomenon. Our new combinatorial framework allows us to prove that the lengths of maximal green sequences for the completed path algebra of the oriented $n$-cycle are precisely the integers in the interval $[2n-1,\binom{n+1}{2}]$.

Figures (18)

• Figure 1: The Hasse diagram of the affine Tamari lattice for $n=3$, with each element represented as a window notation of a $312$-avoiding translation-invariant total order. The darker elements are $312$-avoiding affine permutations; the subposet that they induce is not a lattice.
• Figure 2: A translation-invariant binary in-ordered tree for $n=7$.
• Figure 3: Two different translation-invariant binary in-ordered trees for $n=4$ whose underlying infinite binary trees are isomorphic.
• Figure 4: The cyclic Tamari lattice for $n=3$. Each element is represented by a box containing a TIBIT $T$ (on top), the corresponding $312$-avoiding TITO $\sqsubseteq_T$ (on bottom), and the corresponding $132$-avoiding TITO $\sqsubseteq^T$ (in the middle).
• Figure 5: Applying $\pi_{\mathrm{spine}}^\downarrow$ and $\pi_{\mathrm{spine}}^\uparrow$ to two TIBITs for $n=5$.

Theorems & Definitions (113)

• Proposition 2.1: ReadingBook
• Proposition 2.2
• Definition 3.1
• Example 3.2
• Definition 3.3
• Proposition 3.4: Barkley
• Proposition 3.5: BarkleySpeyerCombinatoricsBarkley
• Definition 3.6
• Lemma 3.7
• proof