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Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra

Florent Hivert, Vincent Pilaud, Ludovic Schwob

TL;DR

We extend the excedance congruence from permutations to the lattice of alternating sign matrices by introducing catalan congruences, i.e., lattice quotients isomorphic to the Stanley lattice on Dyck paths. The authors develop a geometric realization of ASM join-irreducibles as a 3D tetrahedron and build depth/d catalan triangles, walls, and bicolored pipe dreams to classify quotients and relate them to Dyck paths and GT patterns. They prove that for any catalan congruence the maximal elements are covexillary and the minimal elements are $321$-avoiding permutations, and that choosing any representative in each class yields a basis of TL$_n(2)$. They further present a unifying symmetrization framework for join-irreducible posets, linking ASM, Dyck, and Catalan structures and enabling broader interpretations of Temperley–Lieb bases and combinatorial models.

Abstract

We prove that the excedance relation on permutations defined by N. Bergeron and L. Gagnon actually extends to a congruence of the lattice on alternating sign matrices. Motivated by this example, we study all lattice congruences of the lattice on alternating sign matrices whose quotient is isomorphic to the Stanley lattice on Dyck paths, which we call catalan congruences. We prove that the maxima of the congruence classes are always covexillary permutations (and all covexillary permutations appear this way), and that the minimal permutations in each class are always precisely the $321$-avoiding permutations. Finally, we show that any choice of representative permutations in each congruence class yield a basis of the Temperley-Lieb algebra with parameter $2$, vastly generalizing the bases arising from the excedance relation.

Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra

TL;DR

We extend the excedance congruence from permutations to the lattice of alternating sign matrices by introducing catalan congruences, i.e., lattice quotients isomorphic to the Stanley lattice on Dyck paths. The authors develop a geometric realization of ASM join-irreducibles as a 3D tetrahedron and build depth/d catalan triangles, walls, and bicolored pipe dreams to classify quotients and relate them to Dyck paths and GT patterns. They prove that for any catalan congruence the maximal elements are covexillary and the minimal elements are -avoiding permutations, and that choosing any representative in each class yields a basis of TL. They further present a unifying symmetrization framework for join-irreducible posets, linking ASM, Dyck, and Catalan structures and enabling broader interpretations of Temperley–Lieb bases and combinatorial models.

Abstract

We prove that the excedance relation on permutations defined by N. Bergeron and L. Gagnon actually extends to a congruence of the lattice on alternating sign matrices. Motivated by this example, we study all lattice congruences of the lattice on alternating sign matrices whose quotient is isomorphic to the Stanley lattice on Dyck paths, which we call catalan congruences. We prove that the maxima of the congruence classes are always covexillary permutations (and all covexillary permutations appear this way), and that the minimal permutations in each class are always precisely the -avoiding permutations. Finally, we show that any choice of representative permutations in each congruence class yield a basis of the Temperley-Lieb algebra with parameter , vastly generalizing the bases arising from the excedance relation.

Paper Structure

This paper contains 29 sections, 46 theorems, 20 equations, 30 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Two distributive lattices
  3. Meet and join representations in distributive lattices
  4. Dyck paths
  5. Alternating sign matrices
  6. Higher dimensional analogues
  7. The excedance congruence of $\mathbf{ASM}_n$
  8. Lattice morphisms and congruences
  9. The excedance relation on permutations
  10. The excedance congruence on $\mathbf{ASM}$
  11. Catalan congruences of $\mathbf{ASM}_n$
  12. Quotients of distributive lattices
  13. Walls
  14. Depth triangles
  15. Catalan triangles
  16. ...and 14 more sections

Key Result

Theorem 1.1

For the excedance relation on permutations,

Figures (30)

  • Figure 1: A lower set of $\mathcal{D}_5$ and the corresponding Dyck path of semilength $5$.
  • Figure 2: The Stanley lattice $\mathbf{Dyck}_3$ (left) and the ASM lattice $\mathbf{ASM}_3$ (right).
  • Figure 3: The posets $\mathcal{D}_5$ (left) and $\mathcal{T}_5$ (right).
  • Figure 4: An ASM, a CSM, and a HFM, all corresponding to each other.
  • Figure 5: The ASM of \ref{['fig:ASM1']} seen as a lower set $X$ of the tetrahedron poset $\mathcal{T}_5$ (left) or equivalently as the surface separating $X$ from its complement (middle and right).
  • ...and 25 more figures

Theorems & Definitions (103)

  • Theorem 1.1: BergeronGagnon
  • Theorem 1.2
  • Theorem 2.1: Birkhoff
  • Definition 3.1: BergeronGagnon
  • Definition 3.2: BergeronGagnon
  • Theorem 3.3: BergeronGagnon
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 93 more