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

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.

💡 Research Summary

The paper investigates a deep connection between alternating sign matrices (ASMs), lattice congruences, and the Temperley‑Lieb algebra TL(2). It begins by recalling the “excedance relation” introduced by Bergeron and Gagnon, which groups permutations that share the same set of excedance positions and values. In the Bruhat order on permutations, each excedance class forms an interval whose minimal element is a 321‑avoiding permutation and whose maximal element is a covexillary permutation. The authors observe that this relation can be lifted from the Bruhat order to the full ASM lattice, which is known to be the MacNeille completion of the Bruhat order.

The central notion introduced is that of a “Catalan congruence”: a lattice congruence on the ASM lattice whose quotient is isomorphic to the Stanley lattice of Dyck paths (the Catalan lattice). The paper provides a complete classification of such congruences. The key geometric tool is the description of the join‑irreducible elements of the ASM lattice as integer points inside a four‑colored tetrahedral poset Tₙ (the (n‑2)‑th dilate of a 3‑dimensional simplex). Each point corresponds to a bigrassmannian permutation, and lower sets in Tₙ correspond to ASMs. By selecting appropriate sub‑posets of Tₙ that are isomorphic to the triangular poset Dₙ (the join‑irreducible poset of Dyck paths), the authors construct the Catalan congruences. These sub‑posets are encoded combinatorially as “depth triangles”, “Catalan triangles”, and “bicolored pipe dreams”. Depth triangles are shown to be in bijection with doubly‑gapless Gelfand–Tsetlin patterns and with pipe‑dream configurations having a prescribed number of pipes.

The main structural results are:

1. In any Catalan congruence, the maximal element of each congruence class is a covexillary permutation matrix. Moreover, every covexillary permutation appears as a maximal element for at least one Catalan congruence.
2. The minimal element of each class is precisely a 321‑avoiding permutation matrix. While the minimal ASM of a class need not be a permutation matrix, the set of permutations in the class always forms a Bruhat interval with the same minimal and maximal permutations described above.
3. Choosing an arbitrary representative permutation from each class yields a basis of the Temperley‑Lieb algebra TL(2). This generalizes the basis constructed by Bergeron–Gagnon for the specific excedance congruence.

The paper further develops the internal lattice structure of the Catalan triangles, proving that they form a self‑dual distributive lattice. It introduces the notion of “nappes” to describe the geometry of minimal elements and provides explicit bijections between the various combinatorial models (height functions, six‑vertex configurations, fully packed loops, etc.) and the geometric picture of stacks of rhombic tetrahedra that arise from lower sets in Tₙ.

In the final section, the authors define a symmetrization operation on posets that, when applied to the join‑irreducible posets considered, yields new descriptions via systems of inequalities indexed by upper sets. This operation clarifies the relationship between the original ASM lattice and its Catalan quotients.

Overall, the work unifies several strands of combinatorial representation theory: lattice congruences, Catalan combinatorics, and diagrammatic bases of Temperley‑Lieb algebras. By extending the excedance congruence to a whole family of Catalan congruences, the authors provide a flexible framework for constructing TL(2) bases from a wide variety of combinatorial objects, opening avenues for further exploration of higher‑dimensional analogues and deeper algebraic connections.