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

TL;DR

This paper explores lattice congruences on alternating sign matrices, revealing connections to Catalan structures, covexillary and 321-avoiding permutations, and constructing bases for the Temperley-Lieb algebra.

Contribution

It extends the excedance relation to lattice congruences on alternating sign matrices and constructs generalized bases for the Temperley-Lieb algebra.

Findings

Maxima of congruence classes are covexillary permutations.

Minimal permutations in classes are 321-avoiding permutations.

Any choice of representatives yields a basis of the Temperley-Lieb algebra.

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.