Séminaire Lotharingien de Combinatoire, 95B.17 (2026), 12 pp.
Florent Hivert, Vincent Pilaud and Ludovic Schwob
Heaps of rhombic dodecahedra,
catalan Congruences on Alternating Sign Matrices, and Bases of the Temperley-Lieb Algebra
Abstract.
We prove that the excedance relation on permutations of N. Bergeron
and L. Gagnon extends to a congruence of the lattice on alternating
sign matrices.
We therefore study all congruences of the lattice on ASMs whose quotient is isomorphic to the Stanley lattice on Dyck paths.
The maxima of their classes are always covexillary permutations (and all covexillary permutations appear this way), and the minimal permutations in each class are always precisely the 321-avoiding permutations.
Finally, any choice of representative permutations in each class yield a basis of the Temperley-Lieb algebra, vastly generalizing the bases arising from the excedance relation.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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