Monotone triangles are a rich extension of permutations that are in bijection with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains are in bijection with monotone triangles; among these shellings is a family of EL-shellings. The weak order turns out to encode an action of the 0-Hecke monoid of type A on the monotone triangles, generalizing the usual bubble-sorting action on permutations. It also leads to a notion of descent set for monotone triangles, having another natural property: the surjective algebra map from the Malvenuto–Reutenauer Hopf algebra of permutations into quasisymmetric functions extends in a natural way to an algebra map out of the recently-defined Cheballah–Giraudo–Maurice algebra of alternating sign matrices.
Bibliographical noteFunding Information:
The second author was partially supported by NSF, United States of America grant DMS-1601961 . The authors thank Ilse Fischer, Darij Grinberg, John Harding, Brendon Rhoades, John Stembridge and Jessica Striker for helpful discussions, and thank Brendan Pawlowski for sharing his code to compute MacNeille completion of posets. In addition, we are grateful to Roger Behrend for detailed feedback on an earlier draft leading to numerous improvements, including his illuminating example. We also thank the referees for their remarks, which improved the exposition. This work began during the Fall 2017 MSRI semester in Geometric and Topological Combinatorics.