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Monotone triangles are a rich extension of permutations that biject withalternating sign matrices. The notions of weak order and descent sets forpermutations are generalized here to monotone triangles, and shown to enjoymany analogous properties. It is shown that any linear extension of the weakorder gives rise to a shelling order on a poset, recently introduced byTerwilliger, whose maximal chains biject with monotone triangles; among theseshellings are a family of EL-shellings.  The weak order turns out to encode an action of the 0-Hecke monoid of type Aon the monotone triangles, generalizing the usual bubble-sorting action onpermutations. 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 extendsin a natural way to an algebra map out of the recently-definedCheballah-Giraudo-Maurice algebra of alternating sign matrices.

Weak order and descents for monotone triangles