TY - JOUR

T1 - Weak order and descents for monotone triangles

AU - Hamaker, Zachary

AU - Reiner, Victor

N1 - Publisher Copyright:
© 2020

PY - 2020/5

Y1 - 2020/5

N2 - 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.

AB - 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.

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U2 - 10.1016/j.ejc.2020.103083

DO - 10.1016/j.ejc.2020.103083

M3 - Article

AN - SCOPUS:85078464859

SN - 0195-6698

VL - 86

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103083

ER -