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 -