Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups

Abstract

We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.

Lattices of acyclic pipe dreams

“…3.3], D. Jahn and C. Stump describe an algorithm to compute the f -antigreedy facet of a subword complex SC(Q, ω) associated to a linear functional f . The output of our algorithm coincides with the output of their algorithm whenever f is positive on π(Φ + ) and negative on π(Φ − ), see [JS21,Prop. 4.12].…”

“…Our first algorithm to construct pd(π, ω) is a sweeping algorithm, inspired by the algorithm to compute greedy pipe dreams [PP12,PS13]. An extension of this algorithm to subword complexes will be discussed in Section 5.6, and a related algorithm appeared independently in [JS21]. We say that an order on the boxes of the triangular shape is northeast compatible if each box b is before all boxes which are located weakly northeast of b.…”

“…Remark 5.14. We note that the set L(Q, ω) of linear extensions has been considered independently in the work on brick polyhedra by D. Jahn and C. Stump in [JS21]. In particular, L(Q, ω) is completely characterized using Bruhat cones in [JS21, Prop.…”

“…We include a short proof here for self containment. We refer to [JS21] for the description of the notation C + (ω, •). By [JS21, Theorem 3.1] we have…”

“…We now extend the sweeping algorithm of Section 4.1 to construct, from a linear extension π ∈ L(Q, ω), the unique acyclic facet I such that π ∈ L(I). We note that this algorithm is related to an algorithm that was independently described in [JS21], see Remark 5.29.…”