Brick Polyhedra

Throughout this work, we restrict our study to finite Coxeter groups, subword complexes and brick polyhedra of type A.

A Coxeter system (W, S)(W,S) of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A_n $$\end{document}An consists of the Coxeter group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ W := \mathcal {S}{n+1} $$\end{document}W:=Sn+1 of permutations of [n+1][n+1], which acts on the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ { x \in \mathbb {R}^{n+1} \mid x_1 + \dots + x{n+1} = 0 } $$\end{document}{x∈Rn+1∣x1+⋯+xn+1=0} by permuting coordinates. It is finitely generated by simple transpositions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S := { s_p \mid p \in [n] } $$\end{document}S:={sp∣p∈[n]} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ s_p = (p, p + 1) $$\end{document}sp=(p,p+1). The root system is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Phi = { e_p - e_q \mid p \ne q \in [n+1] } $$\end{document}Φ={ep-eq∣p≠q∈[n+1]}, and can be partitioned into positive roots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Phi ^+ = { e_i - e_j \mid 1 \le i < j \le n+1 } $$\end{document}Φ+={ei-ej∣1≤i<j≤n+1} and negative roots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Phi ^- = { e_j - e_i \mid 1 \le i < j \le n+1 } $$\end{document}Φ-={ej-ei∣1≤i<j≤n+1}. The simple roots are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta = { \alpha p:=e_p - e{p+1} \mid p \in [n] } $$\end{document}Δ={αp:=ep-ep+1∣p∈[n]} and the fundamental weights are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \nabla = { \omega _p:=\sum _{q \le p} e_q \mid p \in [n] } $$\end{document}∇={ωp:=∑q≤peq∣p∈[n]}.

Definition 1

(Subword complex [2]) For a Coxeter system (W, S), let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=(q_1,...,q_m)$$\end{document}Q=(q1,...,qm) be a word in the generators S of W and let _w ∈ W_w∈W. The subword complex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q,w)$$\end{document}SC(Q,w) is the simplicial complex whose facets are subsets _I⊆ [m]I⊆[m] such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q{[m]\setminus I}$$\end{document}Q[m]\I is a reduced expression for w. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_J$$\end{document}QJ denotes the subword of Q with positions at J.

We can now define two important functions associated with brick polyhedra.

Definition 2

(Root and Weight Function [5, 9]) Given a facet I of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q,w)$$\end{document}SC(Q,w), the root function is the map defined by r_r \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I,k):=\prod Q{ {1,...,k-1}\setminus I} (\alpha _{q_k})$$\end{document}(I,k):=∏Q{1,...,k-1}\I(αqk). We call \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}(I):= {{ $$\end{document}R(I):={{ r_r \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I,i)\mid i \in I}}$$\end{document}(I,i)∣i∈I}} the root configuration of I. The other function is the weight function , defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (I,k):=\prod Q{ {1,...,k-1}\setminus I} (\omega _{q_k})$$\end{document}ω(I,k):=∏Q{1,...,k-1}\I(ωqk).

Definition 3

(Bruhat cone [11]) The Bruhat cone of a non-empty subword complex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q,w)$$\end{document}SC(Q,w) is defined bywhere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname {Dem}(Q)=\text {max}_{\le _B} {\prod Q_X \mid X \subseteq {1,...,m}}$$\end{document}Dem(Q)=max≤B{∏QX∣X⊆{1,...,m}} denotes the Demazure product of Q [2, Lemma 3.4 (1)], and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le _B,\prec _B$$\end{document}≤B,≺B denote the Bruhat order and its cover relation.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {C}^+(w,\operatorname {Dem}(Q)):=\operatorname {cone}\{ \beta \in \Phi ^+ \mid w \prec _B s_\beta w \le _B \operatorname {Dem}(Q)\}, $$\end{document}C+(w,Dem(Q)):=cone{β∈Φ+∣w≺Bsβw≤BDem(Q)},

Proposition 1

([11]) The Bruhat cone can be computed as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {C}^+(w,\operatorname {Dem}(Q)) = \bigcap _{\text {Facet } J \in \mathcal{S}\mathcal{C}(Q,w)} \operatorname {cone} ~R(J). \end{aligned}$$\end{document}C+(w,Dem(Q))=⋂FacetJ∈SC(Q,w)coneR(J).

Definition 4

(Brick polyhedron [11]) The brick vector of a facet \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I \in \mathcal{S}\mathcal{C}(Q,w)$$\end{document}I∈SC(Q,w) isThe brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q,w)$$\end{document}B(Q,w) is the Minkowski sum of the convex hull of all brick vectors and the Bruhat cone:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ b(I):=-\sum _{k=1}^{m}\omega (I,k). $$\end{document}b(I):=-∑k=1mω(I,k).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q,w):= \text {conv}{b(I)\mid $$\end{document}B(Q,w):=conv{b(I)∣ I facet of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q,w)}+\mathcal {C}^+(w,\operatorname {Dem}(Q))$$\end{document}SC(Q,w)}+C+(w,Dem(Q)).

At first glance, brick polyhedra do not seem natural, but they turn out to have very nice properties related to the combinatorics and geometry of the corresponding subword complex [11]. We aim to relate this to the combinatorics and geometry of _ν _ν-associahedra.