The νν-brick Polyhedron

In this section, we introduce the ν ν-brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q\nu , w\nu )$$\end{document}B(Qν,wν) and provide a useful tool (Corollary 2) to compute the brick vector b(T) of a _ν _ν-tree T.

Definition 12

(_ν _ν-Brick Polyhedron) For a lattice path ν ν the ν ν-brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q\nu , w\nu )$$\end{document}B(Qν,wν) is defined as the brick polyhedron of the ν ν-subword complex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q\nu , w\nu )$$\end{document}SC(Qν,wν).

Convention 1

We use the following convention to draw a pipe dream P(T), a _ν _ν-tree T and the Ferrers diagram simultaneusly in the same figure. We shift the Ferrers diagram slightly in the direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(\epsilon , \epsilon )$$\end{document}-(ϵ,ϵ) for some small _ε > 0_ϵ>0. This is illustrated in Fig. 7. We say a pipe is above (or below) a lattice point in the Ferrers diagram if this is the case in the shifted figure. Additionally, we consider a pipe to be above a point if it lies entirely to the left of the point. This can be seen in Fig. 6.

Fig. 6: Illustration of Convention 1

Proposition 4

For a ν ν-tree T the weight function of the corresponding facet in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}\mathcal{C}(Q\nu , w\nu )$$\end{document}SC(Qν,wν) is given byfor _k∈ [m]k∈[m], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\nu $$\end{document}Qν is of length m and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_k$$\end{document}pk is the lattice point in the Ferrers diagram corresponding to the _k_th letter in Q.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (T,k)= \sum \limits _ {i \text {: pipe }i \text { is above }p_k} e_i$$\end{document}ω(T,k)=∑i: pipeiis abovepkei

Corollary 2

For a _ν _ν-tree T the _i_th entry of the corresponding brick vector is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b(T)_i=-\# \text { lattice points in the Ferrers diagram below pipe } i. \end{aligned}$$\end{document}b(T)i=-#lattice points in the Ferrers diagram below pipei.

Remark 1

This description is essentially the same as the classical description of brick vectors in type A, by counting bricks below pseudolines in a sorting network [10].

Example 2

(ν ν-Brick Polyhedron for ν =ENEEN_ν=ENEEN) The word \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\nu $$\end{document}Qν and element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\nu $$\end{document}wν are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\nu =(s_3,s_2,s_1,s_4,s_3,s_2,s_4,s_3,s_5,s_4)$$\end{document}Qν=(s3,s2,s1,s4,s3,s2,s4,s3,s5,s4), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\nu =s_3s_2s_3s_4$$\end{document}wν=s3s2s3s4. By Corollary 2, the brick vectors are obtained by counting the points below the pipes. For instance, the computation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_4$$\end{document}T4 from Example 1, using Convention 1, is shown in Fig. 7. For instance, the 3rd entry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(T_4)_3=-5$$\end{document}b(T4)3=-5 because there are 5 lattice points in the Ferrers diagrams below pipe 3, as shown in Fig. 7 (right). Counting lattice points below each of the six pipes, we get \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(T_4)=-(10,8,5,7,1,0)$$\end{document}b(T4)=-(10,8,5,7,1,0).

Fig. 7: Computation of the brick vector for the νν-tree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_4$$\end{document}T4, by counting lattice points below pipes in the corresponding pipe dream

The brick vectors of all the ν ν-trees from Example 1 are:The ν ν-brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(Q\nu , w\nu )\subseteq \mathbb {R}^6$$\end{document}B(Qν,wν)⊆R6 is of dimension 4. However, to get a feeling about how it looks like, we can remove the letters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1$$\end{document}s1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_5$$\end{document}s5 from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\nu $$\end{document}Qν. They are contained in every facet (are non-flippable) and give rays in the brick polyhedron. If we call \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{Q\nu }$$\end{document}Qν~ the resulting word \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{Q_\nu }=(s_3,s_2,s_4,s_3,s_2,s_4,s_3,s_4)$$\end{document}Qν~=(s3,s2,s4,s3,s2,s4,s3,s4) then the brick polyhedron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(\widetilde{Q_\nu }, w_\nu )$$\end{document}B(Qν~,wν) is of dimension 3 and is illustrated in Fig. 8 (Left). The Bruhat cone is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^+(w_\nu , \operatorname {Dem}(Q_\nu ))= \operatorname {cone}{\alpha _4, s_3(\alpha _4)}$$\end{document}C+(wν,Dem(Qν))=cone{α4,s3(α4)}.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b(T_0)&= -(10,9,6,5,1,0)&b(T_4)&= -(10,8,5,7,1,0) \\ b(T_1)&= -(10,7,8,5,1,0)&b(T_5)&= -(10,7,6,7,1,0) \\ b(T_2)&= -(10,6,8,5,2,0)&b(T_6)&= -(10,6,6,7,2,0) \\ b(T_3)&= -(10,9,5,6,1,0) \end{aligned}$$\end{document}b(T0)=-(10,9,6,5,1,0)b(T4)=-(10,8,5,7,1,0)b(T1)=-(10,7,8,5,1,0)b(T5)=-(10,7,6,7,1,0)b(T2)=-(10,6,8,5,2,0)b(T6)=-(10,6,6,7,2,0)b(T3)=-(10,9,5,6,1,0)