%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2020}
\articlenumber{31}
\addbibresource{fpsac.bib}

%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

% ========================================AMS packages:
\usepackage{amsmath,amssymb,amsthm,amsfonts}
\usepackage{mathtools}%Mathematical tools to use with amsmath

% ========================================standard packages:
%\usepackage[english]{babel}%Multilingual support for Plain TeX or LaTeX
\usepackage{comment}%Selectively include/exclude portions of text
% \usepackage[hypertex]{hyperref}
%\usepackage[bookmarksdepth=4]{hyperref}%Extensive support for hypertext in LaTeX
\usepackage{bbm}%"Blackboard-style" cm fonts
\usepackage{mathrsfs}%Support for using RSFS fonts in maths (\mathcal fonts via \mathscr)

% ========================================graphics packages:
\usepackage{tikz,graphicx,color}
\usepackage{tikz-cd}%Create commutative diagrams with TikZ
\usepackage{tikz-3dplot}%Coordinate transformation styles for 3d plotting in TikZ
% % for more on tikz libs, see 
% % http://tex.stackexchange.com/questions/42611/list-of-available-tikz-libraries-with-a-short-introduction
\usetikzlibrary{calc}%to make complex coordinate caculations.
\usetikzlibrary{arrows}%Provides various new and costumizable arrow tips
\usetikzlibrary{shapes}%Shape library, used to define shapes other than rectangle, circle and co-ordinate
% \usetikzlibrary{snakes}%curved lines, and are used either between nodes or as a border to a shape , or as independent shapes. DOESN'T COMPILE
\usetikzlibrary{patterns}%This package "defines patterns for filling areas".
\usetikzlibrary{positioning}%a way of positioning one node in respect with several others
\usetikzlibrary{arrows.meta}
\usetikzlibrary{decorations.markings}
\usepackage{epstopdf}%Convert EPS to PDF using Ghostscript
\pdfpageattr{/Group <</S /Transparency /I true /CS /DeviceRGB>>}

% ========================================extra packages:
% \usepackage{enumerate}%Enumerate with redefinable labels. Precedes enumitem
% \usepackage{enumitem}%Control layout of itemize, enumerate, description. Supersedes enumerate
\usepackage[arrow]{xy}%Typesetting Graphs and Diagrams in TeX 
\usepackage{diagbox}%Table heads with diagonal lines
\usepackage{subfig}%Figures broken into subfigures
% \usepackage{chngcntr} %Change the resetting of counters\PG{doesn't compile for me}
\usepackage{arcs}%Draw arcs over and under text. \overarc and \underarc
\usepackage{xcolor}%Driver-independent color extensions for LaTeX and pdfLaTeX
\usepackage{tabu}
\usepackage{booktabs}% for better rules in the table

% ========================================not recommended:
% \usepackage[margin=0.992in]{geometry}
% \usepackage[margin=1.05in]{geometry}
%\usepackage[margin=1.01in]{geometry}

\usepackage{soul}
\usepackage{accents}
% \usepackage[capitalise]{cleveref}
%\usepackage[T1]{fontenc}




% Typography
\usepackage{enumitem}
% \usepackage{enumerate}
\usepackage{letltxmacro}
\usepackage{thmtools,etoolbox}

\def\myarabic#1{\normalfont(\roman{#1})}
\newlist{theoremlist}{enumerate}{1}
\setlist[theoremlist]{label=\myarabic{theoremlisti},ref={\myarabic{theoremlisti}},itemindent=0pt,labelindent=0pt,
leftmargin=*,noitemsep}


\makeatletter
\renewcommand{\p@theoremlisti}{\perh@ps{\thetheorem}}
\protected\def\perh@ps#1#2{\textup{#1#2}}
\newcommand{\itemrefperh@ps}[2]{\textup{#2}}
\newcommand{\itemref}[1]{\begingroup\let\perh@ps\itemrefperh@ps\ref{#1}\endgroup}
\makeatother


\usepackage{nameref}


\newtheorem{theorem}{Theorem}[section]
\newtheorem*{statement}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{example}[theorem]{Example}




\addtotheorempostheadhook[theorem]{\crefalias{theoremlisti}{theorem}}
\addtotheorempostheadhook[lemma]{\crefalias{theoremlisti}{lemma}}
\addtotheorempostheadhook[proposition]{\crefalias{theoremlisti}{proposition}}


% ================mathcal symbols
\def\Acal{\mathcal{A}}\def\Bcal{\mathcal{B}}\def\Ccal{\mathcal{C}}\def\Dcal{\mathcal{D}}\def\Ecal{\mathcal{E}}\def\Fcal{\mathcal{F}}\def\Gcal{\mathcal{G}}\def\Hcal{\mathcal{H}}\def\Ical{\mathcal{I}}\def\Jcal{\mathcal{J}}\def\Kcal{\mathcal{K}}\def\Lcal{\mathcal{L}}\def\Mcal{\mathcal{M}}\def\Ncal{\mathcal{N}}\def\Ocal{\mathcal{O}}\def\Pcal{\mathcal{P}}\def\Qcal{\mathcal{Q}}\def\Rcal{\mathcal{R}}\def\Scal{\mathcal{S}}\def\Tcal{\mathcal{T}}\def\Ucal{\mathcal{U}}\def\Vcal{\mathcal{V}}\def\Wcal{\mathcal{W}}\def\Xcal{\mathcal{X}}\def\Ycal{\mathcal{Y}}\def\Zcal{\mathcal{Z}}

% ================mathbf symbols
\def\abf{\mathbf{a}}\def\bbf{\mathbf{b}}\def\cbf{\mathbf{c}}\def\dbf{\mathbf{d}}\def\ebf{\mathbf{e}}\def\fbf{\mathbf{f}}\def\gbf{\mathbf{g}}\def\hbf{\mathbf{h}}\def\ibf{\mathbf{i}}\def\jbf{\mathbf{j}}\def\kbf{\mathbf{k}}\def\lbf{\mathbf{l}}\def\mbf{\mathbf{m}}\def\nbf{\mathbf{n}}\def\obf{\mathbf{o}}\def\pbf{\mathbf{p}}\def\qbf{\mathbf{q}}\def\rbf{\mathbf{r}}\def\sbf{\mathbf{s}}\def\tbf{\mathbf{t}}\def\ubf{\mathbf{u}}\def\vbf{\mathbf{v}}\def\wbf{\mathbf{w}}\def\xbf{\mathbf{x}}\def\ybf{\mathbf{y}}\def\zbf{\mathbf{z}}

% ================mathfrak symbols
\def\afr{{\mathfrak{a}}}
\def\nfr{{\mathfrak{n}}}
\def\Sfr{{ \mathfrak{S}}}
% ...

% ================mathbb symbols
\def\one{{\mathbbm{1}}}
\def\C{\mathbb{C}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\P{\mathbb{P}}
% ...

% ==============Hat, bar, tilde, par
\def\Ahat{\hat{A}} 
\def\ipar{{(i)}}
\def\kpar{{(k)}}
\newcommand\parr[1]{{({#1})}}
% ...


% ==============Math notation
\def\<{{\langle}}
\def\>{{\rangle}}
\def\e{{\epsilon}}
\def\la{{\lambda}}
\def\l{{\lambda}}
\def\m{{\mu}}
\def\lm{{\l/\m}}
\def\RP{{\R P}}
\def\CP{{\C P}}
\def\multiset#1#2{\left(\!\left({#1\atopwithdelims..#2}\right)\!\right)}
\def\toi{{\xhookrightarrow{i}}}

% ==============Operators
\def\Id{\operatorname{Id}}
\def\Vol{\operatorname{Vol}}
\def\det{{ \operatorname{det}}}
\def\tr{{ \operatorname{tr}}}
\def\Ker{{ \operatorname{Ker}}}
\def\Im{{ \operatorname{Im}}}
\def\diag{{ \operatorname{diag}}}
\def\rank{{ \operatorname{rank}}}
\def\rk{{\mathrm{rk}}}
\def\cork{ \operatorname{cork}}
\def\codim{ \operatorname{codim}}
\def\Vert{{ \operatorname{Vert}}}
\def\op{{ \operatorname{op}}}
\def\type{{\operatorname{type}}}
\def\sh{{ \operatorname{sh}}}
\def\Conv{ \operatorname{Conv}}
\def\Span{ \operatorname{Span}}
\def\proj{ \operatorname{proj}}
\def\sgn{ \operatorname{sgn}}
\def\wt{\operatorname{wt}}
\def\inv{\operatorname{inv}}
\def\Int{\operatorname{int}}
\def\Cone{\operatorname{Cone}}

\def\RR{{\mathbb R}}
\def\CC{{\mathbb C}}
\def\RP{{\RR\mathbb P}}
\def\CP{{\CC\mathbb P}}

% ==============Total positivity
\def\GL{\operatorname{GL}}
\def\SL{\operatorname{SL}}
\def\Fl{\operatorname{Fl}}
\def\Mat{\operatorname{Mat}}
\def\Gr{\operatorname{Gr}}
\def\Grtnn{\Gr_{\ge 0}}
\def\Grtp{\Gr_{>0}}
\def\alt{\operatorname{alt}}
\def\Mattp{\operatorname{Mat}_{>0}}
\def\Mattnn{\operatorname{Mat}_{\ge0}}

\def\Fltnn{\Fl_{\ge0}}

\def\Lk{\operatorname{Lk}}
\def\Lkint{\operatorname{Lk}^\circ}

%notation from affine.tex
\def\n{{\mathfrak n}}
\def\tFl{{\tilde \Fl}}
\def\tW{{\tilde W}}
\def\Fl{{\rm Fl}}
\def\Lk{{\rm Lk}}
\def\codim{{\rm codim}}
\def\Z{{\mathbb Z}}
\def\R{{\mathbb R}}
\def\Gr{{\rm Gr}}
\def\Bound{{\rm Bound}}
\def\Cone{{\rm Cone}}
\def\sp{{\rm span}}
\def\GL{{\rm GL}}
\def\diag{{\rm diag}}
\def\tS{{\tilde S}}
\def\af{{\rm af}}
\def\Inv{{\rm Inv}}
\newcommand\fixit[1]{\textcolor{green}{TL:#1}}

\def\Waff{\tW}
\def\Flaff{\tFl}

\newcommand\notn[1]{\textcolor{blue}{#1}}
\newcommand\pc[1]{\Pi_{#1}^{>0}}
\newcommand\pctnn[1]{\Pi_{#1}^{\geq0}}
\newcommand\pcvar[1]{\Pi_{#1}^\circ}
\def\Grlam{\Gr_\lambda(k,n)}
\def\id{{\operatorname{id}}}
\def\Opm{O_\pm}
\def\Op{O_+}
\def\Om{O_-}
\def\Group{G(\Opm)}
% \def\str{\varphi}
\def\spheref{S_f^\epsilon}
%\def\str{\operatorname{str}}
\newcommand{\Le}{\textup{\protect\scalebox{-1}[1]{L}}}

\newcommand{\smat}[1]{\left[\begin{smallmatrix}
      #1
    \end{smallmatrix}\right]}
\newcommand{\pmat}[1]{\begin{pmatrix}
#1
  \end{pmatrix}}


\def\ub{{\bar u}}

\def\xs{\operatorname{xs}}
\def\ys{\operatorname{ys^{-1}}}
\def\ut{{\tilde u}}

\def\Lie{\operatorname{Lie}\;}
\def\U{{\mathcal U}}
\def\B{{\mathcal B}}
\def\X{\Xcal}
\def\G{\Gcal}
\def\U{\Ucal}
\def\Ft{{\tilde F}}

\def\pj{^{\parr J}}
\def\Uj{U\pj}
\def\Cj{C\pj}
\def\Cuj{C\pj_u}
\def\Phij{\Phi\pj}
\def\gj{g\pj}
\let\uu\u
\def\u{h}
\def\hj{h^{\parr J}}
\def\Lie{\operatorname{Lie}}

\def\bs{\backslash}
\let\ii\i
\def\i{{\mathbf{i}}}
\def\j{{\mathbf{j}}}
\def\t{{\mathbf{t}}}

\def\d#1{\dot{#1}}
\def\ds{\d{s}}
\def\dw{\d{w}}
\def\du{\d{u}}
\def\dv{\d{v}}
\def\da{\d{a}}
\def\db{\d{b}}
\def\dc{\d{c}}
\def\dz{\d{z}}
\def\dr{\d{r}}
\def\dq{\d{q}}
\def\gfr{\mathfrak{g}}
\def\alphacheck{\alpha^\vee}
\def\X{\accentset{\circ}{X}}
\def\Xcl{X}


\def\Rich_#1^#2{\accentset{\circ}{R}_{#1,#2}}
\def\Richcl_#1^#2{R_{#1,#2}}
\def\RRich_#1^#2{\accentset{\circ}{R}_{#1,#2}^\R}%open Richardson over R
\def\RRichcl_#1^#2{R_{#1,#2}^\R}%closed Richardson over R
\def\Rtp_#1^#2{R_{#1,#2}^{>0}}
\def\Rtnn_#1^#2{R_{#1,#2}^{\geq0}}

\def\Gtnn{{G_{\geq0}}}
\def\GBtnn{{(G/B)_{\geq0}}}
\def\GPtnn{{(G/P)_{\geq0}}}

\def\PR_#1^#2{\accentset{\circ}{\Pi}_{#1,#2}}%projected Richardson
\def\PRtp_#1^#2{\Pi_{#1,#2}^{>0}}%projected Richardson: totally positive
\def\PRtnn_#1^#2{\Pi_{#1,#2}^{\geq0}}%projected Richardson: tnn
\def\PRcl_#1^#2{\Pi_{#1,#2}}%projected Richardson: closure
\def\PRR_#1^#2{\accentset{\circ}{\Pi}_{#1,#2}^\R}%projected Richardson over R
\def\PRRcl_#1^#2{\Pi_{#1,#2}^\R}%projected Richardson: closure over R

\def\sf{\operatorname{sf}}
\def\Tsf{T_{\sf}}
\def\bw{{\mathbf{w}}}
\def\bv{{\mathbf{v}}}
\def\bu{{\mathbf{u}}}
\def\ba{{\mathbf{a}}}
\def\bb{{\mathbf{b}}}
\def\bc{{\mathbf{c}}}
\def\wnot{{\mathbf{w_0}}}
\def\woj{w_J}
\def\dwoj{\dw_J}


\def\Fcal{\mathcal{F}}
\def\Qsf{\Fcal_{\sf}}
\def\Qsfs{\Fcal^\ast_{\sf}}
\def\QsfN#1{\Fcal_{\sf}^{#1}}
\def\QsfsN#1{(\Fcal^\ast_{\sf})^{#1}}
\def\Tsf{T^{\sf}}
% \def\Bsf{B^{\sf}}
\def\Bsf{B^{\diamond}}
\def\Usf{U_{\sf}}

\def\Utp{U_{>0}}
\def\Utnn{U_{\geq0}}

\def\HKL{{\operatorname{HKL}}}
\def\maxx{{\operatorname{max}}}
\def\minn{{\operatorname{min}}}
\def\leqJ{\preceq}
\def\lessJ{\prec}
\def\geqJ{\succeq}
\def\gessJ{\succ}
\def\QJfilt#1{Q_J^{\succeq#1}}
\def\QJfilter#1#2{Q_J^{\succeq(#1,#2)}}
\def\QJideal#1#2{Q_J^{\preceq(#1,#2)}}

\def\pd#1{_{(#1)}}
\def\pu#1{^{(#1)}}
% \def\p#1{{(#1)}}

\def\bx{{\mathbf{x}}}
\def\by{{\mathbf{y}}}
\def\bi{{\mathbf{i}}}
\def\bj{{\mathbf{j}}}


\def\Gomp{G_0^\mp}
\def\Gopm{G_0^\pm}
\def\Deltamp{\Delta^\mp}
\def\Deltapm{\Delta^\pm}
\def\Goj{G_0\pj}
\def\K{{\mathbb{K}}}
\def\k{{\mathbbm{k}}}
\def\kast{\k^\ast}


\def\RedMR{\operatorname{Red}}
\def\Red{\operatorname{Red}}
\def\gMR#1#2{{\mathbf{g}}_{#1,#2}}
\def\gMRvw{\gMR\bv\bw}
% \def\gMRdual#1#2{{\mathbf{g}}^\ast_{#1,#2}}
\def\GMRtp#1#2{G^{>0}_{#1,#2}}
% \def\GMRtpdual#1#2{G^{>0,\ast}_{#1,#2}}
\def\GMRtpvw{\GMRtp\bv\bw}
\def\weightL{Y(T)}
\def\rootL{Q_\Phi}
\def\kls{{\operatorname{KLS}}}
\def\Qkls{Q_P(W,W_J)}
\def\leqkls{\preceq_{P}}

\def\flag{{\operatorname{flag}}}
\def\minormap{\Delta^\flag}
\def\Proj{{\mathbb{P}}}

\def\Kast{\K^{\ast}}
\def\GMRsf#1#2{G^{\sf}_{#1,#2}}

\def\Hom{\operatorname{Hom}}


\def\wj{w^J}
\def\TR{T(\R)} 
\def\GR{G(\R)}
\def\BR{B(\R)}
\def\BRm{B_-(\R)}
\def\PAR{P(\R)}
\def\GBR{(G/B)_{\R}}
\def\GPR{(G/P)_{\R}}
\def\conjug{\sigma_\C}

\def\hjmap{\kappa}
\def\hjmp_#1{\hjmap_{#1}}
\def\hjx{\hjmap_x}
\def\Ftmap{\eta}
\def\Ft{\Ftmap}
\def\zetamap{\zeta_{u,v}\pj}
\def\Guvbig{{G_{u,v}\pj}}
\def\pidup{\pi_{\du P_-}}
\def\ut{u_\maxx}
\def\ut{{\tilde u}}
\def\dut{\d{\ut}}
\def\Fcalo{{\Fcal^{\diamond}}}
\def\Go{G^{\diamond}}
\def\Uo{U^{\diamond}}
\def\Uom_#1{U^{\diamond,-}_{#1}}

\def\GQT{G}
\def\BQT{B}
\def\UQT{U}
\def\TQT{T}

\def\Fcalb{{\overline{\Fcal}}}

\def\Rsf_#1^#2{R_{#1,#2}^{\sf}}
\def\eval{\operatorname{eval}}

\def\Rtpv{\Rtp_{v''}^{[\ut,w_0]}}
\def\hjh{\hjmp_h}

\def\xrasim{\xrightarrow{\sim}}

%NOTATION

\def\chr{\operatorname{char}}

\let \oldlabel \label
\let\oldref\ref
\let\oldcref\cref




%%%%%% AFFINE NOTATION:


\def\Lie{\operatorname{Lie}\;}
\def\U{{\mathcal U}}
\def\B{{\mathcal B}}
\def\Xaff{\accentset{\circ}{\Xcal}}
\def\Xaffcl{\Xcal}
\def\Richaff_#1^#2{\accentset{\circ}{\mathcal{R}}_{#1}^{#2}}
\def\G{\Gcal}
\def\Gmin{\G^{\minn}}
\def\Umin{\U^{\minn}}
\def\Bmin{\B^{\minn}}
\def\U{\Ucal}
\def\T{\Tcal}
\def\Caff{{\mathcal{C}}}
\def\GCMaff{\tilde{A}}

\def\bs{\backslash}
\def\Phiaff{\Delta}
\def\re{\operatorname{re}}
\def\im{\operatorname{im}}
\def\Phire{\Phiaff_{\re}}
\def\Phiim{\Phiaff_{\im}}
\def\Waff{{\tilde{W}}}

\def\FPS{{\mathcal{A}}}
\def\CRL{{\rootL^\vee}}
\def\lch{\l}
\let\oldtau\tau
\def\tau_#1{\oldtau_{#1}}
\def\dtau_#1{\dot{\oldtau}_{#1}}
\def\Iaff{\tilde{I}}
\def\minn{{\operatorname{min}}}
\def\Cast{\C^\ast}
\def\hfr{\mathfrak{h}}
\def\gfr{\mathfrak{g}}
\def\hfrast{\mathfrak{h}^\ast}
\def\rank{\operatorname{rank}}
\def\rt{\tilde{r}}
\def\dh{\dot{h}}
\def\dg{\dot{g}}
\def\df{\dot{f}}

\def\tl{\tau_{\lch}}
\def\dtl{\dtau_{\lch}}
\def\tul{\tau_{u\lch}}
\def\twl{\tau_{w\lch}}
\def\dtul{\dtau_{u\lch}}
\def\vtw{v\tau_\lch w^{-1}}
\def\wji{(\wj)^{-1}}
\def\tmin{\tau_\lch^J}



\def\Ical{\mathcal{I}}
\def\tauk{\tau_k}
\def\dtauk{\dtau_k}
\def\leqop{\leq^\op}
\def\Povar_#1{\accentset{\circ}{\Pi}_{#1}}
\def\Povarcl_#1{\Pi_{#1}}
\def\RPovar_#1{\accentset{\circ}{\Pi}^\R_{#1}}
\def\RPovarcl_#1{\Pi^\R_{#1}}
\def\Povtp_#1{\Pi_{#1}^{>0}}
\def\Povtnn_#1{\Pi_{#1}^{\geq0}}

\def\Boundkn{\operatorname{Bound}(k,n)}



\def\FSpro{\pi}
\def\FStra{\varrho}
\def\FSstr{\operatorname{str}}
\def\FSdil{\vartheta}
\def\FSiso{\bar\nu}




\def\Star_#1{\operatorname{Star}_{#1}}
\def\Startnn_#1{\operatorname{Star}^{\geq0}_{#1}}
\def\SCone{\operatorname{SCone}}


\def\FSstr{\operatorname{str}}
\def\FSdil{\vartheta}

\def\Ofg{{\O_{f,g}}}
\def\Og{\O_{g}}
\def\O{\mathcal{O}}

\def\BAcomb{\psi}
\def\BAgeom{\bar\varphi_u}
\def\BAgeo{\varphi_u}

\def\tht{t}





\def\Ng{{N_g}}
\def\dist{\operatorname{dist}_b}
\def\strix{t_1(x)}
\def\strixn{t_1(x_n)}
\def\Bcl{\overline{B}}
\def\pB{\partial B}
\def\epsb{\eps_B}
\def\Sepsb{S_{g}}


\def\conept{c}

\def\Link{\operatorname{Lk}}
\def\Lkx_#1{\Link_{#1}}
\def\Lkxx_#1^#2{\accentset{\circ}{\Link}_{#1}^{#2}}
\def\Lkg{\Lkx_g}
\def\Lkgh{\Lkxx_g^h}
\def\Lkghp{\Lkxx_g^{h'}}
\def\Lktxx_#1^#2{\Link^{>0}_{#1,#2}}
\def\Starxx_#1^#2{\operatorname{Star}_{#1,#2}}
\def\Startxx_#1^#2{\operatorname{Star}^{\geq0}_{#1,#2}}

\def\sctnn_#1{\sc^{\geq0}_{#1}}
\def\sctnng{\sctnn_g}
\def\sctnngh{\sctnn_{g,h}}
\def\sctp_#1^#2{\sc^{>0}_{#1,#2}}
\def\sctpgh{\sctp_g^h}

\def\eps{\varepsilon}
\def\Seps_#1{S_{#1}}


\def\Lktpe_#1^#2{\Link^{>0}_{#1,#2}}
\def\Lktnne_#1{\Link^{\geq0}_{#1}}
\def\Lktnnge{\Lktnne_g}
\def\Lktnnghe{\Lktnne_{g,h}}
\def\Lkge{\Lkx_g(\eps)}
\def\Lkghe{\Lkxx_g^h(\eps)}
\def\Lktnnghpe{\Lktnne_{g,h'}}


\def\Lktp_#1^#2{\Link^{>0}_{#1,#2}}
\def\Lktnn_#1{\Link^{\geq0}_{#1}}
\def\Lktnng{\Lktnn_g}
\def\Lktnngh{\Lktnn_{g,h}}


\def\axref#1{\ref{#1}}
\def\sc{Z}
\def\scg{\sc_g}
\def\sch{\sc_h}
\def\schp{\sc_{h'}}
\def\sco_#1^#2{\accentset{\circ}{\sc}_{#1,#2}}
\def\scogh{\sco_g^h}
\def\sccl_#1^#2{\sc_{#1}^{#2}}
\def\base{0}
\def\baseg{0}
\def\baseh{0}
\def\basehp{0}
\def\sphere{S_{\base}}

\def\Y{\mathcal{Y}}
\def\Yo_#1{\accentset{\circ}{\Y}_{#1}}
\def\Ycl_#1{\Y_{#1}}
\def\Ytnn{\Y^{\geq0}}
\def\Ytp_#1{\Y_{#1}^{>0}}
\def\YQ{Q}
\def\leqY{\leqJ}
\def\lessY{\lessJ}
\def\gessY{\gessJ}
\def\geqY{\geqJ}
\def\gradY{\dim}
\def\YQhat{\widehat{\YQ}}
\def\FSisox#1{\FSiso^{#1}}
\def\YQmin{\YQ_\minn}

\def\dilg{\FSdil_g}
\def\strg(#1){\normg{#1}}
\def\isog{\FSiso_g}
\def\isoga{\FSiso_{g,1}}
\def\isogb{\FSiso_{g,2}}
\def\isoh{\FSiso_h}
\def\isoha{\FSiso_{h,1}}
\def\isohb{\FSiso_{h,2}}

\def\norm#1{\|#1\|}
\def\normg#1{\|#1\|}

\input{boundary_sage.tex}
\input{boundary_figures.tex}

%% define your title in the usual way
\title[Regularity theorem for totally nonnegative flag varieties]{Regularity theorem for totally nonnegative flag~varieties}

\author{Pavel Galashin\thanks{\href{mailto:galashin@math.ucla.edu}{galashin@math.ucla.edu}. Supported by NSF grant DMS--1954121.}\addressmark{1}, Steven N.\ Karp\thanks{\href{mailto:karp.steven@courrier.uqam.ca}{karp.steven@courrier.uqam.ca}. Supported by an NSERC postdoctoral fellowship.}\addressmark{2}, \and Thomas Lam\thanks{\href{mailto:tfylam@umich.edu}{tfylam@umich.edu}. Supported by a von Neumann Fellowship from the Institute for Advanced Study and by NSF grant DMS-1464693.}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} Department of Mathematics, University of California, Los Angeles, CA, USA\\
\addressmark{2} Laboratoire de combinatoire et d'informatique math\'{e}matique, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, Canada\\
\addressmark{3} Department of Mathematics, University of Michigan, Ann Arbor, MI , USA}


%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.}
%\resume{}

\keywords{Total positivity, algebraic group, partial flag variety, Richardson varieties, totally nonnegative Grassmannian, positroid cells.}


\begin{document}
\maketitle

\section{Introduction}\label{sec:introduction}

Let $G$ be a semisimple algebraic group, split over $\R$, and let $P\subset G$ be a parabolic subgroup.  Lusztig~\cite{LusGB} introduced the totally nonnegative part of the partial flag variety $G/P$, denoted $\GPtnn$, which he called a ``remarkable polyhedral subspace''. He conjectured and Rietsch proved~\cite{Rie99} that $\GPtnn$ has a decomposition into open cells. We prove the following conjecture of Williams~\cite{Wil}:

\begin{theorem}\label{thm:main_intro}
The cell decomposition of $\GPtnn$ forms a regular CW complex. Thus the closure
 of each cell is homeomorphic to a closed ball.
\end{theorem}

A special case of particular interest is when $G/P$ is the Grassmannian $\Gr(k,n)$ of $k$-dimensional linear subspaces of $\R^n$. In this case, $\GPtnn$ becomes the \emph{totally nonnegative Grassmannian} $\Grtnn(k,n)$, introduced by  Postnikov~\cite{Pos} as the subset of $\Gr(k,n)$ where all Pl\"ucker coordinates are nonnegative. He gave a stratification of $\Grtnn(k,n)$ into \emph{positroid cells} according to which Pl\"ucker coordinates are zero and which are strictly positive, and conjectured that the closure of each positroid cell is homeomorphic to a closed ball. Postnikov's conjecture follows as a special case of \cref{thm:main_intro}:
\begin{corollary}\label{cor:positroids_intro}
The decomposition of $\Grtnn(k,n)$ into positroid cells forms a regular CW complex. Thus the closure of each positroid cell is homeomorphic to a closed ball.
\end{corollary}

\noindent When $k = 1$, $\Grtnn(1,n)$ is the standard $(n-1)$-dimensional simplex $\Delta_{n-1} \subset \P^{n-1}$, a prototypical example of a regular CW complex.

\subsection{History and motivation}
A matrix is called \emph{totally nonnegative} if all its minors are nonnegative. The theory of such matrices originated in the 1930's~\cite{Schoenberg,GK37}. Later, Lusztig~\cite{LusGB} was motivated by a question of Kostant to consider connections between totally nonnegative matrices and his theory of canonical bases for quantum groups \cite{LusCan}. This led him to introduce the totally nonnegative part $G_{\ge 0}$ of a split semisimple $G$.  Inspired by a result of Whitney~\cite{Whitney}, he defined $G_{\ge 0}$ to be generated by exponentiated Chevalley generators with positive real parameters, and generalized many classical results for $G = \SL_n$ to this setting. He introduced the totally nonnegative part $\GPtnn$ of a partial flag variety $G/P$, and showed~\cite[Section 4]{Lus_intro} that $G_{\geq0}$ and $\GPtnn$ are contractible.

Fomin and Shapiro~\cite{FS} realized that Lusztig's work may be used to address a longstanding problem in poset topology. Namely, the Bruhat order of the Weyl group $W$ of $G$ had been shown to be shellable by Bj\"orner and Wachs~\cite{BW}, and by general results of Bj\"orner~\cite{Bjo2} it followed that there exists a ``synthetic'' regular CW complex whose face poset coincides with $(W,\leq)$ (see \cref{figure_bruhat}). The motivation of~\cite{FS} was to answer a natural question due to Bernstein and Bj\"orner of whether such a regular CW complex exists ``in nature''. Let $U \subset G$ be the unipotent radical of the standard Borel subgroup, and let $U_{\ge 0} := G_{\ge 0} \cap U$ be its totally nonnegative part.  For $G = \SL_n$, $U_{\ge 0}$ is the semigroup of upper-triangular unipotent matrices with all minors nonnegative. The work of Lusztig~\cite{LusGB} implies that $U_{\ge 0}$ has a cell decomposition whose face poset is $(W, \le)$. The space $U_{\ge 0}$ is not compact, but Fomin and Shapiro~\cite{FS} conjectured that taking the link of the identity element in $U_{\ge 0}$, which also has $(W, \le)$ as its face poset, gives the desired regular CW complex. Their conjecture was confirmed by Hersh~\cite{Her}.  Hersh's theorem also follows as a corollary to our proof of \cref{thm:main_intro}, see \cref{sec:U}. 
\begin{corollary}[{\cite{Her}}] \label{cor:Her}
The link of the identity in $U_{\geq0}$ is a regular CW complex.
\end{corollary}
\noindent
For recent related developments, see~\cite{DHM}.
\begin{figure}
\begin{center}~\\[-12pt]
\begin{align*}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.0]
\pgfmathsetmacro{\s}{1.0};
\pgfmathsetmacro{\hd}{1.20};
\pgfmathsetmacro{\vd}{0.84};
\pgfmathsetmacro{\is}{1.68};
\useasboundingbox(0,0)rectangle(-1.32,0)rectangle(1.32,2.40);
\node[inner sep=\is](123)at(0,0){\scalebox{\s}{$\id$}};
\node[inner sep=\is](213)at($(123)+(-\hd,\vd)$){\scalebox{\s}{$s_1$}};
\node[inner sep=\is](132)at($(123)+(\hd,\vd)$){\scalebox{\s}{$s_2$}};
\node[inner sep=\is](312)at($(213)+(0,\vd)$){\scalebox{\s}{$s_2s_1$}};
\node[inner sep=\is](231)at($(132)+(0,\vd)$){\scalebox{\s}{$s_1s_2$}};
\node[inner sep=\is](321)at($(312)+(\hd,\vd)$){\scalebox{\s}{$w_0$}};
\path[semithick](123)edge(213) edge(132) (213)edge(312) edge(231) (132)edge(312) edge(231) (312)edge(321) (231)edge(321);
\end{tikzpicture}\hspace*{60pt}\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzstyle{out1}=[inner sep=0,minimum size=1.8mm,circle,draw=black,fill=black,semithick]
\pgfmathsetmacro{\radius}{0.84};
\draw[thick,fill=teal!30](0:0)circle[radius=\radius];
\node[out1]at(180:\radius){};
\node[out1]at(0:\radius){};
\node[inner sep=0]at(0,0){$w_0$};
\node[inner sep=0]at(0,\radius+0.24){$s_1s_2$};
\node[inner sep=0]at(0,-\radius-0.24){$s_2s_1$};
\node[inner sep=0]at(\radius+0.42,0){$s_1$};
\node[inner sep=0]at(-\radius-0.42,0){$s_2$};
\end{tikzpicture}
\end{align*}\vspace*{-8pt}
\caption{The Bruhat order on $W = S_3$ (after deleting the bottom element) is the face poset of the regular CW complex homeomorphic to a $2$-dimensional ball on the right.\vspace*{-18pt}}
\label{figure_bruhat}
\end{center}
\end{figure}

Meanwhile, Postnikov~\cite{Pos} defined the totally nonnegative Grassmannian $\Grtnn(k,n)$, decomposed it into positroid cells, and showed that each positroid cell is homeomorphic to an open ball. Motivated by work of Fomin and Zelevinsky~\cite{FZ} on double Bruhat cells, he conjectured~\cite[Conjecture 3.6]{Pos} that this decomposition forms a regular CW complex.  It was later realized that the space $\Gr_{\ge 0}(k,n)$ and its cell decomposition coincide with the one studied by Lusztig and Rietsch in the special case that $G/P = \Gr(k,n)$. Williams~\cite[Section 7]{Wil} extended Postnikov's conjecture from $\Grtnn(k,n)$ to $\GPtnn$.

There has been much progress towards proving these conjectures. Williams~\cite{Wil} showed that the face poset of $(G/P)_{\ge 0}$ (and hence of $\Gr_{\ge 0}(k,n)$) is graded, thin, and shellable, and therefore by~\cite{Bjo2} is the face poset of some regular CW complex. Postnikov, Speyer, and Williams~\cite{PSW} showed that $\Gr_{\ge 0}(k,n)$ is a CW complex, and their result was generalized to $(G/P)_{\ge 0}$ by Rietsch and Williams~\cite{RW08}. Rietsch and Williams~\cite{RW10} also showed that the closure of each cell in $(G/P)_{\ge 0}$ is contractible.  
In previous work~\cite{GKL,GKL2}, we showed that the spaces $\Gr_{\ge 0}(k,n)$ and $(G/P)_{\ge 0}$ are homeomorphic to closed balls,  which is the special case of \cref{thm:main_intro} for the top-dimensional cell of $\GPtnn$.  We remark that our proof of \cref{thm:main_intro} uses different methods than those employed in~\cite{GKL,GKL2}, in which we relied on the existence of a vector field on $G/P$ contracting $\GPtnn$ to a point in its interior. Singularities of lower-dimensional positroid cells give obstructions to the existence of a continuous vector field with analogous properties.

Totally positive spaces have attracted a lot of interest due to their appearances in other contexts such as cluster algebras~\cite{FZclust} and the physics of scattering amplitudes~\cite{ABCGPT}. Our original motivation for studying the topology of spaces arising in total positivity was to better understand the \emph{amplituhedra} of Arkani-Hamed and Trnka~\cite{AT}, and more generally the Grassmann polytopes of the third author~\cite{Lam16}. A Grassmann polytope is a generalization of a convex polytope in the Grassmannian $\Gr(k,n)$. For example, the totally nonnegative Grassmannian $\Grtnn(k,n)$ is a generalization of a simplex, while amplituhedra generalize cyclic polytopes~\cite{sturmfels_88}. The faces of a Grassmann polytope are linear projections of closures of positroid cells, and therefore it is essential to understand the topology of these closures in order to develop a theory of Grassmann polytopes.

\subsection{Outline}

We provide some background definitions in \cref{sec:background}. We give a brief overview of our proof of~\cref{thm:main_intro} in \cref{sec:proof}. Finally, in \cref{sec:examples} we consider three examples: the unipotent radical $U_n$, the complete flag variety $\Fl_n$, and the Grassmannian $\Gr(k,n)$. Further details and full proofs of the results stated here appear in our preprint \cite{arxiv}.

\section{Background}\label{sec:background}

In this section we review background on regular CW complex and totally nonnegative partial flag varieties. We refer to \cite{Hat,LW,LusGB} for further details.

\subsection{Regular CW complexes}

Let $X$ be a Hausdorff space. We call a finite disjoint union $X = \bigsqcup_{g\in \YQ}X_g$
a \emph{regular CW complex} if it satisfies the following two properties.
\begin{enumerate}[label={(\arabic*)},leftmargin=*]
\item\label{cond:cw:regularity} For each $g\in \YQ$, there exists a homeomorphism from the closure $\overline{X_g}$ to a closed ball $\Bcl$ which sends $X_g$ to the interior of $\Bcl$.
\item\label{cond:cw:normality} For each $g\in \YQ$, there exists $\YQ'\subset \YQ$ such that $\overline{X_g} = \bigsqcup_{f\in \YQ'} X_f$.
\end{enumerate}
The \emph{face poset} of $X$ is the poset $(\YQ, \leqY)$, where $f \leqY g$ if and only if $X_f \subset \overline{X_g}$.


\def\barI{{I}}
\subsection{Totally nonnegative partial flag varieties}\label{sec:pinning}

Let $\gfr$ denote the Lie algebra of $G$ over $\RR$. We fix Chevalley generators $(e_i,f_i)_{i\in \barI}$ of $\gfr$, so that the elements $h_i:=[e_i,f_i]$ ($i\in\barI$) span the Lie algebra of a split real maximal torus $T$ of $G$. For $i\in\barI$ and $t\in\mathbb{R}$, we define the elements of $G$
\[
x_i(t) := \exp(te_i),\quad y_i(t) := \exp(tf_i).
\]
We also let $\alpha_i^\vee:\mathbb{R}^\ast\to T$ be the homomorphism of algebraic groups whose tangent map takes $1\in\mathbb{R}$ to $h_i$. The $x_i(t)$'s (respectively, $y_i(t)$'s) generate the unipotent radical of a Borel subgroup $B$ (respectively, $B_-$) of $G$, with $B\cap B_- = T$. The data $(T,B,B_-,x_i,y_i;i\in \barI)$ is called a \emph{pinning} for $G$.

We define the \emph{totally nonnegative part} $G_{\ge 0}$ of $G$ as the semigroup generated by all $x_i(t)$'s, $y_i(t)'s$, and $\alpha_i^\vee(t)$'s with $t > 0$. For a parabolic subgroup $P\supset B$, we define the totally nonnegative part $(G/P)_{\ge 0}$ of $G/P$ as the closure of the image of $G_{\ge 0}$ inside $G/P$. For examples in the case $G = \SL_n$, see \cref{sec:examples}.

Rietsch~\cite{Rie99,Rie} established the decomposition
\begin{align}\label{rietsch_decomp}
\GPtnn=\bigsqcup_{g\in Q} \Povtp_g
\end{align}
of $\GPtnn$ into open balls $\Povtp_g$ indexed by the elements $g$ of a certain poset $(Q,\leqJ)$, which is the face poset of $\GPtnn$. When $\GPtnn$ is the totally nonnegative Grassmannian $\Grtnn(k,n)$, this is the positroid cell decomposition of~\cite{Pos} (see \cref{sec:Gr}). 

\section{Stars, links, and the Fomin--Shapiro atlas}\label{sec:proof}

In this section, we outline our proof of \cref{thm:main_intro}. Given $g\in Q$, define the \emph{star} of $g$ in $\GPtnn$ by 
\[
  \Startnn_g:=\bigsqcup_{h\geqY g} \Povtp_h.
\]
We also consider the space $\Lktnng$ (the \emph{link} of $g$), stratified as
\[
\Lktnng=\bigsqcup_{h\gessJ g}\Lktp_g^h.
\]
We show along the way that $\Lktnng$ is a regular CW complex homeomorphic to a closed ball.
%, which will imply~\cref{thm:main_intro}.

At the core of our approach is a collection of (stratification-preserving) homeomorphisms
\begin{align}\label{eq:isog_intro}
\isog: \Startnn_g\xrasim \Povtp_g\times \Cone(\Lktnng),
\end{align}
one for each $g \in Q$ (see \cref{fig:flag}).  Here $\Cone(A):=(A\times \R_{\geq0})/(A\times \{0\})$ denotes the \emph{open cone} over $A$.  The homeomorphisms $\isog$, along with dilation actions $\dilg$ on the cones, are part of the data of what we call a \emph{Fomin--Shapiro atlas}.  Our construction is inspired by similar maps introduced in~\cite{FS} for the unipotent radical $U_{\geq 0}$.

\begin{figure}
\figureFlag
  \caption{\label{fig:flag} The map $\isog$ for the case $G=\SL_3$ and $P=B$ from \cref{ex:flag_intro}.}
\end{figure}

We also introduce the abstract notion of a \emph{totally nonnegative space}, which captures several known combinatorial and geometric properties of $\GPtnn$ used in our proof. This includes the shellability of $Q$ due to Williams~\cite{Wil}, and some topological results~\cite{Rie,KLS} on  Richardson varieties. We prove that every totally nonnegative space that admits a Fomin--Shapiro atlas is a regular CW complex.  Our argument proceeds by induction on the dimension of $\Lktp_g^h$, and depends on a delicate interplay between objects in smooth and topological categories.  We use crucially that the maps \eqref{eq:isog_intro} in a Fomin--Shapiro atlas are restrictions of smooth maps.  On the topological level, we rely on the generalized Poincar\'e conjecture~\cite{Sma,Fre,Per1} combined with some general results on poset topology. We formulate our results in the abstract language of totally nonnegative spaces since we expect that they can be applied in other contexts, such as to totally nonnegative Kac--Moody flag varieties, totally nonnegative double Bruhat cells~\cite{FZ}, spaces of electrical networks~\cite{Lam}, spaces of boundary correlations of planar Ising models~\cite{Ising}, amplituhedra~\cite{AT}, and the totally nonnegative part of the wonderful compactification~\cite{He2}.

The bulk of the proof of \cref{thm:main_intro} is devoted to the construction of the Fomin--Shapiro atlas.  For each $g\in Q$ we give an isomorphism $\BAgeom$ between an open dense subset $\Og\subset G/P$ and a certain subset of the \emph{affine flag variety} $\G/\B$ of the loop group $\G$ associated with $G$. The map $\BAgeom$, which we call an \emph{affine Bruhat atlas}, sends the projected Richardson stratification~\cite{KLS} of $G/P$ to the affine Richardson stratification of its image inside $\G/\B$. The hardest part of the proof consists of showing that the subset $\Og\subset G/P$ contains $\Startnn_g$.

\begin{remark}\label{rmk:HKL}
The map $\BAgeom$ generalizes the map of Snider~\cite{Sni} from $\Gr(k,n)$ to all $G/P$. A different approach to give such a generalization is due to He, Knutson, and Lu~\cite{HKL}, which led them to the notion of a \emph{Bruhat atlas}. See~\cite{Elek} for the definition. Huang~\cite{huang} has independently constructed a map similar to our $\BAgeom$.
%We call our map $\BAgeom$ an affine Bruhat atlas since its target space is always an affine flag variety, while the Bruhat atlases of~\cite{HKL} necessarily involve more general \emph{Kac--Moody flag varieties}.
\end{remark}

\def\faff{{\tilde{f}}}
\def\gaff{{\tilde{g}}}
\def\haff{{\tilde{h}}}
\def\fbac{{\BAcomb(f)}}
\def\gbac{{\BAcomb(g)}}
\def\hbac{{\BAcomb(h)}}

\def\evm{\ev_\infty}
\def\evp{\ev_0}
\def\ev{\bar{\operatorname{ev}}}
\def\Am{\Acal_-}
\def\Ap{\Acal_+}

\def\hG{{\hat \G}}
\def\hB{{\hat \B}}

\def\Uvp{\U^{(I)}}
\def\Uvn{\U^{(I)}_-}

\def\BAgeomx{\BAgeom^\dagger}
\def\BAgeomxx{\BAgeom^\ddagger}

%\tikzset{labl/.style={anchor=south, rotate=90, inner sep=.5mm}}

\def\isogaff{\tilde{\nu}_g}
\def\isogaffa{\tilde{\nu}_{g,1}}
\def\isogaffb{\tilde{\nu}_{g,2}}

\def\NNg{N}

\def\rch{\tilde\rho}
\def\dilgaff{\FSdil_g}

\def\Ogc{\Og^{\C}}
\def\scgc{\sc_g^{\C}}
\def\scoghc{\accentset{\circ}{\sc}_{g,h}^{\C}}

\def\rhou{\rho_u^\vee}
\def\rhour{\rho_{ur}^\vee}

\def\center{C}
\def\centeraff{\tilde{C}}
\def\pone{^1}

\def\BAgeos{\BAgeo'}
\def\BAgeoms{\BAgeom'}

\def\trnc#1#2{{\operatorname{\tr}_{#1}^{#2}}}
\def\trunc#1#2{#1^{\trnc u{#2}}}
\def\trunci#1#2#3{#1^{\trnc u{#2}}_{#3}}
\def\truncij#1#2#3#4{#1^{\trnc u{#2}}_{#3,#4}}
\def\trmin#1#2#3{\Delta^{\trnc u{#2}}_{#3}(#1)}
\def\pivot#1#2{\theta^u_{#1,#2}}


\def\parz{}
\def\BAP{B(\Ap)}
\def\BAM{B_-(\Am)}
\def\BAPP{B\pone(\Ap)}
\def\BAMM{B\pone_-(\Am)}
\def\UAP{U(\Ap)}
\def\UAM{U_-(\Am)}

\def\inveps#1#2{\epsilon_{#1,#2}}
\def\dft{\tilde{f}}
\def\xt{\tilde{x}}
\def\yt{\tilde{y}}
\def\Glnap{\GL_n(\Ap)}
\def\Glnam{\GL_n(\Am)}
\def\Gminp{\Gmin_+}
\def\Gminm{\Gmin_-}

\def\Gminx#1{\Gmin_{(#1)}}
\def\dett#1{\det\,#1}
\def\vall#1{\val\,#1}
\def\Ctr{C}
\def\Slna{\SL_n(\Acal)}
\def\Glona{\GL_n^\circ(\Acal)}
\def\Glnax#1{\GL_n^\parr{#1}(\Acal)}
\def\Glna{\GL_n(\Acal)}
\def\val{\operatorname{val}}
\def\rkk#1#2#3{\rank(#1;#2,#3)}

\def\av{\operatorname{av}}
\def\Wext{\widehat{W}}
\def\Saffn{\tilde{S}_n}
\def\Saffxn#1{\tilde{S}_{#1,n}}
\def\Saffkn{\Saffxn k}
\def\uS#1#2{#1\cdot #2}
\def\uS#1#2{#1 #2}
\def\uk#1{\uS #1{[k]}}

\def\uphi#1#2{#1 \cdot #2}

\def\nchk{{[n]\choose k}}
\def\Cast{\C^\ast}

\def\GR#1{\left[#1\right|}

\newcommand\scalemath[2]{\scalebox{#1}{\mbox{\ensuremath{\displaystyle #2}}}}

\def\sbl#1#2{\scalemath{0.6}{\left[
\begin{array}{c|c}
#1 \\
\hline
#2
\end{array}
\right]}}

\def\sbld#1#2{\left[\scalebox{0.8}{$
\begin{array}{c|c}
#1 \\
\hline
#2
\end{array}$}
\right]}

\def\svec#1#2{\scalemath{0.6}{\left[
\begin{array}{c}
#1 \\
\hline
#2
\end{array}
\right]}}
\def\fm_#1{\Delta^\flag_{#1}}
\def\lex{\leq_{\operatorname{lex}}}

\def\shi{\chi}
\def\dshi{{\ddot{\chi}}}

\section{Examples}\label{sec:examples}

In this section we discuss three examples (in type $A$) of regular CW complexes which are addressed by~\cref{thm:main_intro}. We fix $n \ge 1$, and let $[n] := \{1, 2, \dots, n\}$. For $0 \le k \le n$, let $\binom{[n]}{k}$ denote the set of all $k$-element subsets of $n$. We set $G := \SL_n$. In the setup of \cref{sec:pinning}, we may take $I := [n-1]$, where $x_i(t)$, $y_i(t)$, and $\alpha_i^\vee(t)$ are obtained from the $n\times n$ identity matrix by placing, respectively,
$$
\begin{bmatrix}1 & t \\ 0 & 1\end{bmatrix}\, , \, \begin{bmatrix}1 & 0 \\ t & 1\end{bmatrix}, \text{ and } \begin{bmatrix}t & 0 \\ 0 & t^{-1}\end{bmatrix} \text{ in rows and columns } i, i+1.
$$
Then $B$ and $B_-$ are the subgroups of $G$ of upper- and lower-triangular matrices, respectively, and $G_{\ge 0}$ is the subset of $G$ of matrices whose minors are all nonnegative.

The Weyl group is the symmetric group $S_n$. We let $w_0 := (i \mapsto n+1-i)$ denote the longest permutation in $S_n$. For $w\in S_n$, we let $\dot{w}\in G$ denote the signed permutation matrix which contains a $\pm 1$ in row $w(k)$ and column $k$ for each $k\in [n]$, where the signs are chosen so that the submatrix with rows $\{w(1), \dots, w(k)\}$ and columns $[k]$ has determinant $1$. For example,
$$
\dot{s}_1 = \begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}, \text{ and for $w = s_1s_2$, we have } \dot{w} = \begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}.
$$

\subsection{Unipotent radical $U_n$}\label{sec:U}

Let $U_n$ denote the subgroup of $B$ of all unipotent matrices, and $U_n^{\ge 0} := G_{\ge 0} \cap U_n$ its totally nonnegative part. We have the decomposition into totally positive Bruhat cells~\cite{LusGB}
$$
U_n^{\ge 0} = \bigsqcup_{w\in S_n}\Pi_w^{>0}, \quad \text{ where } \quad \Pi_w^{>0} := U_n^{\ge 0} \cap B_-\dot{w}B_-.
$$
The closure relation on cells is given by the strong Bruhat order.

We can take the link of the identity in $U_n^{\ge 0}$ to be the subset of matrices whose $n-1$ entries immediately above the diagonal sum to $1$. It has a similar cell decomposition indexed by permutations $w\in S_n$ with $w \neq \id$. The natural inclusion $U_n \hookrightarrow G/B_-$ allows us to identify this link with one appearing in the proof of \cref{thm:main_intro} for $G/B_-$, which implies that it is a regular CW complex homeomorphic to a closed ball. This result was conjectured by Fomin and Shapiro~\cite{FS} and proved by Hersh~\cite{Her}, all in general Lie type (cf.\ \cref{cor:Her}).

\begin{example}\label{ex:U}
Let $n = 3$. Then
$$
U_3^{\ge 0} = \left\{\begin{bmatrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1\end{bmatrix} : x, y, z, xz-y \ge 0\right\}.
$$
The permutation $w_0 = s_1s_2s_1 = 321$ indexes the cell where $x, y, z, xz-y > 0$, while $w = s_1s_2 = 231$ indexes the cell where $x,y,z > 0$ and $xz - y = 0$.

The link of the identity of $U_3^{\ge 0}$ consists of all matrices in $U_3^{\ge 0}$ with $x + z = 1$. We can plot this region in the $xy$-plane:
$$
\qquad\begin{tikzpicture}[baseline=(current bounding box.center),scale=4.8]
\tikzstyle{out1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=black,semithick]
\fill[teal!30,domain=0:1,variable=\x](0,0)--plot({\x},{\x*(1-\x)})--(1,0)--cycle;
\path[latex-latex,thick](-0.25,0)edge(1+0.25,0) (0,-0.25)edge(0,0.25+0.25);
\path[semithick](1,-0.06)edge(1,0.06);
\node[inner sep=0]at(1+0.25+0.04,0){$x$};
\node[inner sep=0]at(0,0.25+0.25+0.04){$y$};
\node[inner sep=0]at(1,-0.11){$1$};
\draw[domain=0.5-0.67082:0.5+0.67082,variable=\x,thick]plot({\x},{\x*(1-\x)});
\node[inner sep=0](center)at(0.5,0.12){};
\node[inner sep=0](left)at(0.9,0.4){};
\path[thick](0,0)edge(1,0){};
\node[out1]at(0,0){};
\node[out1]at(1,0){};
\node[inner sep=0]at(0.5,0.1){$w_0$};
\node[inner sep=0]at(0.5,-0.06){$s_2s_1$};
\node[inner sep=0]at(0.5,0.3){$s_1s_2$};
\node[inner sep=0]at(1.06,0.06){$s_1$};
\node[inner sep=0]at(-0.06,0.06){$s_2$};
\end{tikzpicture}\qquad.
$$
We see that this agrees with the regular CW complex of \cref{figure_bruhat}.
\end{example}

\subsection{Complete flag variety $\Fl_n$}\label{sec:flag}

Taking $P = B$, we can identify $G/B$ with the space $\Fl_n$ of {\itshape complete flags} in $\RR^n$, i.e.\ the space of tuples $(V_1, \dots, V_{n-1})$, where $V_k$ is a subspace of $\RR^n$ of dimension $k$ ($1 \le k \le n-1$) and $V_1 \subset \cdots \subset V_{n-1}$. (Here $V_k$ is the subspace spanned by the first $k$ columns of $x\in G/B$.) The totally nonnegative part $\Fl_n^{\ge 0}$ consists of all complete flags which can be represented by an element $x\in G$ with nonnegative initial minors:
$$
\det(x_{I,[k]}) \ge 0 \quad\text{ for all } 1 \le k \le n-1 \text{ and } I \in\textstyle\binom{[n]}{k}.
$$

We have the decomposition \eqref{rietsch_decomp} into totally positive Richardson cells
$$
\Fl_n^{\ge 0} = \bigsqcup_{v\le w \text{ in } S_n}\Pi_{(v,w)}^{>0}, \quad \text{ where } \quad \Pi_{(v,w)}^{>0} := \Fl_n^{\ge 0} \cap B_-\dot{v}B \cap B\dot{w}B.
$$
The dimension of $\Pi_{(v,w)}^{>0}$ is $\ell(w) - \ell(v)$. The closure relation on cells is given by containment of intervals in the strong Bruhat order:
$$
(v,w) \preceq (v',w') \quad \Leftrightarrow \quad v' \le v \le w \le w'.
$$

\def\PRSteven_#1^#2{\Povtp_{(#1,#2)}}
\begin{example}\label{ex:flag_intro}
Let $n=3$. Then $\Fl_3^{\ge 0}$ gives a cell decomposition of a $3$-dimensional ball, see \cref{fig:flag} (left). Let us illustrate the homeomorphism \eqref{eq:isog_intro} for $g:=(s_1,s_2s_1)$. Here $\Povtp_g$ is an open line segment, and $\Startnn_g$ consists of $4$ cells: a line segment $\Povtp_g=\PRSteven_{s_1}^{s_2s_1}$, two open square faces $\PRSteven_{s_1}^{w_0}$ and $\PRSteven_{\id}^{s_2s_1}$, and an open $3$-dimensional ball $\PRSteven_{\id}^{w_0}$. This union is indeed homeomorphic to $\Povtp_g\times \Cone(\Lktnng)$ shown in \cref{fig:flag} (right). Here $\Lktnng$ is a closed line segment whose endpoints are $\Lktp_g^{(s_1,w_0)}$ and $\Lktp_g^{(\id,s_2s_1)}$, and whose interior is $\Lktp_g^{(\id,w_0)}$.
\end{example}


\subsection{Grassmannian $\Gr(k,n)$}\label{sec:Gr}

Fix $1 \le k \le n-1$, and take $P$ to be the subset of $G$ of all matrices whose lower-left $(n-k) \times k$ block is zero. Then we can identify $G/P$ with the {\itshape Grassmannian} $\Gr(k,n)$, i.e.\ the space of $k$-dimensional subspaces of $\RR^n$. (Here $x\in G/P$ corresponds to the $k$-dimensional subspace spanned by the first $k$ columns of $x$.) The totally nonnegative part $\Gr_{\ge 0}(k,n)$ consists of all subspaces which can be represented by an $n\times k$ matrix whose $k\times k$ minors (known as \emph{Pl\"{u}cker coordinates}) are all nonnegative.

In the case of $\Grtnn(k,n)$, we can describe the decomposition \eqref{rietsch_decomp} in terms of \emph{positroid cells} \cite{Pos}. Namely, let $\Boundkn$ denote the set of \emph{bounded affine permutations}, i.e.\ bijections $f : \Z \to \Z$ satisfying
$$
f(i+n) = f(i)+n \text{ and } i \le f(i) \le i+n \text{ for all } i \in \Z, \quad \text{ and } \quad  \sum_{i=1}^n (f(i) - i) = kn.
$$
We write $f$ in \emph{window notation} as $[f(1), \dots, f(n)]$. Given an element of $\Gr(k,n)$ represented by an $n\times k$ matrix $M$, we associate an element $f\in\Boundkn$ as follows: for each $i\in [n]$, we set $f(i) \ge i$ to be minimum such that row $f(i)$ of $M$ is in the span of rows $i, i+1, \dots, f(i)-1$ (where indices are taken modulo $n$). Then the positroid cell $\Pi_f^{>0}$ is defined to be the set of all elements of $\Grtnn(k,n)$ associated to $f$, and
$$
\Grtnn(k,n) = \bigsqcup_{f\in\Boundkn}\Pi_f^{>0}.
$$
The closure relation on cells is given by the dual of the Bruhat order on $\Boundkn$. %The unique top-dimensional cell of $\Grtnn(k,n)$ corresponds to the minimum element $[1+n, \dots, k+n, k+1, \dots, n]$ of $\Boundkn$.

\begin{example}\label{ex:ssa_4}
Let $(k,n) := (2,4)$, and take $g := [2,4,5,7] \in \Bound(2,4)$. Then $\Pi_g^{>0}$ consists of all elements which can be represented by a matrix of the form
$$
\begin{bmatrix}1 & 0 \\ x_1 & 0 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix} \quad \text{ with } x_1, x_3, x_4 > 0.
$$
Now let us describe the map $\isog$ from \eqref{eq:isog_intro} and the dilation action $\dilg$. We have $\Startnn_g = \Pi_g^{>0} \sqcup \Pi_{[3,4,5,6]}^{>0}$, and $\Lktnn_g$ is a point, so $\Cone(\Lktnn_g) \simeq \R$.

First we must fix a subset $I\in\binom{[4]}{2}$ whose Pl\"{u}cker coordinate does not vanish on $\Pi_g^{>0}$. Here we may take any $I\neq\{1,2\}$; let us take $I := \{1,4\}$. This allows us to define the embedding $\bar{\varphi}_I$ of an open dense subset of $\Gr(2,4)$ into the affine flag variety:
$$
\bar{\varphi}_I\left(\begin{bmatrix}1 & 0 \\ x_1 & x_2 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix}\right) = \begin{bmatrix}\frac{1}{z} & 0 & 0 & 0 \\ -x_1 & 1 & 0 & \frac{x_2}{z} \\ -x_3 & 0 & 1 & \frac{x_4}{z} \\ 0 & 0 & 0 & \frac{1}{z}\end{bmatrix}\, , \text{ where $z$ is the formal loop parameter}.
$$
After performing some calculations in the affine flag variety and pulling back the result to $\Gr(2,4)$, we find that
$$
\isog\left(\begin{bmatrix}1 & 0 \\ x_1 & x_2 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix}\right) = \left(\begin{bmatrix}1 & 0 \\ \frac{x_1x_4 - x_2x_3}{x_4} & 0 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix}, \frac{x_2}{x_4}\right).
$$
The first component gives a projection $\Startnn_g \to \Pi_g^{>0}$. The dilation action $\dilg$ is given by
$$
t\cdot\begin{bmatrix}1 & 0 \\ x_1 & x_2 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix} \mapsto \begin{bmatrix}1 & 0 \\ x_1 + (t-1)\frac{x_2x_3}{x_4} & tx_2 \\ x_3 & x_4 \\ 0 & 1\end{bmatrix} \quad \text{ for } t > 0.
$$
\end{example}

\acknowledgements{We thank Sergey Fomin, Patricia Hersh, Alex Postnikov, and Lauren Williams for stimulating discussions. We are also grateful to George Lusztig and Konni Rietsch for their comments on the first version of~\cite{arxiv}.}

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