%=======================================================
% usepackage
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{graphicx}
\usepackage{hyperref}

\usepackage[all]{xy}
\input xy \xyoption{frame}
\xyoption{dvips}

\usepackage[colorinlistoftodos]{todonotes}
\usetikzlibrary{chains,scopes,decorations.markings}

\usepackage[backend=bibtex]{biblatex}
\addbibresource{65-Marberg.bib}

%=========================================================================
% theorem declarations
\newtheorem{theorem}{Theorem}[section]
\newtheorem{thmdef}[theorem]{Theorem-Definition}
\newtheorem{propdef}[theorem]{Proposition-Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{example}[theorem]{Example}
\newtheorem* {theorem*}{Theorem}
\newtheorem* {conjecture*}{Conjecture}
\newtheorem* {claim}{Claim}
\newtheorem* {example*}{Example}
\newtheorem* {fact}{Fact}
\newtheorem* {notation}{Notation}
\newtheorem* {remark}{Remark}
\newtheorem* {examples}{Examples}
\newtheorem* {exercise}{Exercise}
\newtheorem* {remarks}{Remarks}
\xyoption{dvips}



\def\rW{]}
\def\lW{[}

\def\modu{\ (\mathrm{mod}\ }
\def\wh{\widehat}
\def\({\left(}
\def\){\right)}
\newcommand{\sP}{\mathscr{P}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cZ}{\mathcal{Z}}
\def\cS{\mathcal{S}}
\def\cX{\mathcal{X}}
\def\cW{\mathcal{W}}
\def\cY{\mathcal{Y}}
\def\NN{\mathbb{N}}
\def\tr{\mathrm{tr}}
\def\Hom{\mathrm{Hom}}
\def\End{\mathrm{End}}
\def\CC{\mathbb{C}}
\def\RR{\mathbb{R}}
\def\ZZ{\mathbb{Z}}
\def\Aut{\mathrm{Aut}}
\def\X{\mathfrak{X}}
\def\sl{\mathfrak{sl}}
\def\Ind{\mathrm{Ind}}
\def\GL{\mathrm{GL}}
\def\ind{\mathrm{ind}}
\def\Res{\mathrm{Res}}
\def\Inf{\mathrm{Inf}}
\def\ch{\mathrm{ch}}
\def\spanning{\textnormal{-span}} 
\def\Irr{\mathrm{Irr}}
%\def\wt{\widetilde}

\newcommand\wt{\operatorname{wt}}

\def\sh{\mathrm{sh}}
\def\diag{\mathrm{diag}}
\def\ts{\textstyle}
\newcommand{\fkn}{\mathfrak{n}}
\newcommand{\supp}{\mathrm{supp}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\g}{\mathfrak{g}}
\newcommand{\One}{{1\hspace{-.15.5cm} 1}}
\newcommand{\one}{{1\hspace{-.11cm} 1}}
\def\X{\mathcal{X}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\fka}{\mathfrak{a}}
\newcommand{\fkh}{\mathfrak{h}}

\def\FC{\operatorname{FC}}

\def\fk{\mathfrak}
\def\fkm{\mathfrak{m}}

\def\barr{\begin{array}}
\def\earr{\end{array}}
\def\ba{\begin{aligned}}
\def\ea{\end{aligned}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}

\def\Cyc{\mathrm{Cyc}}
\def\Fix{\mathrm{Fix}}

\def\qquand{\qquad\text{and}\qquad}
\def\quand{\quad\text{and}\quad}
\def\qquord{\qquad\text{or}\qquad}
\def\quord{\quad\text{or}\quad}

\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\Sym}{\operatorname{Sym}}

\newcommand{\im}{\operatorname{im}}

\def\inv{\mathrm{Inv}}

\def\I{\mathcal{I}}
\def\M{\mathcal{M}}
\def\cH{\mathcal H}
\def\cM{\mathcal M}
\def\DesR{\mathrm{Des}_R}
\def\DesL{\mathrm{Des}_L}

\def\omdef{\overset{\mathrm{def}}}
\def\uS{\underline S}
\def\hs{\hspace{0.5mm}}

\def\uW{\underline W}
\def\uw{\underline w}

\def\RIGHT{\{\to\}}
\def\LEFT{\{\leftarrow\}}

\def\ds{\displaystyle}

\def\proj{\mathrm{proj}}

\def\id{\mathrm{id}}
\def\PP{\mathbb{P}}
\def\dash{\hs\text{---}\hs}

\def\fkS{\fk S}
\def\tphi{\tilde \phi}
\def\tpsi{\tilde \psi}


\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}

\def\cT{\mathscr{T}}
\def\sU{\mathscr{U}}
\def\cE{\mathcal E}
\def\invol{\textbf{invol}}

\newcommand{\Mon}[1]{\mathscr{F}(#1)}


\def\uL{ \underline{L}}
\def\uR{R}

\def\hs{\hspace{0.5mm}}
\def\simstar{{\hs\hs\sim_{\I_*}\hs\hs}}


\def\fpf{{\tt {FPF}}}

\def\D{\hat D}
\newcommand{\wfpf}{\Theta}
\def\DesF{\mathrm{Des}_\fpf}
\def\DesI{\mathrm{Des}_I}
\def\DesIF{\mathrm{Des}_{F}}
\def\Dfpf{\hat D_{\fpf}}
\def\cfpf{\hat c_{\fpf}}
\def\ilambda{\mu}
\def\flambda{\nu}

\def\Des{\mathrm{Des}}

\def\ellhat{\hat\ell}
\def\Ffpf{\hat F^\fpf}
\def\Sfpf{\hat {\fk S}^\fpf}
\def\ifpf{\iota}

\def\Ifpf{\I_\fpf}

\def\x{\textbf{x}}
\def\y{\textbf{y}}
\def\z{\textbf{z}}
\def\a{\textbf{a}}
\def\b{\textbf{b}}
\def\c{\textbf{c}}
\def\d{\textbf{d}}
\def\e{\textbf{e}}


\newcommand{\xRightarrow}[2][]{\ext@arrow 0359\Rightarrowfill@{#1}{#2}}

\newcommand{\Fl}{\operatorname{Fl}}
\newcommand{\std}{\operatorname{std}}
\newcommand{\Proj}{\operatorname{Proj}}
\newcommand{\dist}{\operatorname{dist}}

\renewcommand{\O}{\operatorname{O}}
\newcommand{\Sp}{\operatorname{Sp}}
\newcommand{\stb}{\operatorname{stb}}
\newcommand{\Ess}{\operatorname{Ess}}
\newcommand{\rank}{\operatorname{rank}}
\renewcommand{\dim}{\operatorname{dim}}

\renewcommand{\r}[1]{\textcolor{red}{#1}}
\renewcommand{\skew}{\operatorname{skew}}
\newcommand{\rk}{\operatorname{rk}}

\def\dbldots{: \hspace{.55mm} : \hspace{.55mm} :}

\def\L{\underline L}
\def\R{R}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\iB}{\hat\cB}
\newcommand{\fB}{\hat\cB^\fpf}

\def\cAfpf{\cA_\fpf}
\def\cRfpf{\hat\cR_\fpf}
\def\iR{\hat\cR}
\def\iHR{\cH\hat\cR}

\def\act{\rtimes}
\newcommand\dact{\hat\circ}
\def\dem{\circ}

\def\invdemspace{\hspace{.85mm}}
\newcommand{\geninvdem}[3]{#2 \invdemspace \dact#1\invdemspace #3}
\newcommand{\idem}[2]{\geninvdem{_*}{#1}{#2}}
\newcommand{\sidem}[2]{\geninvdem{}{#1}{#2}}

\def\F{\mathscr{F}}
\newcommand{\arc}[2]{ \ar @/^#1pc/ @{-} [#2] }
\def\arcstop{\endxy\ }
\def\arcstart{\ \xy<0cm,-.06cm>\xymatrix@R=.1cm@C=.2cm }
\newcommand{\arcstartc}[1]{\ \xy<0cm,-.15cm>\xymatrix@R=.1cm@C=#1cm}

\def\ellhat{\hat\ell}
\def\Ffpf{\hat F^\fpf}
\def\Sfpf{\hat {\fk S}^\fpf}
\def\iS{\hat \fkS}

\def\Luc{\mathrm{Luc}}

\newcommand{\ct}{\operatorname{ct}}

\newcommand{\del}{\operatorname{del}}

\newcommand{\push}{\mathord{\downarrow}}
\newcommand{\pull}{\mathord{\uparrow}}

\newcommand{\ipush}{\mathord{\Downarrow}}
\newcommand{\ipull}{\mathord{\Uparrow}}

\def\sM{\mathscr{M}}
\def\ttS{{\tt S}}

\def\s{\mathbf{s}}
\def\t{\mathbf{t}}
\def\r{\mathbf{r}}
\def\customstar{\hs\hat \bullet_{\mathrm{des}}\hs}

\def\red{\emph{reduce}}
\def\sA{\mathscr{A}}
\def\sB{\mathscr{B}}
\def\sR{\mathscr{R}}
\def\sC{\mathscr{C}}
\def\sE{\mathscr{E}}
\def\ihB{\underline{\isB}}

\def\sI{\mathscr{I}}
\def\tS{\tilde S}
\def\tX{G}


\def\tI{\tilde I}
\def\cG{\mathcal{G}}


\newcommand{\sAq}[1]{\sA_{\bullet,\geq #1}}
\newcommand{\sAp}[1]{\sA_{\leq #1,\bullet}}

\def\sX{\mathscr{X}}
\def\sY{\mathscr{Y}}

\def\u{\mathbf{u}}
\def\v{\mathbf{v}}
\def\isB{\hat{\mathscr{B}}}

\def\cWmin{\cM}

\def\UT{\operatorname{UT}}
\def\traj{\operatorname{traj}}





\def\diagramAB{
\arcstart
{
*{\circ}     & *{\circ}
}
\arcstop}

\def\diagramBA{
\arcstart
{
*{\circ}   \arc{.5}{r}  & *{\circ}
}

\arcstop
}
\def\diagrambAC{
\arcstart
{
*{\bullet}    \arc{.5}{r}  & *{\circ} & *{\circ}
}
\arcstop
}
\def\diagramBaC{
\arcstart
{
*{\circ}    \arc{.5}{r}  & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramcBA{
\arcstart
{
*{\bullet}    \arc{.5}{rr}  & *{\circ} & *{\circ}
}
\arcstop
}




\def\diagramACb{
\arcstart
{
*{\circ}     & *{\circ}  \arc{.5}{r} & *{\bullet}
}
\arcstop
}
\def\diagramAcB{
\arcstart
{
*{\circ}     & *{\bullet}  \arc{.5}{r} & *{\circ}
}
\arcstop
}
\def\diagramCbA{
\arcstart
{
*{\circ}     \arc{.5}{rr}   & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramCBa{
\arcstart
{
*{\circ}     \arc{.5}{rr}   & *{\circ} & *{\bullet}
}
\arcstop
}


\def\diagramBAdc{
\arcstart
{
*{\circ}  \arc{.5}{r}   & *{\circ}    & *{\bullet} \arc{.5}{r}& *{\bullet}
}
\arcstop
}
\def\diagramBaDc{
\arcstart
{
*{\circ}  \arc{.5}{r}   & *{\bullet}    & *{\circ} \arc{.5}{r}& *{\bullet}
}
\arcstop
}
\def\diagramBadC{
\arcstart
{
*{\circ}  \arc{.5}{r}   & *{\bullet}    & *{\bullet} \arc{.5}{r}& *{\circ}
}
\arcstop
}
\def\diagrambAdC{
\arcstart
{
*{\bullet}  \arc{.5}{r}   & *{\circ}    & *{\bullet} \arc{.5}{r}& *{\circ}
}
\arcstop
}
\def\diagrambADc{
\arcstart
{
*{\bullet}  \arc{.5}{r}   & *{\circ}    & *{\circ} \arc{.5}{r}& *{\bullet}
}
\arcstop
}
\def\diagrambaDC{
\arcstart
{
*{\bullet}  \arc{.5}{r}   & *{\bullet}    & *{\circ} \arc{.5}{r}& *{\circ}
}
\arcstop
}


\def\diagramCDab{
\arcstart
{
*{\circ}  \arc{.5}{rr}   & *{\circ} \arc{.5}{rr}   & *{\bullet} & *{\bullet}
}
\arcstop
}
\def\diagramCdAb{
\arcstart
{
*{\circ}  \arc{.5}{rr}   & *{\bullet} \arc{.5}{rr}   & *{\circ} & *{\bullet}
}
\arcstop
}
\def\diagramCdaB{
\arcstart
{
*{\circ}  \arc{.5}{rr}   & *{\bullet} \arc{.5}{rr}   & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramcDAb{
\arcstart
{
*{\bullet}  \arc{.5}{rr}   & *{\circ} \arc{.5}{rr}   & *{\circ} & *{\bullet}
}
\arcstop
}
\def\diagramcDaB{
\arcstart
{
*{\bullet}  \arc{.5}{rr}   & *{\circ} \arc{.5}{rr}   & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramcdAB{
\arcstart
{
*{\bullet}  \arc{.5}{rr}   & *{\bullet} \arc{.5}{rr}   & *{\circ} & *{\circ}
}
\arcstop
}

\def\diagramDbcA{
\arcstart
{
*{\circ}  \arc{.5}{rrr}   & *{\bullet}   & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramDbCa{
\arcstart
{
*{\circ}  \arc{.5}{rrr}   & *{\bullet}   & *{\circ} & *{\bullet}
}
\arcstop
}
\def\diagramdBcA{
\arcstart
{
*{\bullet}  \arc{.5}{rrr}   & *{\circ}   & *{\bullet} & *{\circ}
}
\arcstop
}


\def\diagramDcBa{
\arcstart
{
*{\circ}  \arc{.5}{rrr}   & *{\bullet} \arc{.25}{r}   & *{\circ} & *{\bullet}
}
\arcstop
}

\def\diagramDCba{
\arcstart
{
*{\circ}  \arc{.5}{rrr}   & *{\circ} \arc{.25}{r}   & *{\bullet} & *{\bullet}
}
\arcstop
}
\def\diagramDcbA{
\arcstart
{
*{\circ}  \arc{.5}{rrr}   & *{\bullet} \arc{.25}{r}   & *{\bullet} & *{\circ}
}
\arcstop
}
\def\diagramdcBA{
\arcstart
{
*{\bullet}  \arc{.5}{rrr}   & *{\bullet} \arc{.25}{r}   & *{\circ} & *{\circ}
}
\arcstop
}


