%=======================================================
% usepackage
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{shuffle}
\usepackage{stmaryrd}
\usepackage{enumitem}
\usepackage[noabbrev]{cleveref}
\newcommand\Crefrangeconjunction{\textendash}
\usepackage[belowskip=-15pt,aboveskip=0pt]{caption}


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

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

\usepackage{tikz-cd}

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


\def\cols{\mathsf{cols}}

\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{\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\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}
\def\cyc{\operatorname{cyc}}

\def\fG{\mathcal{G}^{(\beta)}}
\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}}

\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\Pair{\mathrm{Pair}}
\def\pair{\mathrm{pair}}

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

\def\inv{{\mathsf{inv}}}
\def\Inv{\operatorname{Inv}}


\def\I{\mathcal{I}}
\def\M{\mathcal{M}}
\def\cH{\mathcal H}
\def\cM{\mathcal M}
\def\DesR{\operatorname{Des}_R}
\def\DesL{\operatorname{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{Z}_{>0}}
\def\dash{\cdots\cdot\cdot\hs}

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


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

\def\bei{\begin{itemize}}
\def\eei{\end{itemize}}

\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{{\mathsf {FPF}}}
\def\fp{\mathrm{fp}}
\def\D{\mathsf{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{\operatorname{Des}}
\def\NDes{\operatorname{NDes}}
\def\NNeg{\operatorname{NNeg}}
\def\NFix{\operatorname{NFix}}

\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}}

\newcommand{\SO}{\operatorname{SO}}
\renewcommand{\O}{\mathsf{O}}
\newcommand{\Sp}{\mathsf{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=.10cm }
\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\cWmin{\cM}

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


\def\neg{\operatorname{neg}}
\def\Neg{\operatorname{Neg}}
\def\sort{\operatorname{sort}}
\def\offset{\operatorname{offset}}


\def\cV{\mathcal{V}}
\def\NCSM{\textsf{NCSP}}
\def\htb{\operatorname{ht}_B}

\def\XA{\cW_{{\tt embed}}}
\def\HA{\cA_{{\tt hecke}}}
\def\bar{\overline}
\def\Asc{\operatorname{Asc}}

\def\scov{<\hspace{-1.6mm}\vartriangleleft_B}
\def\bcov{<\hspace{-1.6mm}\blacktriangleleft_B}
\def\lleq{\leq \hspace{-1.6mm}\leq}

\def\X{\mathcal{X}}
\def\Y{\mathcal{Y}}
\def\Z{\mathcal{Z}}
\def\wB{\bar1\hs\bar  2\hs\bar 3\cdots  \bar n}
\def\Id{Id}

\def\cRwin{\cR^{\mathrm{win}}}
\def\iF{\hat F}


\def\SL{\mathrm{SL}}
\def\O{\mathrm{O}}


\def\W{W^{\mathsf{BC}}}
\def\DGroup{W^{\mathsf{D}}}
\def\ellD{\ell^{\mathsf{D}}}
\def\wD{w_0^{\mathsf{D}_n}}
\def\HA{\cA_{\mathsf{Hecke}}}
\def\cAB{\cA_{\mathsf{Brion}}}
\def\RSphi{\phi_{\mathrm{RS}}}
\def\bRSphi{\bar\phi_{\mathrm{RS}}}

\def\cAfpfD{\cAfpf^{\mathsf{D}}}
\def\cAD{\cA^{\mathsf{D}}}

\def\Aphi{\iota^{\mathsf{A}}}
\def\BCDphi{\iota^{\mathsf{BC}}}
\def\RSphiB{\phi_{\mathrm{B}}}
\def\bRSphiB{\bar\phi_{\mathrm{B}}}

\def\SGamma{\Gamma^{\mathsf{sym}}}
\def\SSGamma{\Gamma^{\mathsf{skew}}}

\def\Clans{\mathsf{Clans}}

\def\fpf{\mathsf{fpf}}

\def\BC{\mathsf{BC}}
\def\cAk{\cA_k}
\def\cAkfpf{\cA_k^{\fpf}}
\def\yfpf{y^{\fpf}}
\def\gfpf{g^{\fpf}}
\def\shfpf{\sh^{\fpf}}
\def\signs{\mathsf{signs}}

\definecolor{darkred}{rgb}{0.7,0,0} % darkred color
\newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition


\def\AO{\mathsf{AO}}
\def\MAO{\fkm\mathsf{AO}}
\newcommand{\Sym}{\mathsf{Sym}}
\newcommand{\QSym}{\mathsf{QSym}}

\def\wzeta{\bar\zeta}
\def\zetaq{\zeta_{\mathsf{Q}}}
\def\wzetaq{\wzeta_{\mathsf{Q}}}

\def\op{\mathsf{op}}
\def\asc{\operatorname{asc}}
\def\des{\operatorname{des}}


\def\oX{\overset{\to}{X}}
\def\oGamma{\overset{\to}{\Gamma}}
\def\bDelta{\Delta^{(\beta)}}

\def\mQSym{\fkm\QSym}
\def\mSym{\fkm\Sym}

\newcommand{\cO}{\mathcal{D}}
\def\wO{\bar\cO}
\def\wG{\bar\cG}
\def\wP{\bar\cP}
\def\htimes{\mathbin{\bar\otimes}}

\def\bX{\bar X}
\def\Set{\mathsf{Set}}

\def\bare{\bar e}
\def\bGamma{\bar \Gamma}
\def\mL{\cM}
\def\bL{\bar L}
\def\antipode{\mathsf{S}}

\def\wGwt{\mathfrak{m}\mathsf{WGraphs}}
\def\wzetaw{\wzeta_{\mathsf{WGraphs}}}

\def\kam{\overline{\widetilde m}}

\def\align{\mathsf{align}}

\def\SetSSYT{\mathsf{SetSSYT}}
\def\source{\mathsf{source}}

\def\oDelta{\overset{\to}\Delta}
\def\ozetaG{\overset{\to}{\zeta_\cG}}
\def\ozetaP{\overset{\to}{\zeta_\cP}}
\def\ocG{\overset{\to}\cG}
\def\ocP{\overset{\to}\cP}




\def\tm{\widetilde m}
\def\btm{\overline{\widetilde m}}

\def\oT{\mathscr{G}}
\def\inc{\mathsf{inc}}


 








\usepackage{pbox}
\tikzset{every loop/.style={min distance=10 mm, in=60, out=120, looseness=10}}
\tikzset{
dot/.style = {circle, fill, inner sep=2.2,outer sep=0},
% dot/.default = 6pt % size of the circle diameter 
}



\numberwithin{equation}{section}
\allowdisplaybreaks[1]
\UseCrayolaColors

\def\KK{\mathbb{K}}

\def\merge{\mathsf{merge}}

\newcommand{\arxiv}[1]{\href{https://arxiv.org/abs/#1}{\texttt{arXiv:#1}}}

\def\op{\mathrm{decr}}
\def\ozeta{\zeta_{\oG}}

\def\bpartial{\partial^{(\beta)}}
\def\fkG{\mathcal{G}^{(\beta)}}
\def\fkGO{\mathcal{G}^{\mathsf{O}}}
\def\fkGSp{\mathcal{G}^{\mathsf{Sp}}}

\def\MX{M\hspace{-0.5mm}X}
\def\Mat{\mathsf{Mat}}
\def\codim{\mathrm{codim}}
\def\DesV{\operatorname{Des}_V}