%=======================================================
% usepackage
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{shuffle}
\usepackage{tikz-cd}
\usepackage[all]{xy}
\input xy \xyoption{frame}
\usepackage{ytableau}


\usepackage{enumerate,tikz}
\usepackage{color}
\usepackage{mathdots}
\usepackage[vcentermath,enableskew]{youngtab}


\usepackage{mathabx}
\usepackage{mathtools}
\usepackage{array}

\def\PhiBS{\Psi}
\def\ModG{\mathsf{Mod}_G}
\def\ModH{\mathsf{Mod}_H}
\def\TT{{\mathbb T}}
\def\Ad{\mathrm{Ad}}
\def\SS{{\mathbb{S}}}

\def\W{W^{\BC}}
\def\WD{W^{\D}}
\def\e{\mathbf{e}}
\def\abs{\mathsf{abs}}
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}

\newcommand{\dK}[1]{\mathbin{\widesim{#1}{\mathsf{dK}}}}
\newcommand{\K}[1]{\mathbin{\widesim{#1}{\mathsf{K}}}}
\newcommand{\iK}[1]{\mathbin{\widesim{#1}{\mathsf{iK}}}}
\newcommand{\lr}[1]{\mathbin{\lrarrow{#1}}}


 \newcommand{\mleftrightarrow}[1]{\mathbin{\underset{\m}{\overset{#1}{\longleftrightarrow}}}}
 \renewcommand{\nleftrightarrow}[1]{\mathbin{\underset{\n}{\overset{#1}{\longleftrightarrow}}}}


\newcommand{\simRBS}[1]{\mathbin{\widesim{#1}{\rBS}}}
\newcommand{\simCBS}[1]{\mathbin{\widesim{#1}{\cBS}}}

\def\cI{I}
\def\I{\cI}

\def\Ifpf{I^{\mathsf{FPF}}}
\def\onefpf{1_{\mathsf{FPF}}}
\def\mmap{\iota_{\mathsf{asc}}}
\def\nmap{\iota_{\mathsf{des}}}
\def\cGm{\cG^\m}
\def\cGn{\cG^\n}


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


\numberwithin{equation}{section}
%=====================================================================
% 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{hypothesis}[theorem]{Hypothesis}
\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}[theorem]{Exercise}
\theoremstyle{definition}
\newtheorem{problem}[theorem]{Problem}
\theoremstyle{definition}
\newtheorem* {remarks}{Remarks}



\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{\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\M{\mathcal{M}}
\def\cH{\mathcal H}
\def\DesR{\mathrm{Des}_R}
\def\DesL{\mathrm{Des}_L}
\def\DesV{\mathrm{Des}_V}

\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\cF{\mathcal F}
\def\cV{\mathcal V}
\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\ellfpf{\ellhat_\fpf}
\def\Ffpf{\hat F^\fpf}
\def\Sfpf{\hat {\fk S}^\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\HAfpf{\HA^\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\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\Inv{\operatorname{Inv}}
\def\HA{\cA_{\textsf{Hecke}}}
\def\iF{\hat{F}}

\def\F{\tilde F}
\def\tF{F}
\def\QSym{\textsf{QSym}}
\def\Sym{\textsf{Sym}}
\def\zetaq{\zeta_{\QSym}}
\def\Par{\textsf{Par}}
\def\tI{\tilde I}
\def\iPhi{\Upsilon}
\def\fPhi{\Psi}

\def\Refl{\textsf{Ref}}
\def\Cov{\textsf{Cov}}
\def\dotequals{\mathbin{\overset{_\bullet} =}}
\def\V{\textsf{Vir}}

\newcommand{\arc}[2]{ \ar @/^#1pc/ @{-} [#2] }
\def\arcstop{\endxy\ }
\def\arcstart{\ \xy<0cm,-0.5cm>\xymatrix@R=.1cm@C=.5cm }
\newcommand{\arcstartc}[1]{\ \xy<0cm,-.15cm>\xymatrix@R=.1cm@C=#1cm}

\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
{
*{\bullet}  \arc{.5}{rrr}   & *{\circ} \arc{.25}{r}   & *{\bullet} & *{\circ}
}
\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
}





\def\tp{\Theta^+}
\def\tm{\Theta^-}
\def\tpm{\{\Theta^\pm\}}
\def\cF{\mathcal F}
\def\cFp{\cF_n^+}
\def\cFm{\cF_n^-}
\def\h {\mathrm{ht}}
\def\Cyc{\mathrm{Cyc}}
\def\wCyc{\widetilde\Cyc}
\def\sfpf{\sgn_\fpf}
\def\m{\mathfrak{m}}
\def\n{\mathfrak{n}}
\def\Em{E_\m}
\def\En{E_\n}
\def\E{E}
\def\invsim_i{\overset{\mathrm{i}}{\underset{\mathrm{inv}}{\sim}}}



\def\LL{\ZZ[x,x^{-1}]}
\def\H{\mathcal{H}}
\def\A{\mathsf{A}}
\def\BC{\mathsf{BC}}
\def\D{\mathsf{D}}
%\def\one{M}
\def\Ifpf{\cI^{\mathsf{FPF}}}
\def\ellfpf{\ell^{\mathsf{FPF}}}
\def\AbsFix{\mathsf{Fix}^\pm}
\def\cG{\mathcal{G}}
\def\G{\mathbf{G}}
\def\Asc{\mathrm{Asc}}
\def\cP{\mathscr{P}}

\def\WB{W^{\mathsf{BC}}}
\def\WD{W^{\mathsf{D}}}

\def\cFB{\cF^{\mathsf{BC}}}
\def\cFD{\cF^{\mathsf{D}}}

\def\cKB{\cK^{\mathsf{BC}}}
\def\cKD{\cK^{\mathsf{D}}}

\def\SB{S^{\mathsf{BC}}}
\def\SD{S^{\mathsf{D}}}

\def\m{\mathbf{m}}
\def\n{\mathbf{n}}

\def\lch{\mathrm{lch}}
\def\dr{\mathrm{dr}}

\def\row{\mathfrak{row}}
\def\col{\mathfrak{col}}
\def\RSK{\xrightarrow{\mathsf{RSK}}}
\def\PRSK{P_{\mathsf{RSK}}}
\def\QRSK{Q_{\mathsf{RSK}}}
\def\b{\mathsf{ivalue}}
\def\f{\mathsf{frow}}

\def\PMRSK{P_{\m}}
\def\PNRSK{P_{\n}}

\def\fromRSK{\xleftarrow{\hs\mathsf{RSK}\hs}}
\def\from{\leftarrow}

\def\rBS{\mathsf{It}}
\def\cBS{\mathsf{nIt}}

\def\fromRBS{\xleftarrow{\hs\rBS\hs}}
\def\fromCBS{\xleftarrow{\hs\cBS\hs}}

\def\PRBS{P_{\rBS}}
\def\PCBS{P_{\cBS}}

\def\fromMRSK{\xleftarrow{\hs\m\hs}}
\def\fromNRSK{\xleftarrow{\hs\n\hs}}
\def\toNRSK{\xrightarrow{\hs\n\hs}}
\def\toNRSK{\xrightarrow{\hs\n\hs}}

\newcommand{\widesim}[3][1.5]{
  \overset{#2}{\underset{#3}{\scalebox{#1}[1]{$\sim$}}}
}
\newcommand{\lrarrow}[2][1.5]{
  \overset{#2}{\scalebox{#1}[1]{$\longleftrightarrow$}}
}

\newcommand{\ytab}[1]{
\ytableausetup{boxsize = .4cm,aligntableaux=center}
 \begin{ytableau} #1 \end{ytableau}}
\newcommand{\ytabb}[1]{
\ytableausetup{boxsize = .9cm,aligntableaux=center}
 \begin{ytableau} #1 \end{ytableau}}
\newcommand{\yytab}[1]{
\ytableausetup{boxsize = .3cm,aligntableaux=center}
 \begin{ytableau} #1 \end{ytableau}}
\newcommand{\yttab}[1]{
\ytableausetup{boxsize = .6cm,aligntableaux=center}
 \begin{ytableau} #1 \end{ytableau}}

\def\Ad{\mathrm{Ad}}
\def\W{W^{\BC}}
\def\WD{W^{\D}}

\def\TT{{\mathbb T}}


\usepackage[colorinlistoftodos]{todonotes}
\newcommand{\eric}[1]{\todo[size=\tiny,inline,color=green!30]{#1 \\ \hfill --- E.}}
\newcommand{\yifeng}[1]{\todo[size=\tiny,inline,color=blue!30]{#1 \\ \hfill --- Y.}}



\def\KK{\mathbb{K}}
\newcommand{\dom}{\Omega_{\mathrm{dom}}}
\newcommand{\fw}[1]{{\operatorname{fw}\mathopen{}\left(#1\right)\mathclose{}}}
\newcommand{\str}[2]{{\operatorname{\mathfrak{st}}_{#1}\mathopen{}\left(#2\right)\mathclose{}}}
\def\st{{\mathfrak{st}}} %there are two of these; resolve
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\RSYT}{\textup{RSYT}}
\newcommand{\offset}{\mathbf{s}}
\DeclareMathOperator{\rev}{\textup{rev}}
\newcommand{\tsc}{{\underline{c}}}
\DeclareMathOperator{\Dom}{\textup{Dom}}
\newcommand{\lc}{\Gamma}