%=======================================================
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{graphicx}
\usepackage{shuffle}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[bbgreekl]{mathbbol}
\usepackage{mathdots}
\usepackage{ytableau}
\usepackage{tikz-cd}

\usepackage{stmaryrd}

\usepackage[colorinlistoftodos]{todonotes}
\usetikzlibrary{chains,scopes}
\usetikzlibrary{shapes.geometric,positioning}

\DeclareSymbolFontAlphabet{\mathbb}{AMSb}
\DeclareSymbolFontAlphabet{\mathbbl}{bbold}

\newcommand{\SSYT}{\ensuremath\mathrm{SSYT}}
\newcommand{\SYT}{\ensuremath\mathrm{SYT}}
\newcommand{\RPP}{\ensuremath\mathrm{PP}}
\newcommand{\MRPP}{\ensuremath\mathsf{ShPP}}


\newcommand{\WkSetSSYT}{\ensuremath\mathrm{Set}^*\mathrm{SSYT}}

\newcommand{\SetSSYT}{\ensuremath\mathrm{SetSSYT}}
\newcommand{\SetSYT}{\ensuremath\mathrm{SetSYT}}


\newcommand{\SSMT}{\ensuremath\mathrm{SSMT}}
\newcommand{\SMT}{\ensuremath\mathrm{SMT}}

\newcommand{\SetSSMT}{\ensuremath\mathrm{SetSSMT}}
\newcommand{\SetSMT}{\ensuremath\mathrm{SetSMT}}

\newcommand{\WkSetSSMT}{\ensuremath\mathrm{WeakSet}\mathrm{SSMT}}



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

%=========================================================================
% 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}
\newtheorem{problem}[theorem]{Problem}
\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}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem {example}[theorem]{Example}
\theoremstyle{definition}
\newtheorem* {examples}{Examples}
\theoremstyle{definition}
\newtheorem* {exercise}{Exercise}
\theoremstyle{definition}
\newtheorem* {remarks}{Remarks}






\def\modu{\ (\mathrm{mod}\ }
\def\wh{\widehat}
\def\({\left(}
\def\){\right)}
\newcommand{\sP}{\mathscr{P}}
\newcommand{\sQ}{\mathscr{Q}}
\newcommand{\sI}{\mathscr{I}}
\newcommand{\sJ}{\mathscr{J}}
\newcommand{\sK}{\mathscr{K}}
\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{\cZ}{\mathcal{Z}}
\def\cS{\mathcal{S}}
\def\cX{\mathcal{X}}
\def\cY{\mathcal{Y}}
\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}
\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}}
\def\X{\mathcal{X}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\fka}{\mathfrak{a}}
\newcommand{\fkh}{\mathfrak{h}}

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

\def\inv{\mathrm{Inv}}
\def\FPF{\mathrm{FPF}}


\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\fkS{\fk S}

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

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

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

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


\def\fpf{{\textsf {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\Asc{\mathrm{Asc}}



%\def\FF{\mathfrak{F}_\infty}
%\def\II{\mathfrak{I}_\infty}
%\def\SS{\mathfrak{S}_\infty}
\def\FF{\mathbb{F}}

\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{{\tt c}}
\def\t{{\tt t}}
\def\r{{\tt r}}


\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}{\textsf{Sp}}
\newcommand{\OSp}{\textsf{OSp}}
\newcommand{\stb}{\operatorname{stb}}
\newcommand{\Ess}{\operatorname{Ess}}
\newcommand{\rank}{\operatorname{rank}}
\renewcommand{\dim}{\operatorname{dim}}

\renewcommand{\skew}{\operatorname{skew}}
\newcommand{\rk}{\operatorname{rk}}

\def\dbldots{: \hspace{.65mm} : \hspace{.65mm} :}

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

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

\def\cAfpf{\cA^\fpf}
\def\cRfpf{\hat\cR^{\textsf{FPF}}}
\def\cHfpf{\cH^{\textsf{Sp}}}
\def\iH{\cH^{\textsf{O}}}
\def\iR{\hat\cR}
\def\iF{\hat F}
\def\Ffpf{\hat F_\fpf}

\def\iG{\hat G^{\textsf{O}}}
\def\Gfpf{\hat G^{\textsf{Sp}}}


\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,-.15cm>\xymatrix@R=.1cm@C=.3cm }
\newcommand{\arcstartc}[1]{\ \xy<0cm,-.15cm>\xymatrix@R=.1cm@C=#1cm}

\def\ellhat{\ell^{\textsf{O}}}
\def\ellfpf{\ell^{\textsf{Sp}}}

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

\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\red{\emph{reduce}}
\def\sR{\mathscr{R}}
\def\sC{\mathscr{C}}
\def\sW{\mathscr{W}}
\def\sE{\mathscr{E}}
\def\ihB{\underline{\isB}}

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

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

\def\v{\mathbf{v}}
\def\b{\mathfrak{b}}
\def\w{\mathbf{w}}

\def\isB{\hat{\mathscr{B}}}

\newcommand{\gray}[1]{\textcolor{gray}{#1}}

\def\cT{\mathcal{T}}
\def\Fock{\overline{\cK}}

\def\p{\mbox{\textbf{p}}}
\def\h{\mbox{\textbf{h}}}
\def\q{\mathfrak{q}}
\def\bfM{\mbox{\textbf{M}}}
\def\bfSigma{\mathbf{\Sigma}}
\def\sSigma{\mathbbl{\Sigma}}
\def\kk{\mathbbl{k}}
\def\bfR{\mbox{\textbf{R}}}
\def\bfW{\mbox{\textbf{W}}}
\def\M{\mathbb{M}}
\def\FC{\mbox{\textsf{FC}}}
\def\PR{\textsf{Tab}_{PR}}
\def\MR{\mbox{\textsf{MR}}}
\def\RW{\mbox{\textsf{RW}}}
\def\Words{\mathbb{W}}
\def\Tab{\textsf{Tab}_{EG}}
\def\FCTab{\textsf{Tab}_{FC}}
\def\Perm{\textsf{Perm}}
\def\FCPerm{\textsf{Perm}_{FC}}


\def\GrVec{\text{\emph{GrVec}}_\QQ}
\def\FinSet{\text{\emph{FinSet}}}

\def\st{\mathfrak{st}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\Inv{\mathrm{Inv}}


\def\rsk{\xmapsto[]{RSK}}
\def\eg{\xmapsto[]{EG}}

\def\FI{\textsf{FI}}
\def\FB{\textsf{FB}}
\def\OB{\textsf{OB}}
\def\Set{\textsf{Set}}
\def\Mon{\textsf{Mon}}
\def\Comon{\textsf{Comon}}
\def\Bimon{\textsf{Bimon}}
\def\Hopf{\textsf{Hopf}}
\def\Rmod{\textsf{Mod}_R}
\def\kVec{\textsf{Vec}_\kk}
\def\whkVec{\wh{\textsf{Vec}}_\kk}
\def\kGrVec{\textsf{GrVec}_\kk}
\def\RmodStar{\textsf{Mod}^*_R}
\def\GrRmod{\textsf{GrMod}_R}
\def\GrRmodStar{\textsf{GrMod}^*_R}
\def\Free{\textsf{Free}_R}

\def\P{\mathrm{P}}
\def\B{\mathrm{B}}
\def\Q{\mathrm{Q}}
\def\F{\mathscr{F}}
\def\G{\mathscr{G}}
\def\sD{\mathscr{D}}
\def\Mor{\mathrm{Mor}}
\def\L{\mathbb{L}}
\def\bfL{\mathbf{L}}

\def\sL{\mathscr{L}}
\def\v{\mathbf{v}}

\def\Zero{\mathbf{0}}
\def\One{\mathbf{1}}
\def\one{\underline{1}}
\def\E{\mathbf{E}}
\def\antipode{{\tt S}}
\def\coeff{\mathrm{coeff}}
\def\RmodSp{\Rmod\text{-}\Sp}
\def\RmodStarSp{\RmodStar\text{-}\Sp}
\def\FreeSp{\Free\text{-}\Sp}
\def\FreeStarSp{\Free^*\text{-}\Sp}
\def\SetSp{\Set\text{-}\Sp}
\def\RmodOSp{\OSp_R}
\def\Bimon{\textsf{Bimon}}
\def\Triv{\textsf{Triv}}
\def\BB{\mathbb{B}}
\def\AA{\mathbb{A}}

\def\Reduced{\textsf{Reduced}}
\def\Packed{\textsf{Packed}}
\def\RSK{\textsf{RSK}}
\def\Gr{\mathrm{Gr}}
\def\fkt{\fk t}
\def\fku{\fk u}
\def\fkv{\fk v}
\newcommand{\row}{\operatorname{row}}

\def\dfslash{/\hspace{-1mm}/}
\def\dbslash{\backslash\hspace{-1mm}\backslash}
\def\htimes{\mathbin{\hat\otimes}}

\def\fsim{\mathbin{=_\textsf{Sp}}}
\def\fequiv{\mathbin{\equiv_\textsf{Sp}}}
\def\isim{\mathbin{=_\textsf{O}}}
\def\iequiv{\mathbin{\equiv_\textsf{O}}}
\def\simA{=_{\textsf{Br}}}
\def\equivA{\equiv_{\textsf{Br}}}

\def\simKK{\overset{\textsf{K}}\approx}
\def\simCK{\overset{\textsf{K}}\sim}
\def\simICK{\overset{\textsf{O}}\sim}
\def\simIKK{\overset{\textsf{O}}\approx}
\def\simFCK{\overset{\textsf{Sp}}\sim}
\def\simFKK{\overset{\textsf{Sp}}\approx}

\def\shcoprod{\wedge}


\newcommand{\Sym}{\textsf{Sym}}
\newcommand{\NSym}{\textsf{NSym}}
\newcommand{\MPR}{\textsf{MR}}
\newcommand{\QMPR}{\textsf{QMR}}
\newcommand{\NMPR}{\textsf{NMR}}
\newcommand{\QSym}{\textsf{QSym}}
\newcommand{\bQSym}{\textsf{QSym}^{(\beta)}}

\def\zetaq{\zeta_{\textsf{Q}}}

\newcommand{\SymP}{\Sym_{\textsf{P}}}
\newcommand{\SymQ}{\Sym_{\textsf{Q}}}
\newcommand{\Peak}{\textsf{Peak}}
\newcommand{\coPeak}{\Pi\Sym}
\newcommand{\bcoPeak}{\overline{\Pi}\QSym}
\def\bK{\bar{K}}

\def\MGSym{\M\Sym_{Q}}%{\M\Gamma\Sym}
\def\mGSym{\m\Sym_{Q}}%{\m\Gamma\Sym}
\def\omGSym{\m\Sym_{P}}%{\m\bar\Gamma\Sym}
\def\oMGSym{\M\Sym_{P}}%{\M\bar\Gamma\Sym}

\newcommand{\MSymP}{\mathfrak{M}\Gamma\Sym}%{\mathfrak{M}\SymP}
\newcommand{\mSymQ}{\mathfrak{m}\Gamma\Sym}

\newcommand{\MSymQ}{\mathfrak{M}\bar\Gamma\Sym}%{\mathfrak{M}\SymQ}
\newcommand{\mSymP}{\mathfrak{M}\bar\Gamma\Sym}%{\mathfrak{m}\SymP}

\newcommand{\mQSymQ}{\mathfrak{m}\Pi\Sym}
\newcommand{\mQSymP}{\mathfrak{m}\bar{\Pi}\Sym}

\newcommand{\MPeakQ}{\mathfrak{M}\bar{\textsf{P}}\textsf{eak}}
\newcommand{\MPeakP}{\mathfrak{M}\Peak}

\newcommand{\MSym}{\mathfrak{M}\textsf{Sym}}
\newcommand{\MNSym}{\mathfrak{M}\textsf{NSym}}
\newcommand{\MMPR}{\mathfrak{M}\textsf{MR}}


\newcommand{\mNSym}{\mathfrak{m}\textsf{NSym}}
\newcommand{\mMPR}{\mathfrak{m}\textsf{MR}}
\newcommand{\MQSym}{\mathfrak{M}\textsf{QSym}}
\newcommand{\Shuffle}{\textsf{Shuffle}}
\newcommand{\colim}{\varinjlim}
\newcommand{\weight}{\operatorname{wt}}

\def\cW{\mathcal{W}}
\newcommand{\Incr}{\operatorname{Incr}}
\newcommand{\Decr}{\operatorname{Decr}}

\def\BP{\textsf{BP}}
\def\RBP{\textsf{RBP}}
\def\CBP{\textsf{CBP}}

\def\SymTab{\textsf{SymTab}}
\def\SymSub{\textsf{SymSh}}
\def\Bump{{\fk B}}




\def\row{\mathfrak{row}}
\def\col{\mathfrak{col}}
\def\swdiag{\mathfrak{d}_{\textsf{SW}}}
\def\nediag{\mathfrak{d}_{\textsf{NE}}}


\def\columns{\mathsf{columns}}

\def\iarrow{\xleftarrow{\textsf{O}}}
\def\farrow{\xleftarrow{\textsf{Sp}}}

\def\PI{P_{{\textsf O}}}
\def\QI{Q_{{\textsf O}}}
\def\PF{P_{{\textsf{Sp}}}}
\def\QF{Q_{{\textsf{Sp}}}}
\def\unbump{\textsf{uninsert}}
\def\bump{\textsf{insert}}

\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=1](char) {#1};}}


\def\dash{\text{ --- }}
\def\word{\mathfrak{word}}


\def\rowbt{\overset{(i,j)}{\underset{\row}{\leadsto}}}
\def\colbt{\overset{(i,j)}{\underset{\col}{\leadsto}}}

\def\Nil{\mathcal{U}_\infty}


\def\tq{GQ}
\def\kq{KQ}
\def\kp{KP}
\def\tQ{\textsf{GQ}}
\def\tP{\textsf{GP}}
\def\tp{GP}
\def\pf{\operatorname{pf}}
\def\F{F}

\def\Words{\textsf{W}}
\def\PWords{\textsf{PWords}}
\def\WQSym{\textsf{WQSym}}
\def\mWQSym{\mathfrak{m}\textsf{WQSym}}
\def\SetComp{\textsf{SComp}}
\def\mWQSym{\m\textsf{WQSym}}
\def\MWQSym{\textsf{WQSym}}
\def\PSet{\textsf{Set}(\PP)}
\def\MSet{\textsf{Set}(\MM)}

\def\tA{\cA}
\def\tGamma{\Gamma^{(\beta)}}
\def\tL{\cL}
\def\tE{\cE}
\def\tOmega{\Omega^{(\beta)}}
\def\ttheta{\Theta^{(\beta)}}

\def\tR{R^{(\beta)}}
\def\tH{H^{(\beta)}}

\def\oOmega{\bar{\Omega}^{(\beta)}}
\def\oE{\bar{\cE}}
\def\oK{\bar{K}}

 

\newcommand{\mSym}{\mathfrak{m}\textsf{Sym}}
\newcommand{\mQSym}{\mathfrak{m}\textsf{QSym}}



\def\Sm{S^{\fk m}}
\def\SM{S^{\fk M}}

\def\LPSet{\textsf{LPoset}}
\def\mLPSet{\mathfrak{m}\textsf{LPoset}}

\def\XX{\mathbb{X}}

\def\zetaLP{\zeta_{<}}
\def\dzetaLP{\bar{\zeta}_{>}}
\def\uzetaLP{\zeta_{>|<}}

\def\sort{\textsf{sort}}
\def\m{\mathfrak{m}}
\def\M{\mathfrak{M}}

\newcommand{\wtrpp}[1]{\mathrm{wt}(#1)}
\def\wtmrpp{\mathrm{wt}}

\def\D{\textsf{D}}
\def\SD{\textsf{SD}}

\def\equivm{\mathbin{\equiv_\m}}
\def\equivM{\mathbin{<_\M}}

\def\cJM{\cI_\M}

\def\PeakSet{\mathrm{Peak}}
\def\ValSet{\mathrm{Val}}
\def\tpeakR{\mathit{\Pi R}^{(\beta)}}
\def\peakR{\mathit{\Pi R}}
\def\peakL{\mathit{\Pi L}}

\def\GS{G\hspace{-0.2mm}S}
\def\GQ{G\hspace{-0.2mm}Q}
\def\GP{G\hspace{-0.2mm}P}
\def\tGQ{\overline{G\hspace{-0.2mm}Q}}
\def\tGP{\overline{G\hspace{-0.2mm}P}}

\def\bGQ{\GQ^{(\beta)}}
\def\bGP{\GP^{(\beta)}}
\def\bGS{\GS^{(\beta)}}


\def\gq{g\hspace{-0.1mm}q}
\def\gp{g\hspace{-0.1mm}p}

\def\bgq{\gq^{(\beta)}}
\def\bgp{\gp^{(\beta)}}

\def\jq{j\hspace{-0.1mm}q}
\def\jp{j\hspace{-0.1mm}p}

\def\JQ{J\hspace{-0.2mm}Q}
\def\JP{J\hspace{-0.2mm}P}


\def\ocoPeak{\bar{\Pi}\Sym}



\def\cHopf{\m\textsf{CB}_{\ZZ[\beta]}}
\def\fkD{\fk D}
\def\fkM{\fk M}
\def\sign{\mathrm{sign}}

\def\ss{/\hspace{-1mm}/}

\def\OG{\mathrm{OG}}
\def\LG{\mathrm{LG}}

\def\SPart{\textsf{SPart}}
\def\mSPart{\m\textsf{SPart}}
\def\IC{\mathrm{Inn}}
\def\cV{\mathcal{V}}

\def\GSp{\GP^{\Sp}}
\def\GO{\GP^{\textsf{O}}}

\def\MM{\mathbb{M}}
\def\PP{\ZZ_{>0}}
\def\NN{\ZZ_{\geq 0}}
\def\diag{\mathfrak{d}}


\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{C}_{\textsf{peak}}}

\def\SLambda{ \Lambda_{\text{strict}}}
\def\mLambda{\fk m \Lambda}
\def\MLambda{\fk M \Lambda}
\def\mSLambda{\fk m \SLambda}
\def\MSLambda{\fk M \SLambda}

\def\tri{{\raisebox{-0.5pt}{\tikz{\draw (0,0) -- (.18,0) -- (0.18,.18) -- (0,0);}}}}
\def\su{u^\tri}
%\def\sd{d^\tri}
\def \sd{d^\angle}

\def\wtsu{{\wt u}^\tri}
%\def\wtsd{{\wt d}^\tri}
\def\wtsd{{\wt d}^\angle}

\def\A{\cG}
\def\B{\fk{g}}

\def\sA{\cQ}
\def\sB{\fk{p}}

\def\ssA{\cP}
\def\ssB{\fk{q}}

\def\bbgq{\gq^{(-\beta)}}
\def\bbgp{\gp^{(-\beta)}}
\def\bX{X^{(\beta)}}
\def\KOG{\mathrm{KOG}}
\def\KLG{\mathrm{KLG}}

%\usepackage{fullpage}

%\usepackage[backend=biber]{biblatex}
%\addbibresource{shifted_ktheory_hopf.bib}


\newcommand{\ytab}[1]{
\ytableausetup{boxsize = .55cm,aligntableaux=center}
{\small\begin{ytableau}  #1  \end{ytableau}}
}

\newcommand{\ytabsmall}[1]{
\ytableausetup{boxsize = .3cm,aligntableaux=center}
{\small\begin{ytableau}  #1  \end{ytableau}}
}

\newcommand{\JBL}[1]{{\bf \color{red} JBL: #1 }}
