\documentclass[ALCO, Unicode,published]{cedram} \usepackage{tikz} \usetikzlibrary{calc} \usepackage{rotating} \usepackage{xspace} \usepackage{cleveref} \usepackage{multirow} \usepackage{stmaryrd} \usepackage{longtable} \usepackage{needspace} \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{ \setbox0\hbox{${\mathaccent"0362{#1}}^H$} \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$} \ifdim\ht0=\ht2 #3\else #2\fi } \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{ \begingroup \def\mathaccent##1##2{ \let\mathaccent\save@mathaccent \if#32 \let\macc@nucleus\first@char \fi \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$} \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$} \dimen@\wd\tw@ \advance\dimen@-\wd\z@ \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace \divide\@tempdima 10 \advance\dimen@-\@tempdima \ifdim\dimen@>\z@ \dimen@0pt\fi \rel@kern{0.6}\kern-\dimen@ \if#31 \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@} \advance\dimen@0.4\dimexpr\macc@kerna \let\final@kern#2 \ifdim\dimen@<\z@ \let\final@kern1\fi \if\final@kern1 \kern-\dimen@\fi \else \overline{\rel@kern{-0.6}\kern\dimen@#1} \fi } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \if#31 \macc@nested@a\relax111{#1} \else \def\gobble@till@marker##1\endmarker{} \futurelet\first@char\gobble@till@marker#1\endmarker \ifcat\noexpand\first@char A\else \def\first@char{} \fi \macc@nested@a\relax111{\first@char} \fi \endgroup } \makeatletter \newif\ifdots@top \dots@topfalse \newdimen\abax \abax=0cm \newdimen\abay \abay=0cm \newdimen\abah \abah=1cm \newdimen\abav \abav=.85cm \newdimen\abahd \abahd=1.6cm \newdimen\abavd \abavd=1.16875cm \newdimen\abavv \abavv=1.275cm \newdimen\abadts \abadts=.9cm \newdimen\abab \abab=.8cm \newdimen\aban \aban=.2cm \newdimen\abat \abat=.3cm \def\abas{.37} \def\drawbead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\advance\abax\abah} \def\drawupperhalfbead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\fill[white](\abax-.5*\abab,\abay)rectangle++(\abab,.5*\abav);\draw(\abax,\abay)--++(0,.5*\abav);\advance\abax\abah} \def\drawlowerhalfbead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\fill[white](\abax-.5*\abab,\abay+.5*\abav)rectangle++(\abab,.5*\abav);\draw(\abax,\abay+.5*\abav)--++(0,.5*\abav);\advance\abax\abah} \def\drawrighthalfbead{\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\fill[white](\abax-.5*\abab,\abay)rectangle++(.5*\abab,\abav);\draw(\abax,\abay)--++(0,.5*\abav);\draw(\abax,\abay)--++(0,\abav);\draw(\abax-.5*\aban,\abay+.5*\abav)--++(.5*\aban,0);\advance\abax\abah} \def\drawlefthalfbead{\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\fill[white](\abax,\abay)rectangle++(.5*\abab,\abav);\draw(\abax,\abay)--++(0,.5*\abav);\draw(\abax,\abay)--++(0,\abav);\draw(\abax,\abay+.5*\abav)--++(.5*\aban,0);\advance\abax\abah} \def\drawstarbead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\draw[white](\abax,\abay+.5*\abav)node{\small$\bigstar$};\advance\abax\abah} \def\drawwhitebead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=white](\abax,\abay+.5*\abav)circle(.5*\abab);\advance\abax\abah} \def\drawgreybead{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=gray](\abax,\abay+.5*\abav)circle(.5*\abab);\advance\abax\abah} \def\drawnobead{\draw(\abax,\abay)--++(0,\abav);\draw(\abax-.5*\aban,\abay+.5*\abav)--++(\aban,0);\advance\abax\abah} \def\drawx{\draw(\abax,\abay)--++(0,.3*\abav)++(0,.4*\abav)--++(0,.3*\abav);\draw(\abax-.8*\aban,\abay+.5*\abav-.8*\aban)--++(1.6*\aban,1.6*\aban);\draw(\abax-.8*\aban,\abay+.5*\abav+.8*\aban)--++(1.6*\aban,-1.6*\aban);\advance\abax\abah} \def\drawnobeadormark{\draw(\abax,\abay)--++(0,\abav);\advance\abax\abah} \def\drawvdots{\ifdots@top \draw(\abax,\abay)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8); \else \draw(\abax,\abay)++(0,3*\abav/16)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8); \fi \advance\abax\abah} \def\drawvvdots{ \draw(\abax,\abay)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8)++(0,\abav/8)--++(0,\abav/8); \advance\abax\abah} \def\drawhdots{\draw[dotted,thick](\abax+.5*\abahd-.5*\abah-.5*\abadts,\abay+.5*\abav)--++(\abadts,0);\advance\abax\abahd} \def\drawtopleft{\draw(\abax,\abay-.1cm)--++(0,\abat+.1cm)--++(.5*\abah+.1cm,0);\advance\abax\abah} \def\drawtopright{\draw(\abax,\abay-.1cm)--++(0,\abat+.1cm)--++(-.5*\abah-.1cm,0);\advance\abax\abah} \def\drawtopmiddle{\draw(\abax,\abay-.1cm)--++(0,\abat+.1cm)++(-.5*\abah-.1cm,0)--++(\abah+.2cm,0);\advance\abax\abah} \def\drawtopdots{\draw[dotted,thick](\abax+.5*\abahd-.5*\abah-.5*\abadts,\abay+\abat)--++(\abadts,0);\advance\abax\abahd} \def\drawnobeadtwo{\draw(\abax,\abay)--++(0,\abav);\draw(\abax-.5*\aban,\abay+.5*\abav)--++(\aban,0);\draw(\abax,\abay+.5*\abav)node[fill=white,inner sep=.5pt]{\footnotesize2};\advance\abax\abah} \def\drawbeadtwo{\draw(\abax,\abay)--++(0,\abav);\shade[shading=ball,ball color=black](\abax,\abay+.5*\abav)circle(.5*\abab);\draw(\abax,\abay+.5*\abav)node[white,inner sep=.5pt]{\footnotesize2};\advance\abax\abah} \def\@bacussingle#1{ \if b#1\drawbead\fi \if n#1\drawnobead\fi \if *#1\drawstarbead\fi \if ,#1\abax=0cm\advance\abay-\abav\fi \if ;#1\abax=0cm\advance\abay-\abavd\fi \if M#1\drawrighthalfbead\fi \if L#1\drawlefthalfbead\fi \if h#1\drawhdots\fi \if v#1\drawvdots\fi \if w#1\drawvvdots\fi \if l#1\drawtopleft\fi \if m#1\drawtopmiddle\fi \if r#1\drawtopright\fi \if t#1\drawtopdots\fi \if x#1\drawx\fi \if o#1\drawwhitebead\fi \if g#1\drawgreybead\fi \if 2#1\drawbeadtwo\fi \if 3#1\drawnobeadtwo\fi \if s#1\drawnobeadormark\fi \if_#1\advance\abax\abah\fi \if-#1\advance\abax-\abah\fi } \def\@bacustwo#1#2#3.{ \if e#1 \draw(\abax,\abay+.5*\abav)node{\small$\vphantom1\smash{#2}$};\advance\abax\abah \if\space #3\else\@bacus#3.\fi \else \@bacussingle#1 \@bacus#2#3. \fi } \def\@bacus#1#2.{ \if\space #2 \@bacussingle#1 \else\@bacustwo#1#2 .\fi } \def\sabacus(#1,#2){ \abax=0cm\abay=0cm\begin{tikzpicture}[scale=\abas*#1,,baseline={([yshift=-.8ex]current bounding box.center)}]\@bacus#2 .\end{tikzpicture}} \def\abacus(#1){\sabacus(1.2,#1)} \makeatletter \newdimen\g@b@xdim \newdimen\g@tempdim \newdimen\g@boxdim \g@boxdim=13pt \newdimen\g@boxdimx \g@boxdimx=13pt \newdimen\g@boxdimy \g@boxdimy=13pt \newdimen\g@xcoord \newdimen\g@ycoord \newdimen\g@xpos \def\Yboxdim#1{\g@boxdimx=#1\g@boxdimy=#1} \def\Yboxdimx#1{\g@boxdimx=#1} \def\Yboxdimy#1{\g@boxdimy=#1} \newdimen\g@linethick \g@linethick=.3pt \def\Ylinethick#1{\g@linethick=#1} \def\g@linestyle{} \def\Ylinestyle#1{\def\g@linestyle{#1}} \newif\ifg@vcenter \g@vcenterfalse \def\Yvcentermath#1{\ifnum #1=0 \g@vcenterfalse\else\g@vcentertrue\fi} \newif\ifg@stdtext \g@stdtextfalse \newif\ifg@french \g@frenchfalse \newif\ifg@russian \g@russianfalse \newif\ifg@addables \g@addablesfalse \newif\ifg@internals \g@internalstrue \newif\ifg@removables \g@removablestrue \newif\ifg@tabloid \g@tabloidfalse \def\Ystdtext#1{\ifnum #1=0 \g@stdtextfalse\else\g@stdtexttrue\fi} \def\Yaddables#1{\ifnum #1=0 \g@addablesfalse\else\g@addablestrue\fi} \def\Yremovables#1{\ifnum #1=0 \g@removablesfalse\g@internalsfalse\else\g@removablestrue\fi} \def\Yinternals#1{\ifnum #1=0 \g@internalsfalse\else\g@removablestrue\g@internalstrue\fi} \def\g@filcol{white} \def\Yfillcolor#1{\def\g@filcol{#1}} \def\Yfillcolour#1{\def\g@filcol{#1}} \def\g@lincol{black} \def\Ylinecolor#1{\def\g@lincol{#1}} \def\Ylinecolour#1{\def\g@lincol{#1}} \def\g@nodcol{black} \def\Ynodecolor#1{\def\g@nodcol{#1}} \def\Ynodecolour#1{\def\g@nodcol{#1}} \def\g@filopa{1} \def\Yfillopacity#1{\def\g@filopa{#1}} \def\g@scalefactor{1} \def\Yscalefactor#1{\def\g@scalefactor{#1}} \DeclareOption{vcentermath}{\g@vcentertrue} \DeclareOption{stdtext}{\g@stdtexttrue} \DeclareOption{french}{\g@frenchtrue} \DeclareOption{French}{\g@frenchtrue} \DeclareOption{russian}{\g@frenchtrue\g@russiantrue} \DeclareOption{Russian}{\g@frenchtrue\g@russiantrue} \DeclareOption*{\PackageWarning{genyoungtabtikz}{ Unknown option `\CurrentOption' (Known:\MessageBreak `vcentermath', `stdtext', `french', `French', `russian', `Russian'.)}} \ProcessOptions\relax \def\YFrench{\g@frenchtrue\g@russianfalse} \def\Yfrench{\g@frenchtrue\g@russianfalse} \def\YEnglish{\g@frenchfalse\g@russianfalse} \def\Yenglish{\g@frenchfalse\g@russianfalse} \def\YRussian{\g@frenchtrue\g@russiantrue} \def\Yrussian{\g@frenchtrue\g@russiantrue} \newdimen\g@adder \newdimen\g@line \newdimen\g@xshifting \newdimen\g@yshifting \newcount\g@counter \newcount\g@counting \newcounter{yrescount} \newcounter{ystartrow} \newdimen\g@shift \newdimen\g@dep \newif\ifg@islast \newif\ifg@cont \newcount\g@corner \g@corner0 \def\Ycorner#1{\g@corner=#1} \newcounter{g@c@rner}\setcounter{g@c@rner}0 \newcount\g@baselineshifter \newcounter{blockheight} \def\g@lastargtest#1,#2 {\if\space #2 \g@islasttrue\else\g@islastfalse\fi} \def\g@singletest#1#2. {\if\space #2 \g@islasttrue\else\g@islastfalse\fi} \def\g@countrows#1,#2 {\if\space #2 \g@counter=0\else\g@countrows#2 \advance\g@counter1\fi} \def\g@strip(#1^ ){#1} \def\g@rowsinblk(#1^#2){\if\space#2\g@counting=1\else\g@counting=\g@strip(#2)\fi} \def\g@countrowsblock#1,#2 {\if\space #2 \g@counter=-1\else\g@countrowsblock#2 \fi\g@rowsinblk(#1^ )\advance\g@counter\g@counting} \def\g@setbaselineshifter{\ifg@vcenter\ifg@french\g@baselineshifter=-1\else\g@baselineshifter=1\fi\else\g@baselineshifter=0\fi} \def\g@setblk(#1^ ){\setcounter{blockheight}{#1}} \def\g@emptyblock(#1^#2){\if\space#2\setcounter{blockheight}1\else\g@setblk(#2)\fi\setlength{\g@adder}{\value{blockheight}\g@boxdimy} \ifg@french\draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord)rectangle++(#1*\g@boxdimx,\value{blockheight}*\g@boxdimy); \draw[\g@lincol,line width=\g@linethick,\g@linestyle,xstep=\g@boxdimx,ystep=\g@boxdimy,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord)grid++(#1*\g@boxdimx,\value{blockheight}*\g@boxdimy); \advance\g@ycoord\g@adder \else\draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa,fill opacity=\g@filopa,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord+\g@boxdimy-\value{blockheight}*\g@boxdimy)rectangle++(#1*\g@boxdimx,\value{blockheight}*\g@boxdimy); \draw[\g@lincol,line width=\g@linethick,\g@linestyle,xstep=\g@boxdimx,ystep=\g@boxdimy,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord+\g@boxdimy-\value{blockheight}*\g@boxdimy)grid++(#1*\g@boxdimx,\value{blockheight}*\g@boxdimy); \advance\g@ycoord -\g@adder\fi} \def\g@yng(#1,#2){\g@emptyblock(#1^ )\g@lastargtest#2, \ifg@islast\g@emptyblock(#2^ )\else\g@yng(#2)\fi} \def\tyng(#1,#2,#3){\g@xpos=0cm\g@ycoord=0cm\g@xshifting=#1\g@yshifting=#2\g@lastargtest#3, \ifg@islast\g@emptyblock(#3^ )\else\g@yng(#3)\fi} \def\yngxy(#1,#2,#3){\settoheight\g@shift{1}\settodepth\g@dep{1}\g@countrowsblock#3, \g@setbaselineshifter \tikz[xscale=#1,yscale=#2,baseline=.5*#2*\g@boxdimy-.5*\g@shift-.5*#2*\g@counter*\g@baselineshifter*\g@boxdimy, rotate=\ifg@russian45\else0\fi,every node/.style={scale=\g@scalefactor}]{\tyng(0cm,0cm,#3)}} \def\yngs(#1,#2){\yngxy(#1,#1,#2)} \def\yngx(#1,#2){\yngxy(#1,1,#2)} \def\yngy(#1,#2){\yngxy(1,#1,#2)} \def\yng(#1){\yngxy(1,1,#1)} \def\g@reducec@rner(#1){\ifnum\value{g@c@rner}=#1\addtocounter{g@c@rner}{-#1}\g@reducec@rner(#1)\else\ifnum\value{g@c@rner}>#1\addtocounter{g@c@rner}{-#1}\g@reducec@rner(#1)\else\ifnum\value{g@c@rner}<0\addtocounter{g@c@rner}{#1}\g@reducec@rner(#1)\fi\fi\fi} \def\g@setc@rner(#1){\setcounter{g@c@rner}\g@corner\ifnum#1=0\else\g@reducec@rner(#1)\fi} \def\g@neblockresboxes(#1,#2,#3^#4){ \if\space#4\setcounter{blockheight}1\else\g@setblk(#4)\fi \setlength{\g@adder}{\value{blockheight}\g@boxdimy} \ifg@french \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord)rectangle++(#3*\g@boxdimx,\value{blockheight}*\g@boxdimy); \draw[\g@lincol,line width=\g@linethick,\g@linestyle,xstep=\g@boxdimx,ystep=\g@boxdimy,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord)grid++(#3*\g@boxdimx,\value{blockheight}*\g@boxdimy); \else \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord+\g@boxdimy-\value{blockheight}*\g@boxdimy)rectangle++(#3*\g@boxdimx,\value{blockheight}*\g@boxdimy); \draw[\g@lincol,line width=\g@linethick,\g@linestyle,xstep=\g@boxdimx,ystep=\g@boxdimy,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos,\g@ycoord+\g@boxdimy-\value{blockheight}*\g@boxdimy)grid++(#3*\g@boxdimx,\value{blockheight}*\g@boxdimy); \fi \setcounter{yrescount}{#2}\g@counter=#3 \ifg@addables \foreach\x in{1,...,\g@counter} {\addtocounter{yrescount}1\ifnum\value{yrescount}=#1\setcounter{yrescount}0\fi} \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+\g@counter*\g@boxdimx+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord-\g@boxdimy+\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \foreach\x in{1,...,\g@counter} {\addtocounter{yrescount}{-1}\ifnum\value{yrescount}=-1\addtocounter{yrescount}#1\fi} \fi \advance\g@counter-1 \ifg@internals \foreach\y in {1,...,\value{blockheight}} { \ifg@french \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord-\g@boxdimy+\y*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \else \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord+\g@boxdimy-\y*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \fi \ifnum\g@counter>0 \foreach\x in{1,...,\g@counter} { \addtocounter{yrescount}1\ifnum\value{yrescount}=#1\setcounter{yrescount}0\fi \ifg@french \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+\x*\g@boxdimx+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord-\g@boxdimy+\y*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \else \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+\x*\g@boxdimx+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord+\g@boxdimy-\y*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \fi } \fi \foreach\x in{0,...,\g@counter} {\addtocounter{yrescount}{-1}\ifnum\value{yrescount}=-1\addtocounter{yrescount}#1\fi} } \else \ifg@removables \ifnum\value{blockheight}>1 \foreach\y in {2,...,\value{blockheight}} {\addtocounter{yrescount}{-1}\ifnum\value{yrescount}=-1\addtocounter{yrescount}#1\fi} \else \addtocounter{yrescount}0 \fi \ifnum\g@counter>0 \foreach\x in{1,...,\g@counter} {\addtocounter{yrescount}1\ifnum\value{yrescount}=#1\setcounter{yrescount}0\fi} \fi \ifg@french \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+\g@counter*\g@boxdimx+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord-\g@boxdimy+\value{blockheight}*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \else \draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+\g@counter*\g@boxdimx+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord+\g@boxdimy-\value{blockheight}*\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$}; \fi \foreach\x in{0,...,\g@counter} {\addtocounter{yrescount}{-1}\ifnum\value{yrescount}=-1\addtocounter{yrescount}#1\fi} \else \foreach\y in {1,...,\value{blockheight}} {\addtocounter{yrescount}{-1}\ifnum\value{yrescount}=-1\addtocounter{yrescount}#1\fi} \fi \fi \setcounter{ystartrow}{\value{yrescount}} \ifg@french\advance\g@ycoord\g@adder\else\advance\g@ycoord -\g@adder\fi } \def\g@ungres(#1,#2,#3,#4){\setcounter{ystartrow}{#2}\g@neblockresboxes(#1,#2,#3^ ) \g@lastargtest#4, \ifg@islast\g@neblockresboxes(#1,\value{ystartrow},#4^ )\else\g@ungres(#1,\value{ystartrow},#4)\fi} \def\tyngres(#1,#2,#3,#4){\g@setc@rner(#3)\g@xpos=0cm\g@ycoord=0cm\g@xshifting=#1\g@yshifting=#2\g@lastargtest#4, \ifg@islast\g@neblockresboxes(#3,\value{g@c@rner},#4^ )\else\g@ungres(#3,\value{g@c@rner},#4)\fi \ifg@addables\draw[\g@nodcol,xshift=\g@xshifting,yshift=\g@yshifting](\g@xpos+.5*\g@boxdimx,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord-\g@boxdimy+\g@boxdimy)node{$\vphantom1\smash{\theyrescount}$};\fi } \def\yngresxy(#1,#2,#3,#4){\settoheight\g@shift{1}\settodepth\g@dep{1}\g@countrowsblock#4, \g@setbaselineshifter \tikz[xscale=#1,yscale=#2,baseline=.5*#2*\g@boxdimy-.5*\g@shift-.5*#2*\g@counter*\g@baselineshifter*\g@boxdimy, rotate=\ifg@russian45\else0\fi,every node/.style={scale=\g@scalefactor}]{\tyngres(0cm,0cm,#3,#4)}} \def\yngress(#1,#2,#3){\yngresxy(#1,#1,#2,#3)} \def\yngresx(#1,#2,#3){\yngresxy(#1,1,#2,#3)} \def\yngresy(#1,#2,#3){\yngresxy(1,#1,#2,#3)} \def\yngres(#1,#2){\yngresxy(1,1,#1,#2)} \def\g@youngbox#1.{ \ifg@tabloid \draw[\g@lincol,fill=\g@filcol,line width=\g@linethick,\g@linestyle,fill opacity=\g@filopa](\g@xcoord,\g@ycoord)--++(\g@boxdimx,0)++(0,\g@boxdimy)--++(-\g@boxdimx,0); \else \draw[\g@lincol,fill=\g@filcol,line width=\g@linethick,\g@linestyle,fill opacity=\g@filopa](\g@xcoord,\g@ycoord)rectangle+(\g@boxdimx,\g@boxdimy); \fi \ifg@stdtext \node[\g@nodcol] at(0.5*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord){\vphantom1\smash{#1}}; \else \node[\g@nodcol] at(0.5*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*\g@boxdimy+\g@ycoord){$\vphantom1\smash{#1}$}; \fi \advance\g@xcoord\g@boxdimx } \def\g@newline{ \g@xcoord=\g@xpos \ifg@french \advance\g@ycoord\g@line \else \advance\g@ycoord -\g@line \fi \setlength{\g@line}{\g@boxdimy} } \def\g@youngboxingbrack#1>#2.{ \g@youngbox#1. \if \space #2\else\g@youngboxing#2.\fi } \def\g@youngboxingcmdbrack#1>#2.{ #1 \if \space #2\else\g@youngboxing#2.\fi } \def\g@youngboxingcmd#1#2.{ \if <#1 \g@youngboxingcmdbrack#2. \else #1 \if \space #2\else \g@youngboxing#2.\fi \fi } \def\g@youngboxing#1#2.{ \if ,#1 \g@newline \if \space #2\else\g@youngboxing#2.\fi \else \if !#1 \g@youngboxingcmd#2. \else \if <#1 \g@youngboxingbrack#2. \else \g@youngbox#1. \if \space #2\else\g@youngboxing#2.\fi \fi \fi \fi } \def\tyoung(#1,#2,#3){ \g@xpos=#1 \g@xcoord=#1 \g@ycoord=#2 \setlength{\g@line}{\g@boxdimy} \g@youngboxing#3 . } \def\youngxy(#1,#2,#3){\settoheight\g@shift{1}\settodepth\g@dep{1}\g@countrows#3, \g@setbaselineshifter \tikz[baseline=.5*#2*\g@boxdimy-.5*\g@shift-.5*#2*\g@counter*\g@baselineshifter*\g@boxdimy,xscale=#1,yscale=#2, rotate=\ifg@russian45\else0\fi,every node/.style={scale=\g@scalefactor}]{\tyoung(0cm,0cm,#3)} } \def\youngs(#1,#2){\youngxy(#1,#1,#2)} \def\youngx(#1,#2){\youngxy(#1,1,#2)} \def\youngy(#1,#2){\youngxy(1,#1,#2)} \def\young(#1){\youngxy(1,1,#1)} \def\tyoungtabloid(#1,#2,#3){\g@tabloidtrue\tyoung(#1,#2,#3)\g@tabloidfalse} \def\youngtabloidxy(#1,#2,#3){\g@tabloidtrue\youngxy(#1,#2,#3)\g@tabloidfalse} \def\youngtabloids(#1,#2){\youngtabloidxy(#1,#1,#2)} \def\youngtabloidx(#1,#2){\youngtabloidxy(#1,1,#2)} \def\youngtabloidy(#1,#2){\youngtabloidxy(1,#1,#2)} \def\youngtabloid(#1){\youngtabloidxy(1,1,#1)} \def\g@emptybigbox#1#2{ \ifg@french \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa](\g@xcoord,\g@ycoord)rectangle++(#1*\g@boxdimx,#2*\g@boxdimy); \else \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa](\g@xcoord,\g@ycoord+\g@boxdimy-#2*\g@boxdimy)rectangle++(#1*\g@boxdimx,#2*\g@boxdimy); \fi \setlength{\g@adder}{#1\g@boxdimx}\advance\g@xcoord\g@adder \setlength{\g@line}{#2\g@boxdimy} } \def\hdts{abracadabra} \def\vdts{arbacadarba} \def\ddts{abracadarba} \def\edts{arbacadabra} \def\g@abcbigbox#1#2#3.{ \ifg@french \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa](\g@xcoord,\g@ycoord)rectangle++(#1*\g@boxdimx,#2*\g@boxdimy); \ifx#3\hdts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+#2*.5*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,0); \else \ifx#3\vdts \draw[thick,dotted](\g@xcoord+.5*#1*\g@boxdimx,\g@ycoord+.25*\g@boxdimy)--++(0,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\ddts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+.25*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\edts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord-.25*\g@boxdimy+#2*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,-#2*\g@boxdimy+.5*\g@boxdimy); \else \ifg@stdtext \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*#2*\g@boxdimy+\g@ycoord){\vphantom1\smash{#3}}; \else \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*#2*\g@boxdimy+\g@ycoord){$\vphantom1\smash{#3}$}; \fi \fi \fi \fi \fi \else \draw[\g@lincol,line width=\g@linethick,\g@linestyle,fill=\g@filcol,fill opacity=\g@filopa](\g@xcoord,\g@ycoord+\g@boxdimy-#2*\g@boxdimy)rectangle++(#1*\g@boxdimx,#2*\g@boxdimy); \ifx#3\hdts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+\g@boxdimy-#2*.5*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,0); \else \ifx#3\vdts \draw[thick,dotted](\g@xcoord+.5*#1*\g@boxdimx,\g@ycoord+1.25*\g@boxdimy-#2*\g@boxdimy)--++(0,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\ddts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+.75*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,-#2*\g@boxdimy+.5*\g@boxdimy); \else \ifx#3\edts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+1.25*\g@boxdimy-#2*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifg@stdtext \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+\g@boxdimy-0.5*#2*\g@boxdimy+\g@ycoord){\vphantom1\smash{#3}}; \else \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+\g@boxdimy-0.5*#2*\g@boxdimy+\g@ycoord){$\vphantom1\smash{#3}$}; \fi \fi \fi \fi \fi \fi \setlength{\g@adder}{#1\g@boxdimx}\advance\g@xcoord\g@adder \setlength{\g@line}{#2\g@boxdimy} } \def\g@emptybignobox#1#2{ \setlength{\g@adder}{#1\g@boxdimx}\advance\g@xcoord\g@adder \setlength{\g@line}{#2\g@boxdimy} } \def\g@abcbignobox#1#2#3.{ \ifg@french \ifx#3\hdts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+#2*.5*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,0); \else \ifx#3\vdts \draw[thick,dotted](\g@xcoord+.5*#1*\g@boxdimx,\g@ycoord+.25*\g@boxdimy)--++(0,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\ddts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+.25*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\edts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord-.25*\g@boxdimy+#2*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,-#2*\g@boxdimy+.5*\g@boxdimy); \else \ifg@stdtext \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*#2*\g@boxdimy+\g@ycoord){\vphantom1\smash{#3}}; \else \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+0.5*#2*\g@boxdimy+\g@ycoord){$\vphantom1\smash{#3}$}; \fi \fi \fi \fi \fi \else \ifx#3\hdts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+\g@boxdimy-#2*.5*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,0); \else \ifx#3\vdts \draw[thick,dotted](\g@xcoord+.5*#1*\g@boxdimx,\g@ycoord+1.25*\g@boxdimy-#2*\g@boxdimy)--++(0,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifx#3\ddts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+.75*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,-#2*\g@boxdimy+.5*\g@boxdimy); \else \ifx#3\edts \draw[thick,dotted](\g@xcoord+.25*\g@boxdimx,\g@ycoord+1.25*\g@boxdimy-#2*\g@boxdimy)--++(#1*\g@boxdimx-.5*\g@boxdimx,#2*\g@boxdimy-.5*\g@boxdimy); \else \ifg@stdtext \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+\g@boxdimy-0.5*#2*\g@boxdimy+\g@ycoord){\vphantom1\smash{#3}}; \else \node[\g@nodcol] at(0.5*#1*\g@boxdimx+\g@xcoord,-.5*\g@dep+\g@boxdimy-0.5*#2*\g@boxdimy+\g@ycoord){$\vphantom1\smash{#3}$}; \fi \fi \fi \fi \fi \fi \setlength{\g@adder}{#1\g@boxdimx}\advance\g@xcoord\g@adder \setlength{\g@line}{#2\g@boxdimy} } \def\g@emptybox{\g@emptybigbox11} \def\g@emptynobox{\g@emptybignobox11} \def\g@emptytallbox#1{\g@emptybigbox1#1} \def\g@emptytallnobox#1{\g@emptybignobox1#1} \def\g@emptywidebox#1{\g@emptybigbox#11} \def\g@emptywidenobox#1{\g@emptybignobox#11} \def\g@abcbox#1.{\g@abcbigbox11#1.} \def\g@abcnobox#1.{\g@abcbignobox11#1.} \def\g@abcwidebox#1#2.{\g@abcbigbox#11#2.} \def\g@abcwidenobox#1#2.{\g@abcbignobox#11#2.} \def\g@abctallbox#1#2.{\g@abcbigbox1#1#2.} \def\g@abctallnobox#1#2.{\g@abcbignobox1#1#2.} \def\g@boxingcmdbrack#1>#2.{ #1\g@boxing#2. } \def\g@boxingbignoboxbrack#1#2#3>#4.{ \g@abcbignobox#1#2.\g@boxing#3. } \def\g@boxingbigboxbrack#1#2#3>#4.{ \g@abcbigbox#1#2.\g@boxing#3. } \def\g@boxingwidenoboxbrack#1#2>#3.{ \g@abcwidenobox#1#2.\g@boxing#3. } \def\g@boxingwideboxbrack#1#2>#3.{ \g@abcwidebox#1#2.\g@boxing#3. } \def\g@boxingtallnoboxbrack#1#2>#3.{ \g@abctallnobox#1#2.\g@boxing#3. } \def\g@boxingtallboxbrack#1#2>#3.{ \g@abctallbox#1#2.\g@boxing#3. } \def\g@boxingnoboxbrack#1>#2.{ \g@abcnobox#1.\g@boxing#2. } \def\g@boxingboxbrack#1>#2.{ \g@abcbox#1.\g@boxing#2. } \def\g@boxingcmda#1#2.{ \if ,#1 \g@newline\g@boxing#2. \else \if ;#1 \g@boxingbox#2. \else \if :#1 \g@boxingnobox#2. \else \if |#1 \g@boxingtallbox#2. \else \if /#1 \g@boxingtallnobox#2. \else \if _#1 \g@boxingwidebox#2. \else \if ^#1 \g@boxingwidenobox#2. \else \if `#1 \g@boxingbigbox#2. \else \if '#1 \g@boxingbignobox#2. \else \if !#1 \g@boxingcmd#2. \else \if <#1 \g@boxingcmdbrack#2. \else #1\g@boxing#2. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingbignoboxa#1#2#3#4.{ \if ,#3 \g@emptybigbox#1#2\g@newline\g@boxing#4. \else \if ;#3 \g@emptybignobox#1#2\g@boxingbox#4. \else \if :#3 \g@emptybignobox#1#2\g@boxingnobox#4. \else \if |#3 \g@emptybignobox#1#2\g@boxingtallbox#4. \else \if /#3 \g@emptybignobox#1#2\g@boxingtallnobox#4. \else \if _#3 \g@emptybignobox#1#2\g@boxingwidebox#4. \else \if ^#3 \g@emptybignobox#1#2\g@boxingwidenobox#4. \else \if `#3 \g@emptybignobox#1#2\g@boxingbigbox#4. \else \if '#3 \g@emptybignobox#1#2\g@boxingbignobox#4. \else \if !#3 \g@emptybignobox#1#2\g@boxingcmd#4. \else \if <#3 \g@boxingbignoboxbrack#1#2#4. \else \g@abcbignobox#1#2#3.\g@boxing#4. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingbigboxa#1#2#3#4.{ \if ,#3 \g@emptybigbox#1#2\g@newline\g@boxing#4. \else \if ;#3 \g@emptybigbox#1#2\g@boxingbox#4. \else \if :#3 \g@emptybigbox#1#2\g@boxingnobox#4. \else \if |#3 \g@emptybigbox#1#2\g@boxingtallbox#4. \else \if /#3 \g@emptybigbox#1#2\g@boxingtallnobox#4. \else \if _#3 \g@emptybigbox#1#2\g@boxingwidebox#4. \else \if ^#3 \g@emptybigbox#1#2\g@boxingwidenobox#4. \else \if `#3 \g@emptybigbox#1#2\g@boxingbigbox#4. \else \if '#3 \g@emptybigbox#1#2\g@boxingbignobox#4. \else \if !#3 \g@emptybigbox#1#2\g@boxingcmd#4. \else \if <#3 \g@boxingbigboxbrack#1#2#4. \else \g@abcbigbox#1#2#3.\g@boxing#4. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingwidenoboxa#1#2#3.{ \if ,#2 \g@emptywidenobox#1\g@newline\g@boxing#3. \else \if ;#2 \g@emptywidenobox#1\g@boxingbox#3. \else \if :#2 \g@emptywidenobox#1\g@boxingnobox#3. \else \if |#2 \g@emptywidenobox#1\g@boxingtallbox#3. \else \if /#2 \g@emptywidenobox#1\g@boxingtallnobox#3. \else \if _#2 \g@emptywidenobox#1\g@boxingwidebox#3. \else \if ^#2 \g@emptywidenobox#1\g@boxingwidenobox#3. \else \if `#2 \g@emptywidenobox#1\g@boxingbigbox#3. \else \if '#2 \g@emptywidenobox#1\g@boxingbignobox#3. \else \if !#2 \g@emptywidenobox#1\g@boxingcmd#3. \else \if <#2 \g@boxingwidenoboxbrack#1#3. \else \g@abcwidenobox#1#2.\g@boxing#3. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingwideboxa#1#2#3.{ \if ,#2 \g@emptywidebox#1\g@newline\g@boxing#3. \else \if ;#2 \g@emptywidebox#1\g@boxingbox#3. \else \if :#2 \g@emptywidebox#1\g@boxingnobox#3. \else \if |#2 \g@emptywidebox#1\g@boxingtallbox#3. \else \if /#2 \g@emptywidebox#1\g@boxingtallnobox#3. \else \if _#2 \g@emptywidebox#1\g@boxingwidebox#3. \else \if ^#2 \g@emptywidebox#1\g@boxingwidenobox#3. \else \if `#2 \g@emptywidebox#1\g@boxingbigbox#3. \else \if '#2 \g@emptywidebox#1\g@boxingbignobox#3. \else \if !#2 \g@emptywidebox#1\g@boxingcmd#3. \else \if <#2 \g@boxingwideboxbrack#1#3. \else \g@abcwidebox#1#2.\g@boxing#3. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingtallnoboxa#1#2#3.{ \if ,#2 \g@emptytallnobox#1\g@newline\g@boxing#3. \else \if ;#2 \g@emptytallnobox#1\g@boxingbox#3. \else \if :#2 \g@emptytallnobox#1\g@boxingnobox#3. \else \if |#2 \g@emptytallnobox#1\g@boxingtallbox#3. \else \if /#2 \g@emptytallnobox#1\g@boxingtallnobox#3. \else \if _#2 \g@emptytallnobox#1\g@boxingwidebox#3. \else \if ^#2 \g@emptytallnobox#1\g@boxingwidenobox#3. \else \if `#2 \g@emptytallnobox#1\g@boxingbigbox#3. \else \if '#2 \g@emptytallnobox#1\g@boxingbignobox#3. \else \if !#2 \g@emptytallnobox#1\g@boxingcmd#3. \else \if <#2 \g@boxingtallnoboxbrack#1#3. \else \g@abctallnobox#1#2.\g@boxing#3. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingtallboxa#1#2#3.{ \if ,#2 \g@emptytallbox#1\g@newline\g@boxing#3. \else \if ;#2 \g@emptytallbox#1\g@boxingbox#3. \else \if :#2 \g@emptytallbox#1\g@boxingnobox#3. \else \if |#2 \g@emptytallbox#1\g@boxingtallbox#3. \else \if /#2 \g@emptytallbox#1\g@boxingtallnobox#3. \else \if _#2 \g@emptytallbox#1\g@boxingwidebox#3. \else \if ^#2 \g@emptytallbox#1\g@boxingwidenobox#3. \else \if `#2 \g@emptytallbox#1\g@boxingbigbox#3. \else \if '#2 \g@emptytallbox#1\g@boxingbignobox#3. \else \if !#2 \g@emptytallbox#1\g@boxingcmd#3. \else \if <#2 \g@boxingtallboxbrack#1#3. \else \g@abctallbox#1#2.\g@boxing#3. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingnoboxa#1#2.{ \if ,#1 \g@emptynobox\g@newline\g@boxing#2. \else \if ;#1 \g@emptynobox\g@boxingbox#2. \else \if :#1 \g@emptynobox\g@boxingnobox#2. \else \if |#1 \g@emptynobox\g@boxingtallbox#2. \else \if /#1 \g@emptynobox\g@boxingtallnobox#2. \else \if _#1 \g@emptynobox\g@boxingwidebox#2. \else \if ^#1 \g@emptynobox\g@boxingwidenobox#2. \else \if `#1 \g@emptynobox\g@boxingbigbox#2. \else \if '#1 \g@emptynobox\g@boxingbignobox#2. \else \if !#1 \g@emptynobox\g@boxingcmd#2. \else \if <#1 \g@boxingnoboxbrack#2. \else \g@abcnobox#1.\g@boxing#2. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingboxa#1#2.{ \if ,#1 \g@emptybox\g@newline\g@boxing#2. \else \if ;#1 \g@emptybox\g@boxingbox#2. \else \if :#1 \g@emptybox\g@boxingnobox#2. \else \if |#1 \g@emptybox\g@boxingtallbox#2. \else \if /#1 \g@emptybox\g@boxingtallnobox#2. \else \if _#1 \g@emptybox\g@boxingwidebox#2. \else \if ^#1 \g@emptybox\g@boxingwidenobox#2. \else \if `#1 \g@emptybox\g@boxingbigbox#2. \else \if '#1 \g@emptybox\g@boxingbignobox#2. \else \if !#1 \g@emptybox\g@boxingcmd#2. \else \if <#1 \g@boxingboxbrack#2. \else \g@abcbox#1.\g@boxing#2. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxingcmd#1.{ \if \space #1 \else \g@boxingcmda#1. \fi } \def\g@boxingbignobox#1#2#3.{ \if \space #3 \g@emptybignobox#1#2 \else \g@boxingbignoboxa#1#2#3. \fi } \def\g@boxingbigbox#1#2#3.{ \if \space #3 \g@emptybigbox#1#2 \else \g@boxingbigboxa#1#2#3. \fi } \def\g@boxingwidenobox#1#2.{ \if \space #2 \g@emptywidenobox#1 \else \g@boxingwidenoboxa#1#2. \fi } \def\g@boxingwidebox#1#2.{ \if \space #2 \g@emptywidebox#1 \else \g@boxingwideboxa#1#2. \fi } \def\g@boxingtallnobox#1#2.{ \if \space #2 \g@emptytallnobox#1 \else \g@boxingtallnoboxa#1#2. \fi } \def\g@boxingtallbox#1#2.{ \if \space #2 \g@emptytallbox#1 \else \g@boxingtallboxa#1#2. \fi } \def\g@boxingnobox#1.{ \if \space #1 \g@emptynobox \else \g@boxingnoboxa#1. \fi } \def\g@boxingbox#1.{ \if \space #1 \g@emptybox \else \g@boxingboxa#1. \fi } \def\g@boxinga#1#2.{ \if ,#1 \g@newline\g@boxing#2. \else \if ;#1 \g@boxingbox#2. \else \if :#1 \g@boxingnobox#2. \else \if |#1 \g@boxingtallbox#2. \else \if /#1 \g@boxingtallnobox#2. \else \if _#1 \g@boxingwidebox#2. \else \if ^#1 \g@boxingwidenobox#2. \else \if `#1 \g@boxingbigbox#2. \else \if '#1 \g@boxingbignobox#2. \else \if !#1 \g@boxingcmd#2. \else \g@boxingbox#1#2. \fi \fi \fi \fi \fi \fi \fi \fi \fi \fi } \def\g@boxing#1.{ \if \space #1\else\g@boxinga#1.\fi } \def\tgyoung(#1,#2,#3){\g@xpos=#1\g@xcoord=#1\g@ycoord=#2\setlength{\g@line}{\g@boxdimy} \g@boxing#3 . } \def\gyoungxy(#1,#2,#3){\settoheight\g@shift{1}\settodepth\g@dep{1}\g@countrows#3, \g@setbaselineshifter\tikz[baseline=.5*#2*\g@boxdimy-.5*\g@shift-.5*#2*\g@counter*\g@baselineshifter*\g@boxdimy,xscale=#1,yscale=#2, rotate=\ifg@russian45\else0\fi,every node/.style={scale=\g@scalefactor}] {\tgyoung(0cm,0cm,#3)}} \def\gyoungs(#1,#2){\gyoungxy(#1,#1,#2)} \def\gyoungx(#1,#2){\gyoungxy(#1,1,#2)} \def\gyoungy(#1,#2){\gyoungxy(1,#1,#2)} \def\gyoung(#1){\gyoungxy(1,1,#1)} \makeatother \datepublished{2025-10-31} \begin{document} \equalenv{cory}{coro} \equalenv{propn}{prop} \newenvironment{pf}{\begin{proof}}{\end{proof}} \Crefname{prop}{Proposition}{Propositions} \Crefname{theo}{Theorem}{Theorem} \Crefname{lemm}{Lemma}{Lemma} \Crefname{coro}{Corollary}{Corollary} \newcommand\xcite[2]{\textup{\textbf{\cite[#2]{#1}}}} \newcommand\jdom{\mathrel{{\ooalign{\hss$\smash{\dom}$\hss\cr\hidewidth\raisebox{1.0pt}{${\blacktriangleright}$}\hidewidth}}}} \newcommand\jdomby{\mathrel{{\ooalign{\hss$\smash{\domby}$\hss\cr\hidewidth\raisebox{1.0pt}{${\blacktriangleleft}$}\hidewidth}}}} \newcommand\jdoms{\blacktriangleright} \newcommand\jdomsby{\blacktriangleleft} \newcommand\njdom{\mathrel{{\ooalign{\hss$\smash{\ndom}$\hss\cr\hidewidth\raisebox{1.0pt}{${\blacktriangleright}$}\hidewidth}}}} \newcommand\njdomby{\mathrel{{\ooalign{\hss$\smash{\ndomby}$\hss\cr\hidewidth\raisebox{1.0pt}{${\blacktriangleleft}$}\hidewidth}}}} \newcommand\njdoms{\not\blacktriangleright} \newcommand\njdomsby{\not\blacktriangleleft} \renewcommand\bottomfraction{0.7} \newcommand\rem[2]{\operatorname{rem}_{#1}(#2)} \newcommand\nor[2]{\operatorname{nor}_{#1}(#2)} \newcommand\blu{\Yfillcolour{blue!25!white}} \newcommand\wht{\Yfillcolour{white}} \newcommand\albe{\al\be} \newcommand\alga{\al\ga} \newcommand\gaal{\ga\al} \newcommand\bega{\be\ga} \newcommand\hand[1]{\operatorname{hand}(#1)} \newcommand\lel{\operatorname{ll}} \newcommand{\resi}{\operatorname{res}} \newcommand{\fx}{x} \newcommand{\fy}{y} \newcommand{\rez}{\widehat{\resi}} \newcommand\sw[1]{#1^\leftrightarrow} \newcommand\wt[1]{\operatorname{wt}(#1)} \newcommand\lra\longrightarrow \newcommand\Hom{\operatorname{Hom}} \newcommand\tricky{exceptional\xspace} \newcommand\untricky{unexceptional\xspace} \newcommand\Tricky{Exceptional\xspace} \newcommand\rt[1]{\rotatebox{90}{$#1$}} \newcommand\bp[2]{\big(#1\ \big|\ #2\big)} \newcommand\tjs{\stackrel{\mathrm{JS}}\longrightarrow} \newcommand\js{Jantzen--Schaper\xspace} \newcommand\ee[1]{\mathrm{e}_{#1}} \newcommand\ff[1]{\mathrm{f}_{#1}} \newcommand\eed[2]{\mathrm{e}_{#1}^{(#2)}} \newcommand\ffd[2]{\mathrm{f}_{#1}^{(r)}} \newcommand\dm{^\diamond} \newcommand\bi{\overline i} \newcommand\bj{\overline j} \newcommand\bip{bipartition\xspace} \newcommand\bips{bipartitions\xspace} \newcommand\zez{\bbz/e\bbz} \newcommand\xr{nucleus\xspace} \newcommand\dl[3]{\operatorname{a}_{#2}^{(#3)}{}{\downarrow}^{#1}} \newcommand\dw[1]{\downarrow^{#1}} \newcommand\ddu[3]{\downarrow^{#1,#2}\uparrow_{#3}} \newcommand\spe[1]{\operatorname{S}_{#1}} \newcommand\wspe[1]{{\operatorname{S}}^\leftrightarrow_{#1}} \newcommand\hspe[1]{\hat{\operatorname{S}}_{#1}} \newcommand\hwspe[1]{\hat{\operatorname{S}}^\leftrightarrow_{#1}} \newcommand\smp[1]{\operatorname{D}_{#1}} \newcommand\dspe[1]{\operatorname{S}'(#1)} \newcommand\dsmp[1]{\operatorname{D}'(#1)} \newcommand\dn[2]{[\spe{#1}\nolinebreak:\nolinebreak\smp{#2}]} \newcounter{tablecase} \newcommand\sss{\mathfrak S_} \newcommand{\gs}{\DOTSB\geqslant} \newcommand{\ls}{\DOTSB\leqslant} \newcommand\nls{\DOTSB\nleqslant} \newcommand\ngs{\DOTSB\ngeqslant} \renewcommand\geq{\DOTSB\geqslant} \renewcommand\leq{\DOTSB\leqslant} \renewcommand\ge{\DOTSB\geqslant} \renewcommand\le{\DOTSB\leqslant} \newcommand\bbf{\mathbb{F}} \newcommand\bbn{\mathbb{N}} \newcommand\bbz{\mathbb{Z}} \newcommand\al\alpha \newcommand\be\beta \newcommand\ga\gamma \newcommand\de\delta \newcommand\ep\epsilon \newcommand\ka\kappa \newcommand\la\lambda \newcommand\lset[2]{\left\{\left.#1\ \right|\ \smash{#2}\right\}} \newcommand\rset[2]{\left\{\smash{#1}\ \left|\ #2\right.\right\}} \newcommand\card[1]{\lvert#1\rvert} \newcommand\vn\varnothing \newcommand\sm\setminus \newcommand{\dom}{\trianglerighteqslant} \newcommand{\doms}{\vartriangleright} \newcommand{\ndom}{\ntrianglerighteqslant} \newcommand{\ndoms}{\not\vartriangleright} \newcommand{\domby}{\trianglelefteqslant} \newcommand{\domsby}{\vartriangleleft} \newcommand{\ndomby}{\ntrianglelefteqslant} \newcommand{\ndomsby}{\not\vartriangleleft} \renewcommand\iff{if and only if\xspace} \newcommand\abd{abacus display\xspace} \newcommand\abds{\abd{}s\xspace} \newcommand\calc{\mathcal{C}} \newcommand\fkm{\mathfrak{m}} \newcommand\fkn{\mathfrak{n}} \newcommand\fkp{\mathfrak{p}} \newcommand\ppmod[1]{\ (\operatorname{mod}\,#1)} \newcommand{\bibi}[8]{\bibitem[#2]{#1} {#3}, `{#4}', \textit{#5} \textbf{#6} ({#7}), {#8}.} \newcommand{\bibbook}[8]{\bibitem[#2]{#1} {#3}, \textit{#4}, {#5} \textbf{#6}, {#7}, {#8}.} \newcommand{\bibshort}[5]{\bibitem[#2]{#1} {#3}, `{#4}', {#5}.} \newcommand{\bibbookshort}[5]{\bibitem[#2]{#1} {#3}, \textit{#4}, {#5}.} \def\caselabel{xx} \newcommand\nextcase{\stepcounter{tablecase}\thetablecase.} \DeclareRobustCommand\rnextcase[1]{\refstepcounter{tablecase}\thetablecase.\label{#1}} \newcommand\ol[1]{\widebar{#1}} \newcommand\hhh{\mathcal{H}_} \newcommand\thhh{\hat{\mathcal{H}}_} \title[Weight \texorpdfstring{$3$}{3} blocks of Iwahori--Hecke algebras of type \texorpdfstring{$B$}{B}]{Decomposition numbers for weight \texorpdfstring{$3$}{3} blocks\\of Iwahori--Hecke algebras of type \texorpdfstring{$B$}{B}} \keywords{Iwahori-Hecke algebra, decomposition numbers, Specht modules} \subjclass{20C08, 20C30, 05E10} \author[M. Fayers]{\firstname{Matthew} \lastname{Fayers}} \address{Queen Mary University of London} \email{m.fayers@qmul.ac.uk} \author[L. Putignano]{\firstname{Lorenzo} \lastname{Putignano}} \address{Universit\`a degli Studi di Firenze} \email{lorenzo.putignano@unifi.it} \begin{abstract} Let $B$ be a weight-$3$ block of an Iwahori--Hecke algebra of type $B$ over any field. We develop the combinatorics of $B$ to prove that the decomposition numbers for $B$ are all $0$ or~$1$. \end{abstract} \maketitle \section{Introduction} In the representation theory of the symmetric groups, a fundamental problem is to determine the \emph{decomposition numbers} in characteristic $p$: the composition multiplicities of $p$-modular simple representations in $p$-modular reductions of ordinary simple representations. A complete solution to this problem seems far out of reach, but solutions are known in various special cases. One of these results (due to the first author~\cite{mfwt3}) says that the decomposition numbers in a $p$-block of weight $3$ in characteristic $p\gs5$ are all either $0$ or $1$; this in particular allows these decomposition numbers to be computed quickly using the Jantzen--Schaper formula. The decomposition number problem naturally extends to the Iwahori--Hecke algebras of type $A$, where we seek the composition multiplicities of the simple modules in the Specht modules. Many of the known results for symmetric groups have been proved for the Iwahori--Hecke algebras as well, including the result for blocks of weight~$3$. More recently, this problem has been extended to the Iwahori--Hecke algebras $\hhh n$ of type~$B$. These were first systematically studied by Dipper, James and Murphy~\cite{djm}, who introduced appropriate definitions of Specht modules labelled by \bips of~$n$. This naturally leads to a combinatorial approach to the corresponding decomposition number problem, underpinned by the combinatorics of \bips. The first author's combinatorial definition of the weight of a \bip, and hence the weight of a block~\cite{mfweight}, allows us to approach this problem systematically by addressing blocks of a fixed weight. As in type $A$, the blocks of weight $0$ are precisely the simple blocks, and blocks of weight $1$ admit a very simple description (analogous to symmetric group blocks with cyclic defect groups). The first author~\cite{mftypebwt2} gave a combinatorial formula for decomposition number for blocks of weight $2$, analogous to Richards's formula~\cite{rich} for type~$A$. In this paper we address blocks of weight $3$. Our main result is as follows. \begin{thm}\label{main} Suppose $\bbf$ is a field, $q\in\bbf$ is non-zero, and $k_1,k_2\in\bbz$. Let $\hhh n$ denote the Iwahori--Hecke algebra of type $B$ over $\bbf$ with parameters $q,q^{k_1},q^{k_2}$, and let $B$ be a block of $\hhh n$ of weight $3$. Then the decomposition numbers for $B$ are all either $0$ or~$1$. \end{thm} We note that (unlike in type $A$) \Cref{main} applies even when $\bbf$ has characteristic~$2$ or $3$. In particular, following the proof of \Cref{main} we deduce the following interesting result. \begin{cory} The decomposition numbers for a weight 3 block of $\hhh n$ are independent of the characteristic of the field $\mathbb{F}$. \end{cory} We conjecture that this result extends to all weight $3$ blocks of \emph{Ariki--Koike algebras}; these are higher-level Hecke algebras generalising the Iwahori--Hecke algebras of types~$A$ and $B$. A particular case of this conjecture (the case of `tree blocks') has already been proved by Lyle and Ruff in~\cite[Corollary A.5]{lyru}. (Note: this result does not appear in the published version of the paper, but in the arXiv version that we cite here.) The proof of \Cref{main} is by induction on $n$, using the Brundan--Kleshchev branching rules which govern induction and restriction between $\hhh n$ and $\hhh m$, for~\mbox{$m\la^{(a)}_{r+1}$), while a node not in $\la$ is an addable node of $\la$ if it can be added to $[\la]$ to give a Young diagram. We depict $[\la]$ by drawing the Young diagram of $\la^{(1)}$ above the Young diagram of~$\la^{(2)}$ (using the English convention). Accordingly, we will say that a node $(r,c,a)$ is \emph{above} or \emph{higher than} a node $(s,d,b)$ if either $a0$ on the abacus such that in component $a$ there is a bead at position $x$ but no bead at position $x-1$. Given such a $x$, the residue of the corresponding removable node is $x+e\bbz$. Similarly, addable nodes correspond to positions $x$ such that there is a bead at position $x-1$ but not at position $x$. More generally, rim hooks in component $a$ of $\la$ correspond to pairs of positions $x,y$ where $x>y$ and there is a bead at position $x$ but not at position $y$ in component $a$. The residue of the hand node of the corresponding rim hook is $x+e\bbz$, the leg length is the number of beads between positions $x$ and $y$ (not including position $x$), and removing the rim hook corresponds to moving the bead at position $x$ to position $y$. \subsection{Restricted \bips}\label{kleshbipsec} In~\cite{arma}, Ariki and Mathas introduced a set of multipartitions to provide a combinatorial labelling for simple modules of Ariki--Koike algebras. They called these \emph{Kleshchev multipartitions}, but here we prefer the term \emph{restricted multipartitions} introduced by Brundan--Kleshchev~\cite{bk}. We now recall the definition here, specialising to the case of \bips. We will use the original recursive definition of Ariki--Mathas, though we note that a more efficient definition was found by Ariki, Kreiman and Tsuchioka~\cite{akt}. The definition of restricted \bips depends on our fixed $\ka\in(\zez)^2$. Take a \bip $\la$ and a residue $i\in\zez$. Define the \emph{$i$-signature} of $\la$ to be the sequence of signs obtained by reading the Young diagram of $\la$ from top to bottom, writing $+$ for each addable $i$-node and $-$ for each removable $i$-node. The \emph{reduced $i$-signature} is the subsequence obtained by repeatedly deleting adjacent pairs $-+$. The removable nodes corresponding to the $-$ signs are called the \emph{normal} $i$-nodes of $\la$, and the highest of these (if there are any) is the \emph{good} $i$-node. The addable nodes corresponding to the $+$ signs in the reduced $i$-signature are the \emph{conormal} $i$-nodes of $\la$, and the lowest of these (if there are any) is the \emph{cogood} $i$-node. For any \bip $\mu$ and any $i\in\zez$, we write $\nor i\mu$ for the number of normal $i$-nodes of $\mu$. \begin{example} Suppose $e=3$, $\ka=(\ol0,\ol1)$ and $\la=\bp{3,2,1^2}{2^3}$. The Young diagram of $\la$ is shown below, with the residues of the addable and removable nodes indicated. \[ \gyoung(;;;2:0,;;0:1,;:2,0,:2,,;;:0,;;,;;0,:1) \] If we let $i=\ol0$, then we see that the $i$-signature of $\la$ is $+--+-$. So the reduced $i$-signature is $+--$. So there are two normal $i$-nodes $(2,2,1)$ and $(3,2,2)$, with $(2,2,1)$ being the good $i$-node. There is a unique conormal (and therefore cogood) $i$-node $(1,4,1)$. \end{example} Now given two \bips $\la$ and $\mu$, write $\mu\stackrel i\longrightarrow\la$ if $\mu$ is obtained from $\la$ by removing the good $i$-node for $i\in\zez$, or equivalently if $\la$ is obtained from $\mu$ by adding the cogood $i$-node. We write $\mu\longrightarrow\la$ to mean that $\mu\stackrel i\longrightarrow\la$ for some $i$, and let $\longleftrightarrow$ be the equivalence relation generated by $\longrightarrow$. Now define the set of restricted \bips (with respect to $\ka$) to be the \bips in the $\longleftrightarrow$-class containing $\bp\vn\vn$. The definition of restricted \bips derives from the theory of crystals for highest-weight representations of quantum groups. It follows from this theory (in particular, the uniqueness of highest-weight vectors) that the only restricted \bip with no normal nodes of any residue is $\bp\vn\vn$. So to test whether a \bip $\la$ is restricted, one can repeatedly remove arbitrarily-chosen good nodes until there are no good nodes remaining; then $\la$ is restricted \iff the resulting \bip is $\bp\vn\vn$. We also make an observation which will be helpful later on. Say that a partition~$\la$ is \emph{$e$-restricted} if $\la_r-\la_{r+1}1, \\ T_0T_1T_0T_1&=T_1T_0T_1T_0, \\ T_iT_{i+1}T_i&=T_{i+1}T_iT_{i+1}&&\text{for }1\ls in$ (and in fact most of the particular cases of blocks that we consider only arise if $q$ is an $e$th root of unity for~$e\ls n$). So with our fixed pair $\ka=(\ka_1,\ka_2)\in\zez^2$, we will assume henceforth that $Q_1=q^{k_1}$ and $Q_2=q^{k_2}$, where $(k_1,k_2)$ is any bicharge for $\ka$. The representation theory of $\hhh n$ is based on the theory of \emph{Specht modules}. For each \bip $\la$ of $n$, we write $\spe\la$ for the corresponding Specht module, as defined in Mathas's survey article~\cite{mathsurv}. The Specht modules are precisely the cell modules with respect to a particular cellular basis of $\hhh n$, and a central problem in the representation theory of $\hhh n$ is to determine their composition factors. To do this, we need a classification of the simple $\hhh n$-modules. If $\la$ is a restricted \bip, then $\spe\la$ has a unique simple quotient $\smp\la$, and the simple modules $\smp\la$ arising in this way give a complete irredundant set of simple $\hhh n$-modules. Thus the problem of determining the composition factors of the Specht modules amounts to determining the \emph{decomposition numbers} $\dn\la\mu$, where $\la,\mu$ are \bips of $n$ with $\mu$ restricted. A fundamental result on decomposition numbers is the following. \begin{propn}[\xcite{djm}{Theorem 6.5}]\label{basicdecomp} If $\la$ and $\mu$ are \bips of $n$ with $\mu$ restricted, then $\dn\mu\mu=1$, while if $\dn\la\mu>0$ then $\la\dom\mu$. \end{propn} \subsection{Blocks and weight} An important tool for studying the decomposition number problem is the classification of blocks, since $\dn\la\mu=0$ unless $\spe\la$ and $\smp\mu$ lie in the same block. It is a consequence of the general theory of cellular algebras that a Specht module always belongs to a single block, and we will abuse notation by saying that $\la$ lies in a block $B$ to mean that $\spe\la$ lies in $B$. Of course, if $\la$ is restricted this means that $\smp\la$ lies in this block too, so to classify the blocks of $\hhh n$ it is sufficient to determine when two Specht modules lie in the same block. The block classification for $\hhh n$ is a special case of the result for Ariki--Koike algebras proved by Lyle and Mathas. \begin{propn}[\xcite{lyma}{Theorem 2.11}]\label{blockclass} Suppose $\la$ and $\mu$ are \bips of $n$. Then $\spe\la$ and $\spe\mu$ lie in the same block of $\hhh n$ \iff $\la$ and $\mu$ have the same content. \end{propn} As explained in~\cite[3.2]{mfweight}, this result can also be formulated in terms of the abacus: if we choose a bicharge $(k_1,k_2)$ for $\ka$ with $k_1,k_2$ sufficiently large, and construct the \abds for $\la$ and $\mu$ with this bicharge, then $\la$ and $\mu$ have the same content \iff for each $i$ the number of beads on runner $i$ (across both components) is the same in each \abd. This leads to another useful way to determine whether two \bips belong to the same block. If $\la$ is a \bip, then for each $i\in\zez$ we define $\de_i(\la)$ to be the number of removable $i$-nodes of $\la$ minus the number of addable $i$-nodes of $\la$. Now we have the following result. \begin{propn}[\xcite{mfweight}{Proposition 3.2}]\label{samede} Suppose $\la$ and $\mu$ are \bips of $n$. Then~$\la$ and $\mu$ lie in the same block \iff $\de_i(\la)=\de_i(\mu)$ for all $i\in\zez$. \end{propn} As a consequence we can define $\de_i(B)$ for a block $B$, meaning $\de_i(\la)$ for any \bip $\la$ in $B$. If $a\in\bbz$, then we may write $\de_a(B)$ to mean $\de_{a+e\bbz}(B)$. In~\cite{mfweight}, the first author introduced the notion of the \emph{weight} $\wt\la$ of a \bip~$\la$. Since the weight of~$\la$ depends only on the content of $\la$, it is the same for all $\la$ in a given block, and therefore one may define the weight of a block $B$ to be the weight of any \bip in $B$. The weight of $B$ is a non-negative integer which measures the complexity of $B$, and a fruitful approach to the representation theory of $\hhh n$ is to consider blocks of a given small weight. We will not give the definition of weight in terms of content here, but we will use the results from~\cite[1.3.5]{mftypebwt2} showing how to calculate the weight of a \bip using the abacus. We will introduce the necessary notation in \Cref{class3sec}. We will also use the following lemma several times. \begin{lemma}[\xcite{mfweight}{Lemma 3.6}]\label{36lem} Suppose $\la$ is a \bip, and that $\mu$ is a \bip obtained by removing $u$ removable $i$-nodes from $\la$. Then \[ \wt\mu=\wt\la+u(\de_i(\la)-u). \] \end{lemma} This paper is motivated by the following result. \begin{thm}\label{w012} Suppose $B$ is a block of $\hhh n$ of weight $0$, $1$ or $2$, and $\la,\mu$ are \bips in $B$ with $\mu$ restricted. Then $\dn\la\mu\ls1$. \end{thm} \begin{pf} The result for blocks of weight $0$ follows from~\cite[Theorem 4.1]{mfweight}, and the result for weight $1$ is~\cite[Theorem 4.2]{mfweight}. For blocks of weight $2$, the result is given in~\cite[Theorems 3.18 \& 4.13]{mftypebwt2}. \end{pf} Our aim in this paper is to prove the same result for blocks of weight $3$. In fact for blocks of weight at most $2$ explicit formul\ae{} for the decomposition numbers are known; for weight $3$ we just show that the decomposition numbers are at most $1$ (which is enough to enable the decomposition numbers to be computed quickly using the \js formula). This is analogous to the situation for Hecke algebras of type $A$~\cite{mfwt3}. \subsection{Conjugation and component-switching}\label{conjswsec} The set of \bips of $n$ has very natural symmetries, given by conjugation or by simply interchanging components. These symmetries provide useful tools for studying decomposition numbers, which we describe here. We begin with conjugation, which relates to automorphisms. Let $\theta$ be the automorphism defined defined by mapping $T_0\mapsto q^{k_1+k_2}T_0^{-1}$ and $T_i\mapsto-qT_i^{-1}$ for $i\gs1$. For any $\hhh n$-module $M$, define the module $M^\theta$ by twisting the $\hhh n$-action by $\theta$. Obviously if $M$ is simple then so is $M^\theta$, and so we can define a bijection $\mu\mapsto\mu\dm$ from the set of restricted \bips to the set of regular \bips by $(\smp\mu)^\theta\cong\smp{(\mu\dm)'}$. Understanding what the automorphism $\theta$ does to Specht modules gives us more information on decomposition numbers, as follows. \begin{propn}\label{conjdual} Suppose $\la$ and $\mu$ are \bips of $n$ with $\mu$ restricted. Then $\dn\la\mu=\dn{\la'}{(\mu\dm)'}$. \end{propn} \begin{pf} Our argument is based on~\cite[Sections 4 and 5]{mathastilt}, where Mathas recalls the \emph{dual Specht module} $\dspe\la$ for each $\la$. The dual Specht modules are related to the usual Specht modules in two ways. On the one hand, one can check that $\dspe\la$ is isomorphic to $(\spe\la)^\theta$. To see this, we recall how $\spe\la$ is constructed: a certain element $m_\la\in\hhh n$ is defined, and $\spe\la$ is defined to be the right ideal generated by $m_\la$, modulo its intersection with the two-sided ideal generated by the elements $m_\nu$ for $\nu\doms\la$. The dual Specht module $\dspe\la$ is defined similarly using a different element $n_\la$. But one can check that $\theta(m_\la)$ equals $n_\la$ times an invertible element of $\hhh n$, so that the (right or two-sided) ideal generated by $\theta(m_\la)$ coincides with the (right or two-sided) ideal generated by $n_\la$. As a consequence, we obtain $\dn\la\mu=[\dspe\la:(\smp\mu)^\theta]$ for all $\la,\mu$. On the other hand, Mathas considers contragredient duality $M\mapsto M^\circledast$ induced by the antiautomorphism of $\hhh n$ that fixes each $T_i$. In~\cite[Corollary 5.7]{mathastilt} Mathas shows that $\dspe\la\cong(\spe{\la'})^\circledast$. Since contragredient duality preserves composition multiplicities, and the simple modules are self-dual (which follows from the cellularity of $\hhh n$), we find that $[\dspe\la:\smp\mu]=\dn{\la'}\mu$ for every $\la,\mu$. The result follows. \end{pf} \begin{cory}\label{basicdecomp2} Suppose $\la$ and $\mu$ are \bips of $n$, with $\mu$ restricted. Then $\dn{\mu\dm}\mu=1$, while if $\dn\la\mu>0$ then $\mu\dm\dom\la$. \end{cory} \begin{pf} This follows from \Cref{basicdecomp} and \Cref{conjdual} using the fact that conjugation reverses the dominance order on \bips. \end{pf} Later in the paper we will need to be able to calculate the bijection ${}\dm$ combinatorially. Recall the relations $\stackrel i\longrightarrow$ and $\stackrel i\Longrightarrow$ from \Cref{kleshbipsec}. \begin{propn}\label{dmbij} Suppose $\la$ and $\mu$ are restricted \bips. Then $\mu\stackrel i\longrightarrow\la$ \iff $\mu\dm\stackrel i\Longrightarrow\la\dm$. \end{propn} \begin{pf} The Brundan--Kleshchev modular branching rules show that the relation $\stackrel i\longrightarrow$ can be described algebraically via \[ \mu\stackrel i\lra\la\qquad\text{\iff}\qquad\Hom_{\hhh{n-1}}(\smp\mu,\ee i\smp\la)\neq0, \] where $\ee i$ is the $i$th restriction functor (see \Cref{branchsec} below). The definition of $\ee i$ and the automorphism $\theta$ shows that $(\ee iM)^\theta=\ee{i'}(M^\theta)$ for any module $M$, where as above $i'=\ka_1+\ka_2-i$. Hence \[ \Hom_{\hhh{n-1}}(\smp\mu,\ee i\smp\la)\neq0\qquad\text{\iff}\qquad\Hom_{\hhh{n-1}}(\smp{(\mu\dm)'},\ee{i'}\smp{(\la\dm)'})\neq0, \] and therefore \[ \mu\stackrel i\lra\la\qquad\text{\iff}\qquad(\mu\dm)'\stackrel{i'}\lra(\la\dm)', \] and from \Cref{kleshbipsec} this is equivalent to $\mu\dm\stackrel i\Longrightarrow\la\dm$. \end{pf} Now we come to component-switching, where we don't need to say as much. If $\la$ is a \bip, let $\sw\la$ denote the \bip $\bp{\la^{(2)}}{\la^{(1)}}$ obtained by switching the two components of $\la$. Note that although the isomorphism type of $\hhh n$ only requires the unordered pair $(\ka_1,\ka_2)$, the definition of the Specht module (and the associated combinatorics of residues, blocks, restricted \bips etc) depends on the ordered pair $(\ka_1,\ka_2)$. We let $\wspe\la$ denote the Specht module constructed using the \bip $\sw\la$ and the ordered pair $(\ka_2,\ka_1)$. \begin{lemma}\label{wspelen} Suppose $\la$ is a \bip of $n$. Then $\spe\la$ and $\wspe\la$ have the same composition factors (with multiplicity). \end{lemma} \begin{pf} This uses a specialisation argument. Let $\hat q,\hat Q_1,\hat Q_2$ be algebraically independent indeterminates over $\bbf$, and consider the Iwahori--Hecke algebra $\thhh n$ over $\bbf(\hat q,\hat Q_1,\hat Q_2)$ with parameters $\hat q,\hat Q_1,\hat Q_2$. Then $\thhh n$ is semisimple by~\cite[Main Theorem]{arikiss}, with the Specht modules being the simple modules. If we define the Specht module $\hspe\la$ for $\thhh n$ using the \bip $\la$ and the parameters $\hat q,\hat Q_1,\hat Q_2$, and we define the Specht module $\hwspe\la$ using the \bip $\sw\la$ and the parameters $\hat q,\hat Q_2,\hat Q_1$, then $\hspe\la$ and $\hwspe\la$ have the same character (a recursive formula for characters of semisimple Iwahori--Hecke algebras of type $B$ is given by Pfeiffer~\cite{pf}), and are therefore isomorphic. The $\hhh n$-modules $\spe\la$ and $\wspe\la$ are obtained by specialising $\hat q$ to $q$, $\hat Q_1$ to $q^{k_1}$ and $\hat Q_2$ to $q^{k_2}$, where $\ka_1=\ol{k_1}$, $\ka_2=\ol{k_2}$. It is a standard result on specialisation that specialisations of isomorphic modules have the same composition factors. \end{pf} We remark that since the simple modules are labelled by a different set of partitions in each case, $\spe\la$ and $\wspe\la$ have the same composition factors but the labels of these may not agree. \Cref{wspelen} allows us to reduce the work we do in considering different types of blocks: given a block $B$ of $\hhh n$, we can study the same block by replacing $\ka$ with $(\ka_2,\ka_1)$ and replacing each \bip $\la$ in $B$ with $\sw\la$. The truth of \Cref{main} for~$B$ is then independent of which way we view $B$. \subsection{Branching rules}\label{branchsec} Now we summarise some essential background on the Brundan--Kleshchev branching rules; the most comprehensive reference for these is~\cite{bk}. Recall that the content of a \bip is the multiset of the residues of its nodes, and that two \bips lie in the same block \iff they have the same content. So we may define the content $\calc_B$ of a block $B$ to be the content of any \bip in $B$. Now suppose $M$ is an $\hhh n$-module lying in a block $B$, and $i\in\zez$. If there is a block $A$ of $\hhh{n-1}$ such that $\calc_A$ is obtained by removing a copy of $i$ from $\calc_B$, then we define $\ee iM$ to be the block component of $M\downarrow_{\hhh{n-1}}$ lying in $A$; otherwise, we set $\ee iM=0$. The functors $\ee i$ are defined for all $n$, so we can define powers $\ee i^r$. In fact, it is possible to define \emph{divided powers}: for any $\hhh n$-module $M$ and for any $r$ there is an $\hhh{n-r}$-module $\eed irM$ such that $\ee i^rM\cong(\eed irM)^{\oplus r!}$. For any non-zero module $M$, we set $\ep_iM=\max\lset{r}{\ee i^rM\neq0}$. The following results are essential tools in the study of decomposition numbers. Recall that we write $\nor i\mu$ for the number of normal $i$-nodes of a \bip $\mu$ and $\rem i\la$ for the number of removable $i$-nodes of $\la$. \begin{propn}\label{brchrules}\, \begin{enumerate} \item Suppose $\la$ is a \bip of $n$. Then $\eed ir\spe\la$ has a filtration in which the factors are the Specht modules $\spe\nu$, for all \bips $\nu$ that can be obtained from $\la$ by removing $r$ removable $i$-nodes. In particular, $\ep_i\spe\la=\rem i\la$, and $\eed i{\ep_i\spe\la}\spe\la$ is isomorphic to the Specht module $\spe{\la^-}$, where $\la$ is the \bip obtained by removing all the removable $i$-nodes from $\la$. \item Suppose $\mu$ is a restricted \bip of $n$. Then $\ep_i\smp\mu=\nor i\mu$ and $\eed i{\ep_i\smp\mu}\smp\mu\cong\smp{\mu^-}$, where $\mu^-$ is the \bip obtained by removing all the normal $i$-nodes from $\mu$. \end{enumerate} \end{propn} \subsection{The cyclotomic \js formula}\label{jssec} The cyclotomic \js formula introduced by James and Mathas~\cite{jmjs} will be a very useful tool. We describe a special case here, specialising to the case of Hecke algebras of type $B$. First we note that because every field is a splitting field for $\hhh n$, we can extend the field $\bbf$ without affecting the representation theory. So we assume that the subfield of~$\bbf$ generated by $q$ is a proper subfield, and we fix an element $x$ of $\bbf$ lying outside this subfield. Let $\hat q$ be an indeterminate over $\bbf$, and let $R=\bbf[\hat q^{\pm1}]$. Choose a bicharge $(k_1,k_2)$ for $\ka$ for which $|k_1-k_2|>n$. Let $\fkp$ be the prime ideal in $R$ generated by $\hat q-q$. We want to consider the Iwahori--Hecke algebra of type $B$ over $R$ with parameters $\hat q,\hat Q_1,\hat Q_2$, where \[ \hat Q_1=\hat q^{k_1},\qquad\hat Q_2=\hat q^{k_2}+x(\hat q-q). \] Now given a node $(r,c,a)\in\bbn^2\times\{1,2\}$ define \[ \rez(r,c,a) = \hat{q}^{c-r}\hat Q_a\in R. \] Now suppose $\la$ and $\nu$ are \bips of $n$ with $\la\doms\nu$. Let $G(\la,\nu)$ be the set of all pairs $(L,N)$ such that \begin{itemize} \item $L$ is a rim hook of $\la$ and $N$ a rim hook of $\nu$, \item $[\la]\sm L=[\nu]\sm N$, and \item $\resi(\hand L)=\resi(\hand N)$. \end{itemize} Given a pair $(L,N)\in G(\la,\nu)$, define $\epsilon_{LN} = (-1)^{\lel(L)-\lel(N)}$, and let \[ j_{\la\nu} = \prod_{(L,N)\in G(\la,\nu)}\big(\rez(\hand L)-\rez(\hand N)\big)^{\epsilon_{LN}}. \] Now for any pair of \bips $(\la,\mu)$ with $\mu$ restricted, we define \[ J_{\la\mu} = \sum_{\nu\domsby\la}\nu_{\mathfrak{p}}(j_{\la\nu})\dn\nu\mu. \] The following statement is a special case of the \js formula~\cite[Theorem~4.6]{jmjs}. \begin{thm}[\xcite{jmjs}{Theorem 4.6}] \label{jansch} Suppose $\la$ and $\mu$ are \bips of $n$ with $\mu$ restricted. Then the decomposition number $\dn\la\mu$ is at most $J_{\la\mu}$, and is non-zero if and only if $J_{\la\mu}$ is non-zero. \end{thm} Our particular choice of $\hat q,\hat Q_1,\hat Q_2$ means that the values $\nu_{\mathfrak{p}}(j_{\la\nu})$ will be easy to compute, and will not depend on the characteristic of $\bbf$ for the cases we will need to consider. In particular, suppose $(r,c,a)$ and $(s,d,b)$ are nodes. Then it is easy to check that: \begin{itemize} \item if $a\neq b$, then \[ \nu_{\fkp}(\rez(r,c,a)-\rez(s,d,b))= \begin{cases} 1&\text{if }\resi(r,c,a)=\resi(s,d,b), \\ 0&\text{otherwise}; \end{cases} \] \item if $a=b$, then \[ \nu_{\fkp}(\rez(r,c,a)-\rez(s,d,b))= \begin{cases} 1&\text{if }c-r=d-s\pm e, \\ 0&\text{if }c-r\nequiv d-s\ppmod e. \end{cases} \] \end{itemize} We can use the \js formula to refine \Cref{basicdecomp,basicdecomp2}, by using a coarser order than the dominance order. We write $\la\jdoms\nu$ if $\la\doms\nu$ and $\nu_{\fkp}(j_{\la\nu})\neq0$, and we extend $\jdoms$ reflexively and transitively to give a partial order $\jdom$, which we call the \emph{\js dominance} order; note that this order depends on our fixed parameters $e$ and $\ka$. It is easy to see that the usual dominance order $\dom$ is a refinement of $\jdom$, and that conjugation of \bips reverses the \js dominance order. Moreover, \Cref{jansch} allows us to strengthen \Cref{basicdecomp,basicdecomp2}. \begin{propn}\label{basicdecompjs} Suppose $\la$ and $\mu$ are \bips with $\mu$ restricted and $\dn\la\mu>0$, then $\mu\dm\jdom\la\jdom\mu$. \end{propn} \begin{pf} The fact that $\la\jdom\mu$ is immediate from \Cref{jansch}. This then implies that $\mu\dm\jdom\la$, using \Cref{conjdual} and the fact that the \js dominance order is reversed by conjugation. \end{pf} With this result in mind, we will use the \js order almost exclusively from now on. Although this order is harder to work with, it will help us to narrow down the cases we need to consider. Note that when trying to decide whether $\la\jdom\mu$, the usual dominance order can be used as a ``first pass'': if $\la\ndom\mu$, then certainly $\la\njdom\mu$. To examine the \js order more closely, it will be helpful to be able to visualise it on the abacus. Take two \bips $\la,\nu$, and take the \abds for these \bips. In order to obtain $\nu_{\fkp}(j_{\la\nu})\neq0$, we need to be able to get from the \abd for $\la$ to the \abd for $\nu$ by moving a bead from position $b_1$ to position $c_1$ on component $i_1$, and a bead from position $c_2$ to position~$b_2$ on component $i_2$, where $i_1,i_2\in\{1,2\}$, $b_1-c_1=b_2-c_2>0$, and $b_1\equiv b_2\ppmod e$. We then have $\la\jdoms\nu$ \iff either $b_1>b_2$ or $i_1\rem i\la$, then $\ep_i\smp\mu>\ep_i\spe\la$, and therefore $\smp\mu$ cannot be a composition factor of $\spe\la$. Suppose instead that $\nor i\mu=\rem i\la=r$, say. Then $r=\ep_i\spe\la=\ep_i\smp\mu$. Now $\ee i^{(r)}(\spe\la)\cong\spe{\la^-}$, and $\ee i^{(r)}(\smp\mu)\cong\smp{\mu^-}$, where $\la^-$ is obtained by removing all the removable $i$-nodes from $\la$, and $\mu^-$ is obtained by removing all the normal $i$-nodes from $\mu$. Therefore \[ \dn{\la^-}{\mu^-}=[\ee i^{(r)}\spe\la:\smp{\mu^-}]=\sum_\nu\dn\la\nu[\ee i^{(r)}\smp\nu:\smp{\mu^-}]\gs\dn\la\mu, \] summing over restricted \bips $\nu$ of $n$. Because $r=\rem i\la$, we obtain $r\gs\de_i(\la)$. So by \Cref{36lem} the weight of $\la^-$ is at most $3$. If $\la^-$ has weight~$3$, then by the inductive assumption $\dn{\la^-}{\mu^-}\ls1$. If $\la^-$ has weight less than $3$, then $\dn{\la^-}{\mu^-}\ls1$ from \Cref{w012}. Either way, we obtain $\dn\la\mu\ls1$. \item We show that the number $r=\rem i\la$ is at most $\max\{1,\de_i(B)+1\}$; then in particular $r\ls\nor i\mu$, and we can use part (1) of the \lcnamecref{simpledn0}. Suppose for a contradiction that $r\gs2$ and $r\gs\de_i(B)+2$, and let $\la^-$ be obtained by removing all the removable $i$-nodes from $\la$. Then by \Cref{36lem} the weight of $\la^-$ equals $3-r(r-\de_i(B))$. Since by assumption $r\gs2$ and $r-\de_i(B)\gs2$, this makes the weight of $\la^-$ negative, a contradiction.\qedhere \end{enumerate} \end{pf} We deduce two very useful corollaries. \begin{cory}\label{casei} Suppose $B$ is a weight $3$ block of $\hhh n$ with $\de_i(B)\ls0$ for all $i\in\zez$, and that \Cref{main} holds with $n$ replaced by any smaller integer. Then \Cref{main} holds for $B$. \end{cory} \begin{pf} Suppose $\la,\mu$ are \bips in $B$ with $\mu$ restricted. Then $\smp\mu\downarrow_{\hhh{n-1}}\neq0$, so $\ee i\smp\mu\neq0$ for some $i\in\zez$. This means that $\nor i\mu\gs1$, so $\nor i\mu\gs\max\{1,\de_i(B)+1\}$, and therefore $\dn\la\mu\ls1$ by \Cref{simpledn0}(2). \end{pf} \begin{cory}\label{untricky} Suppose $B$ is a weight $3$ block of $\hhh n$, and that \Cref{main} holds with $n$ replaced by any smaller integer. If $\la$ is a \bip in $B$ and $i\in\zez$ such that $\de_i(B)\gs1$ and $\la$ has no addable $i$-nodes, then $\dn\la\mu\ls1$ for all $\mu$. \end{cory} \begin{pf} The assumption that $\la$ has no addable $i$-nodes means that $\rem i\la=\de_i(B)$, while $\nor i\mu\gs\de_i(B)$. So $\nor i\mu\gs\max\{1,\rem i\la\}$, and the result follows from \Cref{simpledn0}(1). \end{pf} So to prove \Cref{main} by induction using \Cref{untricky}, it suffices to consider only those $\la$ that have at least one addable $i$-node for \emph{every} $i$ for which $\de_i(B)\gs1$. Call such $\la$ \emph{\tricky}. The next few lemmas will show that in most cases $B$ has no \tricky \bips. \begin{lemma}\label{trickycore} Suppose $B$ is a core block, and that $\de_i(B)\gs1$ for some $i\in\zez$. Then~$B$ contains no \tricky \bips. \end{lemma} \begin{pf} Suppose $\la$ is a \bip in $B$. Then we claim that $\la$ has no addable $i$-nodes. Assume for a contradiction that $\la$ has at least one addable $i$-node. Then because $\de_i(B)\gs1$, we know that $\la$ has at least two removable $i$-nodes. The two components of $\la$ are both $e$-cores, and an $e$-core cannot have addable and removable nodes of the same residue, so the addable $i$-nodes of $\la$ are in one component and the removable $i$-nodes in the other. We assume that the removable $i$-nodes are in component $1$ (the other case is similar). This means that in an \abd for $\la$ there are at least two positions $b$ on runner $i$ such that there is a bead in position $b$ but not in position $b-1$. So there are at least two more beads on runner $i$ than on runner $i-1$ in component~$1$ (or at least three more, in the case $i=\ol0$). Similarly, in component $2$ of the \abd, there is at least one position $b$ on runner $i-1$ such that there is a bead at position $b$ but no bead at position $b+1$. So there are more beads on runner $i-1$ than on runner $i$ in component $2$ (or at least as many beads on runner $i-1$ as on runner~$i$, in the case $i=\ol0$). We deduce that $\ga_i(\la)-\ga_{i-1}(\la)\gs3$. But the assumption that $B$ is a core block gives $\ga_i(\la)-\ga_{i-1}(\la)\ls2$, a contradiction. \end{pf} In view of \Cref{trickycore}, \Cref{main} holds for core blocks. These blocks have been extensively studied; a closed formula for the decomposition numbers is given in~\cite{lyle}. We consider non-core blocks from now on. \begin{lemma}\label{k3} Suppose $\de_i(B)\gs3$ for some $i\in\zez$. Then $B$ contains no \tricky \bips. \end{lemma} \begin{pf} If $\la$ is a \bip in $B$, then we claim that $\la$ cannot have an addable $i$-node: if it did, then by \Cref{36lem} the weight of a \bip obtained by adding an addable $i$-node to $\la$ would be $3-(\de_i(\la)+1)$, which is negative, a contradiction. \end{pf} \begin{lemma}\label{kij} Suppose there are $i,j\in\zez$ such that $\de_i(B),\de_j(B)\gs1$, and $j\neq i,i\pm1$. Then $B$ contains no \tricky \bips. \end{lemma} \begin{pf} Suppose $\la$ is an \tricky \bip in $B$. Then $\la$ has both an addable $i$-node $\fkm$ and an addable $j$-node $\fkn$. Define \bips $\mu,\nu$ by $[\mu]=[\la]\cup\{\fkm\}$ and $[\nu]=[\mu]\cup\{\fkn\}$. \Cref{36lem} then gives $\wt\mu=3-(\de_i(B)+1)\ls1$. Because $j\neq i,i\pm1$, adding the $i$-node to $\la$ does not affect the addable or removable $j$-nodes, so $\de_j(\mu)=\de_j(\la)$. Hence $\wt\nu=\wt\mu-(\de_j(B)+1)\ls-1$, a contradiction. \end{pf} \begin{lemma}\label{consec12} Suppose there are $i,j\in\zez$ such that $i\neq j$, $\de_i(B)\gs1$ and $\de_j(B)\gs2$. Then $B$ contains no \tricky \bips. \end{lemma} \begin{pf} If $j\neq i\pm1$ then the result follows from \Cref{kij}, so assume that $j=i+1$ (the case $j=i-1$ is similar). Suppose $\la$ is an \tricky \bip in $B$. Then $\la$ has both an addable $i$-node $\fkm$ and an addable $(i+1)$-node $\fkn$. Define \bips $\mu,\nu$ by $[\mu]=[\la]\cup\{\fkm\}$ and $[\nu]=[\mu]\cup\{\fkn\}$. \Cref{36lem} now gives $\wt\mu=3-(\de_i(B)+1)\ls1$. Now from~\cite[Lemma 3.1]{mfweight} we obtain $\de_{i+1}(\mu)=\de_{i+1}(\la)-1$, so that $\wt\nu=\wt\mu-\de_{i+1}(B)\ls-1$, and again we have a contradiction. \end{pf} These lemmas mean that there are only four types of non-core block $B$ in which there can be \tricky \bips. These are exactly analogous to the four types of weight $3$ blocks of symmetric groups considered in~\cite[Section 4]{jmsmalld}. These types can be defined according to the values $\de_i(B)$, as follows. \begin{enumerate}[label=\Roman*.] \item $\de_i(B)\ls0$ for all $i\in\zez$. \item There is $i\in\zez$ such that $\de_i(B)=1$ while $\de_j(B)\ls0$ for all $j\neq i$. \item There is $i\in\zez$ such that $\de_i(B)=\de_{i+1}(B)=1$ while $\de_j(B)\ls0$ for all $j\neq i,i+1$. \item There is $i\in\zez$ such that $\de_i(B)=2$ while $\de_j(B)\ls0$ for all $j\neq i$. \end{enumerate} We know from \Cref{casei} that \Cref{main} holds for blocks of type I. To prove \Cref{main}, we need to show that if $B$ is a block of $\hhh n$ of type II, III or IV and if \Cref{main} holds with $n$ replaced by any smaller integer, then it holds for $B$. We address the three types of block in \Cref{type2sec,type3sec,type4sec}. \section{Blocks of type II}\label{type2sec} Throughout this section, we assume $B$ is a block of $\hhh n$ and $i\in\bbz$ is such that $\de_i(B)=1$ while $\de_j(B)\ls0$ for all $j\nequiv i\ppmod e$. Our aim is to show that if \Cref{main} holds with $n$ replaced by any smaller integer, then it holds for $B$. \subsection{The abacus for a block of type II} Let $\xi$ be the \xr of $B$. Whenever we refer to the \abd for $\xi$, we will mean the $(k_1+1,k_2-1)$-\abd, and whenever we refer to the residues of nodes for $\xi$, we mean with respect to \mbox{$(\kappa_1+1,\kappa_2-1)$}. With this convention in mind, we have $\de_i(\xi)=1$ while $\de_j(\xi)\ls0$ for $j\nequiv i\ppmod e$. This means that $\xi$ has exactly one removable node, which has residue~$i$ (because by \Cref{36lem} a \bip of weight $0$ cannot have an addable and a removable node of the same residue). We will assume that this removable node lies in component~$1$ (in the opposite case we can replace $\ka$ with $(\ka_2,\ka_1)$ and appeal to the results in \Cref{conjswsec}). By considering the \abd for $\xi$ and using the fact that $\xi$ has $(\ka_1+1,\ka_2-1)$-weight $0$, we find that there are integers $i\ls j\ls k\ls l\ls e+i-2$ such that in component $1$ of $\xi$ there is a removable $i$-node, and addable nodes of residues $j+1$ and $l+1$, while in component $2$ there is just an addable node of residue $k+1$ (so in fact $\ka_1=\widebar{j+l+1-i}$, and $\ka_2=\widebar{k+2}$). We therefore have $Z=\{\ol i,\dots,\ol j\}\cup\{\widebar{k+1},\dots,\ol l\}$. The fact that $|Z|\gs2$ implies that either $i1$ for some $\la$. Then $\nor i\mu=1$, and there is some $x\in Z$ for which $\dn{\al_x}\mu$, $\dn{\be_x}\mu$, $\dn{\ga_x}\mu$ are all non-zero; in particular, $\mu\dm\jdom\al_x,\be_x,\ga_x\jdom\mu$. \end{lemma} \begin{pf} If $\la$ is such that $\dn\la\mu>1$, then $\la$ is \tricky, so $\la\in\{\al_x,\be_x,\ga_x\}$ for some $x$. The conclusion that $\mu$ has only one normal node comes from \Cref{simpledn0}(2). Now we claim that $\dn{\al_x}\mu,\dn{\be_x}\mu,\dn{\ga_x}\mu\gs1$, which gives the result using \Cref{basicdecompjs}. Each of $\al_x,\be_x,\ga_x$ has two removable $i$-nodes, and the branching rules show that there are \bips $\hat\al_x,\hat\be_x,\hat\ga_x$ of $n-1$ such that \begin{align*} \ee i\spe{\al_x}&\sim\spe{\hat\al_x}+\spe{\hat\be_x}, \\ \ee i\spe{\be_x}&\sim\spe{\hat\al_x}+\spe{\hat\ga_x}, \\ \ee i\spe{\ga_x}&\sim\spe{\hat\be_x}+\spe{\hat\ga_x}. \end{align*} Because $\nor i\mu=1$, there is a restricted \bip $\tilde\mu$ such that $\ee i\smp\mu\cong\smp{\tilde\mu}$. Now (analogously to the proof of~\cite[Proposition 2.8(2)]{mfwt3}) we obtain \[ \dn{\al_x}\mu+\dn{\hat\ga_x}{\tilde\mu}=\dn{\be_x}\mu+\dn{\hat\be_x}{\tilde\mu}=\dn{\ga_x}\mu+\dn{\hat\al_x}{\tilde\mu}. \] Since by assumption the decomposition numbers for weight $3$ blocks of $\hhh{n-1}$ are all $0$ or $1$, this means that the decomposition numbers $\dn{\al_x}\mu$, $\dn{\be_x}\mu$, $\dn{\ga_x}\mu$ differ by at most $1$. So if one of them is greater than $1$, then they are all positive. \end{pf} \Cref{domcond} greatly restricts the set of restricted \bips $\mu$ that we need to consider. To apply \Cref{domcond} we look at the \js dominance order among the \tricky \bips, which is easy to check given the explicit description above: if $\epsilon$ is any of the symbols $\al,\be,\ga$, then \[ \epsilon_i\jdoms\epsilon_{k+1}\jdoms\epsilon_{k+2}\jdoms\cdots\jdoms\epsilon_l\jdoms\epsilon_{i+1}\jdoms\epsilon_{i+2}\jdoms\cdots\jdoms\epsilon_j. \] In particular, the most dominant $\ga_x$ is $\ga_i$. So to show that $\dn\la\mu\ls1$ for all $\mu$, we need only consider \bips $\mu$ for which: \begin{enumerate} \item\label{muklesh} $\mu$ is restricted; \item\label{gadom} $\ga_i\jdom\mu$; and \item\label{normi} $\nor i\mu=1$, and $\nor j\mu=0$ for all $j\neq i$. \end{enumerate} Analysis of the possible abacus configurations enables a classification of the \bips $\mu$ satisfying (\ref{gadom}) and (\ref{normi}). By working through the different \bips in~$B$, we can check that the only \bips $\mu$ satisfying these conditions are $\dl i{i-1}2$ and~$\dl ii2$. (Note that if $i1$ for some $\la$. \begin{lemma}\label{domcond3} Suppose $\mu$ is a restricted \bip in $B$, and $\dn\la\mu>1$ for some~$\la$. Then $\nor i\mu=\nor{i+1}\mu=1$, and at least three of the decomposition numbers $\dn\albe\mu$, $\dn\alga\mu$, $\dn\gaal\mu$ and $\dn\bega\mu$ are positive. Hence $\mu\dm\jdom\alga,\gaal\jdom\mu$. \end{lemma} \begin{pf} In the same way as in the proof of \Cref{domcond}, we can apply the restriction functor $\ee i$ to show that $\mu$ has only one normal $i$-node and that if either of the decomposition numbers $\dn\albe\mu$ and $\dn\gaal\mu$ is greater than $1$ then they are both positive, while if either of $\dn\alga\mu$ and $\dn\bega\mu$ is greater than $1$ then they are both positive. But in the same way we can use $\ee{i+1}$ to show that $\nor{i+1}\mu=1$ and either $\dn\albe\mu$ and $\dn\alga\mu$ are both positive or $\dn\gaal\mu$ and $\dn\bega\mu$ are both positive. Combining these statements shows that at least three of the given decomposition numbers are positive. Now \Cref{basicdecompjs}, together with the \js dominance order on the \bips $\albe,\alga,\gaal,\bega$ gives the second statement. \end{pf} So to prove our main result for $B$ we can focus on \bips $\mu$ in $B$ for which: \begin{enumerate} \item\label{isres3} $\mu$ is restricted; \item\label{aggadm} $\alga,\gaal\jdom\mu$; \item\label{mdagga} $\mu\dm\jdom\alga,\gaal$; and \item\label{allnorm3} $\nor i\mu=\nor{i+1}\mu=1$, and $\nor j\mu=0$ for $j\neq i,i+1$. \end{enumerate} We begin by finding the \bips $\mu$ for which conditions (\ref{aggadm}) and (\ref{allnorm3}) are satisfied. A careful analysis of the possible abacus configurations shows that this happens in the cases given in the following table. \begin{longtable}{|c|c|} \hline $\mu$&conditions \\ \hline $\dl{i+1}{m}1$&$l1$ for some $\la$. By \Cref{simpledn0}(2) we note that such \bips have exactly two normal $i$-nodes. \begin{lemma}\label{domcond4} Suppose $\mu$ is a restricted \bip in $B$, and $\dn\la\mu>1$ for some $\la$. Then the decomposition numbers $\dn{\be_1}\mu,\dn{\be_2}\mu,\dn{\be_3}\mu,\dn{\be_4}\mu$ are all positive; in particular $\mu\dm\jdom\be_4\jdoms\be_3\jdoms\be_2\jdoms\be_1\jdom\mu$. \end{lemma} \begin{pf} Let $\la$ be a \bip such that $\dn\la\mu>1$. Then $\la=\be_x$ for some $x\in\{1,2,3,4\}$. We follow the same idea as in \Cref{domcond} and \Cref{domcond3} with the only difference that here we need to apply the restriction functor $\ee i$ to $\spe{\be_1},\dots,\spe{\be_4}$ 'twice', namely we apply the functor $\ee i^2$ described in \Cref{branchsec}. By the branching rules we find that there are \bips $\hat\be_1,\hat\be_2,\hat\be_3,\hat\be_4,\tilde\mu$ of $n-2$ such that \begin{align*} \ee i^2\spe{\be_1}&\sim(\spe{\hat\be_1})^2+(\spe{\hat\be_2})^2+(\spe{\hat\be_3})^2, \\ \ee i^2\spe{\be_2}&\sim(\spe{\hat\be_1})^2+(\spe{\hat\be_2})^2+(\spe{\hat\be_4})^2, \\ \ee i^2\spe{\be_3}&\sim(\spe{\hat\be_1})^2+(\spe{\hat\be_3})^2+(\spe{\hat\be_4})^2, \\ \ee i^2\spe{\be_4}&\sim(\spe{\hat\be_2})^2+(\spe{\hat\be_3})^2+(\spe{\hat\be_4})^2, \\ \ee i^2\smp{\mu}&\sim\smp{\tilde\mu}. \end{align*} Analogously as in~\cite[Proposition 2.10(4)]{mfwt3}, we find \begin{align*} \dn{\be_1}\mu+\dn{\hat\be_4}{\tilde\mu}&=\dn{\be_2}\mu+\dn{\hat\be_3}{\tilde\mu}\\ &=\dn{\be_3}\mu+\dn{\hat\be_2}{\tilde\mu}\\ &=\dn{\hat\be_4}\mu+\dn{\hat\be_1}{\tilde\mu}, \end{align*} and we conclude by induction that if one of the decomposition numbers $\dn{\be_x}\mu$ is greater than $1$, then they are all positive. The last sentence follows from \Cref{basicdecompjs}. \end{pf} Summing up, it remains to consider those \bips $\mu$ in $B$ such that \begin{enumerate} \item\label{isres4} $\mu$ is restricted; \item\label{bejsdm} $\be_1\jdom\mu$; \item\label{mjsdbe} $\mu\dm\jdom\be_4$; and \item\label{allnorm4} $\nor i\mu=2$, and $\nor j\mu=0$ for $j\neq i$. \end{enumerate} Firstly we list all possible \bips $\mu$ which satisfy (\ref{bejsdm}) and (\ref{allnorm4}) and in which both components are $e$-restricted (which is a necessary condition for $\mu$ to be restricted, by \Cref{resterest}). \needspace{4em} \begin{longtable}{|c|c|} \hline $\mu$&conditions \\ \hline $\dl i{m}1$&$l