The Murphy operators in the Hecke algebra $H_n$ of type $A$ are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of $H_n$. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, $\CC$. Symmetric functions of the Murphy operators are also central in $H_n$. I define geometrically a homomorphism from $\CC$ to the centre of each algebra $H_n$, and find an element in $\CC$, independent of $n$, whose image is the $m$th power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of $\CC$ in terms of the Murphy operators, and to demonstrate relations among other natural skein elements. \phantom{some stuff to separate the keywords from the abstract} \noindent {\em Keywords}: skein theory; Murphy operators; power sums; symmetric functions; annulus; Hecke algebras.
boxedeps.tex
  %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%
 %%%%%               boxedeps.tex 
 %%%%% MACROS FOR FIGURE INSERTS OF EPSF NORM  %%%%%
 %%%%%   (EPSF = Encapsulated PostScript File)
  %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%  
 %%%  AUTHOR: Laurent Siebenmann
  %%    lcs@topo.math.u-psud.fr
  %%  
 %%%  VERSIONS: Feb 1991 -- Nov 1995
  %%  Revisions bi-annual approx.
  %%  
 %%%  SOMMAIRE: boxedeps.tex d\'efinit des macro-commandes
  %%    qui permettent d'int\'egrer dans un document TeX des 
  %%    objets graphiques d\'ecrits par fichier de norme EPSF,
  %%    tout en accordant a chacun le statut d'une bo\^ite TeX ayant 
  %%    les bonnes dimensions.  La (seule!) contribution unique 
  %%    de ce fichier est de faire cela d'une fa{\c}con universelle.
  %%    C'est a dire de fa{\c}con \`a pouvoir commod\'ement 
  %%    servir avec tout pilote d'imprimante de norme 
  %%    PostScript --- malgr\'e l'absence d'une norme 
  %%    pour \special. 
  %%  
 %%%% ACTIVATION:
  %%  by a driver-by-driver protocol
  %%  see \SetTexturesEPSFSpecial 
  %%  and its companions below.
  %%  
 %%%% DOCUMENTATION:
  %%  --- see boxedeps.doc
  %%  The complete utility is called BoxedEPSF but for 
  %%  portability the nickname boxedeps is useful. 
  %%
 %%%%  POSTINGS: anonymous ftp 
  %%  ---  ftp matups.math.u-psud.fr [192.54.146.4] 
  %%  is the master copy in 1995.
  %%  ---  ftp.uni-stuttgart.de 
  %%  ---  ftp ftp.TeX.AC.UK
  %%  ---  ftp ftp.SHSU.edu
  %%  Locate the full package on these three 
  %%  "CTAN" servers by using (while connected) the command
  %%        QUOTE SITE INDEX boxedeps

 %%%%  RECENT ADDITIONS: \SetepsfEPSFSpecial simultaneously serves
   %%  dvips, the Y&Y drivers, DirectTeX, CMacTeX, OzTeX>=1.7
   %%  \SetOldOzTeXEPSFSpecial  %% OzTeX 1.42--1.6x 

 
 \chardef\EPSFCatAt\the\catcode`\@
 \catcode`\@=11

 \chardef\C@tColon\the\catcode`\:
 \chardef\C@tSemicolon\the\catcode`\;
 \chardef\C@tQmark\the\catcode`\?
 \chardef\C@tEmark\the\catcode`\!
 \chardef\C@tDqt\the\catcode`\"

 \def\PunctOther@{\catcode`\:=12
   \catcode`\;=12 \catcode`\?=12 \catcode`\!=12 \catcode`\"=12}
 \PunctOther@

 %%%% MESSAGES (separate macro best for Europe)
  %%  
  \def\ms@g{\immediate\write16}

 %%%% Prevent double input
 \let\EX@\expandafter
 
 \ifx\BoxedArtundefined\relax
   \ms@g{}\ms@g{ !!!  Bad news.
     \noexpand\BoxedEPSF or \noexpand\BoxedArt macros already input.}%
   \ms@g{ !!! Hit RETURN to exit, or  c  and RETURN to forge bravely on.}%
   {\endlinechar=-1\gdef\Temp{c}%
   \global\read16 to \temp}%
     \ifx\temp\Temp 
     \else
       \EX@\EX@\EX@\end
     \fi
 \else
    \global\let\BoxedArtundefined\relax
 \fi


 
 %%temporarily suppress Plain's logging of allocations
 \let\wlog@ld\wlog 
 \def\wlog#1{\relax} 

 %% New for TOOLS
 %\newif\ifIN@
 \newdimen\XShift@ \newdimen\YShift@ 
 \newtoks\Realtoks
 
 %%% New for BoxedEPSF
 %
 \newdimen\Wd@ \newdimen\Ht@
 \newdimen\Wd@@ \newdimen\Ht@@
 %
 \newdimen\TT@
 \newdimen\LT@
 \newdimen\BT@
 \newdimen\RT@
 %
 \newdimen\XSlide@ \newdimen\YSlide@ 
 %
 \newdimen\TheScale  %% secretly scale in mils: 1pt= 1mil 
 \newdimen\FigScale  %% secretly scale in mils: 1pt= 1mil 
 %
 \newdimen\ForcedDim@@

 \newtoks\EPSFDirectorytoks@
 \newtoks\EPSFNametoks@
 \newtoks\BdBoxtoks@
 \newtoks\LLXtoks@  %% useful info for Oz
 \newtoks\LLYtoks@  
 \newtoks\URXtoks@  %% useful for epsf.tex syntax
 \newtoks\URYtoks@  

  
 \newif\ifRepeat@
 \newif\ifNotIn@
 \newif\ifForcedDim@
 \newif\ifForceOn@
 \newif\ifForcedHeight@
 \newif\ifPSOrigin
 \newif\ifPCfmt@
 \newif\ifPCfmtOK@


 \newread\EPSFile@ 
 \newwrite \epsf@log
 \def\WrLog@{\immediate\write\epsf@log}

 \gdef\InitLog@{%
   \def\epsf@log@name{epsf.lst}%
   \count255=\time\divide\count255 by 60
   \edef\temp@{\the\count255}%
   \multiply\count255 by -60\relax 
   \advance\count255 by \time
   \immediate\openout\epsf@log \epsf@log@name
   \WrLog@{ \pct@@\space boxedeps.tex log, date 
   \the\day-\the\month-\the\year, 
   time \temp@:\the\count255.}\WrLog@{}%
   \ms@g{}\ms@g{ **** See file \epsf@log@name\space
    for list of EPSFs integrated by boxedeps.tex}%
   \global\let\InitLog@\relax}

 \def\noArtList{\global\let\InitLog@\relax}
 \let\noEPSFList\noArtList
 \def\noArtReports{\global\let\ReportEPSF@\relax}
 \let\noEPSFReports\noArtReports
 
 %%%% WORD-PROCESSING MACROS
  %%
  %%% \IN@0#1@#2@ : Is 1st exp of #1 in 1st exp of #2 ??
   %% Answer in \ifIN@
 \newif\ifIN@\def\IN@{\EX@\INN@\EX@}
  \long\def\INN@0#1@#2@{\long\def\NI@##1#1##2##3\ENDNI@
    {\ifx\m@rker##2\IN@false\else\IN@true\fi}%
     \EX@\NI@#2@@#1\m@rker\ENDNI@}
  \def\m@rker{\m@@rker}

  %%%  \SPLIT@0#1@#2@  :  Split 1st exp of #2 at 1st exp of #1
   %%  \Initialtoks@ , \Terminaltoks@ will contain pieces
  \newtoks\Initialtoks@  \newtoks\Terminaltoks@
  \def\SPLIT@{\EX@\SPLITT@\EX@}
  \def\SPLITT@0#1@#2@{\def\TTILPS@##1#1##2@{%
     \Initialtoks@{##1}\Terminaltoks@{##2}}\EX@\TTILPS@#2@}

 %%%% MACROS TO TRIM  \ForeTrim@0#1@ and \Trim@0#1@  
   %% result appears in \Trimtoks@
   %% LIMITATION: assume no multiple spaces to trim

  \newtoks\Trimtoks@

  %%% \ForeTrim@0#1@ trims initial space of first erpansion of #1
   %% #1 of form \the\toks0 or \mymacro
 \def\ForeTrim@{\EX@\ForeTrim@@\EX@}
 \def\ForePrim@0 #1@{\Trimtoks@{#1}}
 \def\ForeTrim@@0#1@{\IN@0\m@rker. @\m@rker.#1@%
     \ifIN@\ForePrim@0#1@%
     \else\Trimtoks@\EX@{#1}\fi}
   %%\m@rker expands here to \m@@rker since spot initial,
   %% so no confusuion with \m@rker

  %%% \Trim@0#1@ trims init and terminal spaces 
   %% Same syntax.
   %% Warns if internal spaces found.
   %% 
  \def\Trim@0#1@{%
    \ForeTrim@0#1@%
    \IN@0 @\the\Trimtoks@ @%
    \ifIN@ 
      \SPLIT@0 @\the\Trimtoks@ @\Trimtoks@\Initialtoks@
      \IN@0\the\Terminaltoks@ @ @%
      \ifIN@ \else \Trimtoks@{FigNameWithSpace}\fi
    \fi }


  %%%% MATH MACROS (provisional)
    %% use dimen registers for reals; unit 1pt
    %% (numerical dimension arguments OK unless contrary noted)

  %%%% One needs the point token seq (pt with cat 12) USES dimen 0
   \newtoks\pt@ks
   \def\getpt@ks 0.0#1@{\pt@ks{#1}}
   \dimen0=0pt\relax\EX@\getpt@ks\the\dimen0@

   %%% Convert dimen to "decimal multiplier"% USES dimens 0,2
  %\newtoks\Realtoks% the output!
  \def\Real#1{%
    \dimen2=#1%
      \SPLIT@0\the\pt@ks @\the\dimen2@%%  lop off the points
       \Realtoks=\Initialtoks@ %\showthe\Realtoks
            }

   %%% Multiplication 
      % USES dimens 0,2,4,6; preserves args; output \Product
   \newdimen\Product
   \def\Mult#1#2{%
     \dimen4=#1\relax
     \dimen6=#2%
     \Real{\dimen4}%
     \Product=\the\Realtoks\dimen6%
        }

   %%% Inverse 
     % USES dimens 0; preserves arg; output \Inverse
 \newdimen\Inverse
 \newdimen\hmxdim@ \hmxdim@=8192pt%halfmaxdimen
 \def\Invert#1{%
  \Inverse=\hmxdim@
  \dimen0=#1%
  \divide\Inverse \dimen0%
  \multiply\Inverse 8}

 %%% \Rescale#1#2#3  % USES dimens 0,2,4,6
  %%  alters dimen register #1 by ratio #2/#3 
  %%  where #2,#3 can be raw dimensions OR dimen registers
   \def\Rescale#1#2#3{% Adequate accuracy. Can improve. 
              \divide #1 by 100\relax
              \dimen2=#3\divide\dimen2 by 100 \Invert{\dimen2}% 
              \Mult{#1}{#2}%
              \Mult\Product\Inverse 
              #1=\Product}

 %%% \Scale#1 scales dimen register #1 
  %%  by dimen register real \TheScale; USES dimen 0
  \def\Scale#1{\dimen0=\TheScale 
      \divide #1 by  1280 %% 1280*5120*10=1000*2^16 
      \divide \dimen0 by 5120  
      \multiply#1 by \dimen0 
      \divide#1 by 10   %% max size of #1 about 32000/10 pt
     }
 
 %%% SCRUNCHING BOXES AND SHIFTING CONTENTS
  %% TeX has to do this in general
  %% since some drivers do not let 
  %% one do it readily using Postscript

 \newbox\scrunchbox

 %%% \Scrunched#1 puts #1 in an hbox
  %%    then in effect zeros the dimensions of this box
 \def\Scrunched#1{{\setbox\scrunchbox\hbox{#1}%
   \wd\scrunchbox=0pt
   \ht\scrunchbox=0pt
   \dp\scrunchbox=0pt
   \box\scrunchbox}}

  %%% \Shifted@#1 puts #1 in \hbox 
   %% then locates basepoint to bottom left corner
   %% then translates ink only by \XShift@,\YShift@
   %% with Postscript convention
   %% For simplicity use only on scrunched boxes
  %\newdimen\XShift@ 
  %\newdimen\YShift@ 
 \def\Shifted@#1{%
   \vbox {\kern-\YShift@
       \hbox {\kern\XShift@\hbox{#1}\kern-\XShift@}%
           \kern\YShift@}}

  %%% \cBoxedEPSF#1 the main macro
   %%  component macros are explained in order below

 \def\cBoxedEPSF#1{{\leavevmode 
    %% double brace for amstex \allign, \alligned, ...
   \ReadNameAndScale@{#1}%
   \SetEPSFSpec@
   \InitLog@ 
   \ReadEPSFile@ \ReadBdB@x  
   %% Calculations
     \TrimFigDims@ 
     \CalculateFigScale@  
     \ScaleFigDims@
     \SetInkShift@
   \hbox{$\mathsurround=0pt\relax
         \vcenter{\hbox{%
             \FrameSpider{\hskip-.4pt\vrule}%
             \vbox to \Ht@{\offinterlineskip\parindent=\z@%
                \FrameSpider{\vskip-.4pt\hrule}\vfil 
                \hbox to \Wd@{\hfil}%
                \vfil
                \InkShift@{\EPSFSpecial{\EPSFSpec@}{\FigSc@leReal}}%
             \FrameSpider{\hrule\vskip-.4pt}}%
         \FrameSpider{\vrule\hskip-.4pt}}}%
     $\relax}%
    \CleanRegisters@ 
    \ReportEPSF@ 
    }}
 
 \def\tBoxedEPSF#1{\setbox4\hbox{\cBoxedEPSF{#1}}%
     \setbox4\hbox{\raise -\ht4 \hbox{\box4}}%
     \box4
      }

 \def\bBoxedEPSF#1{\setbox4\hbox{\cBoxedEPSF{#1}}%
     \setbox4\hbox{\raise \dp4 \hbox{\box4}}%
     \box4
      }

  \let\BoxedEPSF\cBoxedEPSF% default setting

  %% Some compatibility with BoxedArt.tex
   %
   \let\BoxedArt\BoxedEPSF

  %% Some compatibility with Sweet-teX
   %
  \def\gLinefigure[#1scaled#2]_#3{%
        \BoxedEPSF{#3 scaled #2}}
    
  %% Some compatibility with Rokicki's dvips
   %
  \let\EPSFbox\bBoxedEPSF \let\EPSFfile\bBoxedEPSF
  
  \def\EPSFxsize{\afterassignment\ForceW@\ForcedDim@@}
      \def\ForceW@{\ForcedDim@true\ForcedHeight@false}
  
  \def\EPSFysize{\afterassignment\ForceH@\ForcedDim@@}
      \def\ForceH@{\ForcedDim@true\ForcedHeight@true}

  \def\EmulateRokicki{%
       \let\epsfbox\bBoxedEPSF \let\epsffile\bBoxedEPSF
       \let\epsfxsize\EPSFxsize \let\epsfysize\EPSFysize} 
 
 %%% \ReadNameAndScale@#1
  %
 \def\ReadNameAndScale@#1{\IN@0 scaled@#1@% DOUBLE BARRELED
   \ifIN@\ReadNameAndScale@@0#1@%
   \else \ReadNameAndScale@@0#1 scaled\DefaultMilScale @%
   \fi}
  
 \def\ReadNameAndScale@@0#1scaled#2@{% HELPER MACRO
    \let\OldBackslash@\\%
    \def\\{\OtherB@ckslash}%
    \edef\temp@{#1}%
    \Trim@0\temp@ @%
    \EPSFNametoks@\EX@{\the\Trimtoks@ }%
    \FigScale=#2 pt%
    \let\\\OldBackslash@
    }
 
 \def\SetDefaultEPSFScale#1{%
      \global\def\DefaultMilScale{#1}}

 \SetDefaultEPSFScale{1000}


 %%% \ReadEPSFile@
  %
 \def \SetBogusBbox@{%
     \global\BdBoxtoks@{ BoundingBox:0 0 100 100 }%
     \global\def\BdBoxLine@{ BoundingBox:0 0 100 100 }%
     \ms@g{ !!! Will use placeholder !!!}%
     }

 {\catcode`\%=12\relax\gdef\P@S@{%!}\gdef\pct@@{%%}}
   %% %! min sign of PS file

 %%% Opening and closing
 \def\ReadEPSFile@{%\show\EPSFSpec@%
   \openin\EPSFile@\EPSFSpec@
   \relax  %necessary to prevent precocious expansion of \ifeof
  \ifeof\EPSFile@
   \ms@g{}%
   \ms@g{ !!! EPS FILE \the\EPSFDirectorytoks@
       \the\EPSFNametoks@\space WAS NOT FOUND !!!}%
   \SetBogusBbox@
  \else
   \ReadEPSFile@@
  \fi
  \closein\EPSFile@ 
   }
 
 %% \AdvanceLcnt@ for PC EPSF case only
 \global\let\AdvanceLcnt@\relax %% initialize
 \def\AdvanceLcnt@@{%
   \advance\count255 1 \ifnum \count255>50 \Lcnt@ms@g \fi}
  
 \def\Lcnt@ms@g{\ms@g{}%
  \ms@g{ !!! PC binary EPS preview probably before BoundingBox.}%
  \ms@g{ !!! To accellerate, use utility to move all previews after \pct@@ EOF.}%
  \global\let\AdvanceLcnt@\relax
  \global\let\AdvanceLcnt@@\relax %% in all, one test, one complaint
  }

 %%% Header tests
 \def\ReadEPSFile@@{%
   \begingroup%%
   \chardef\other=12 \chardef\cmt=14 \catcode"0C=9 \catcode"7F=\cmt 
   \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
   \catcode"00=\cmt \catcode`\\=\cmt \catcode`\!=12 \catcode`\:=12 
   %% next to recognize PC-type header
   \global\PCfmt@false%% initialize
   \catcode"C5=\other \catcode"D0=\other \catcode"D3=\other \catcode"C6=\cmt 
   \global\read\EPSFile@ to \HdLine@ 
   \IN@0\P@S@ @\HdLine@ @%
   \ifIN@ %% %! accepted as %!PS so do BdBox search loop!!
     %\message{*********}%
     %\showthe\catcode`\%
     %\showthe\catcode`\!
     %\show\HdLine@
     %\show\P@S@
     \let\temp@\ReadEPSFile@@@
   \else
     \def\temp@@{^^c5^^d0^^d3}%
     \ifx\temp@@\HdLine@  %% sign of PC binary PostScript file header
       \global\PCfmt@true
       \count255=0 %% will be line counter
       \let\AdvanceLcnt@\AdvanceLcnt@@
       \let\temp@\ReadEPSFile@@@
       \ifPCfmtOK@\else %%
         \ms@g{}%
         \ms@g{ !!! PC binary EPS file header encountered. Designated }%
         \ms@g{ !!! driver may not accept this and similar EPS files.}%
         \relax\global\PCfmtOK@true %% just one warning
       \fi
     \else
       \ms@g{}%
       \ms@g{ !!! \the\EPSFNametoks@\space is not PostScript.}%
       \ms@g{ !!! It should begin with the "\P@S@" or PC binary header.}%
       \SetBogusBbox@
     \fi
   \fi
   \temp@
  \endgroup%\relax
  }

 %%% BoundingBox search loop
 \def\ReadEPSFile@@@{\Repeat@true
   \loop   
   \ifeof\EPSFile@\relax\Repeat@false 
     \ms@g{}%
     \ms@g{ !!! BoundingBox NOT FOUND IN %
       \the\EPSFDirectorytoks@\the\EPSFNametoks@\space!!! }%
     \SetBogusBbox@
   \else
     \ReadEPSFile@@@@
   \fi
   \ifRepeat@
   \AdvanceLcnt@ %% is \relax unless PC EPSF; >50 => complaint
   \repeat
   \global\BdBoxtoks@\expandafter{\BdBoxLine@}%
   }

 %%% BoundingBox or HiResBoundingBox
  %% HiRes normally on line following if present at all
 \def\ReadEPSFile@@@@{%
   \global\read\EPSFile@ to \BdBoxLine@
   %\BdBoxtoks@\EX@{\BdBoxLine@}\ms@g{}\ms@g{ *  \the\BdBoxtoks@}%%%%
   \IN@0BoundingBox@\BdBoxLine@ @%
   \ifIN@
     %\ms@g{}\ms@g{ *  SCORE!!}
     \Repeat@false
     \read\EPSFile@ to \temp@
     \IN@0HiResBoundingBox:@\temp@ @%
     \ifIN@
       \global\let\BdBoxLine@\temp@
     \fi
   \fi}

  %%% \ReadBdB@x
   % Rmk For simplicity 0 not used in syntax 
   %  of \ReadBdB@x@,  \ReadBdB@x@@ 
  \def\ReadBdB@x{% PART 0
   \EX@\ReadBdB@x@\the\BdBoxtoks@ @}
  
  \def\ReadBdB@x@#1BoundingBox:#2@{% PART 1
    \ForeTrim@0#2@%
    \IN@0atend@\the\Trimtoks@ @%
       \ifIN@ \Trimtoks@={0 0 100 100 }%
         \ms@g{}%
         \ms@g{ !!! BoundingBox not found in %
         \the\EPSFDirectorytoks@\the\EPSFNametoks@\space !!!}%
         \ms@g{ !!! It must not be at end of EPSF !!!}%
         \ms@g{ !!! Will use placeholder !!!}%
       \fi%% cf \SetBogusBbox@
    \EX@\ReadBdB@x@@\the\Trimtoks@ @%
   }
    
  \def\ReadBdB@x@@#1 #2 #3 #4@{% PART 2
      \Wd@=#3bp\advance\Wd@ by -#1bp%
      \Ht@=#4bp\advance\Ht@ by-#2bp%
       \Wd@@=\Wd@ \Ht@@=\Ht@ %% useful info for Clark
       \LLXtoks@={#1}\LLYtoks@={#2}%% useful for Oz
       \URXtoks@={#3}\URYtoks@={#4}%% useful for epsf.tex
      \ifPSOrigin\XShift@=-#1bp\YShift@=-#2bp\fi 
     }

  %%% \SetEPSFDirectory 
   %
   \def\G@bbl@#1{}
   \bgroup
     \global\edef\OtherB@ckslash{\EX@\G@bbl@\string\\}
   \egroup

  \def\SetEPSFDirectory{%  Part 1
           \bgroup\PunctOther@\relax
           \let\\\OtherB@ckslash
           \SetEPSFDirectory@}

 \def\SetEPSFDirectory@#1{% Part 2
    \edef\temp@{#1}%
    \Trim@0\temp@ @%  result in \Trimtoks@
    \global\toks1\EX@{\the\Trimtoks@ }\relax
    \egroup
    \EPSFDirectorytoks@=\toks1
    }

 \def\SetEPSFSpec@{%
     \bgroup
     \let\\=\OtherB@ckslash
     \global\edef\EPSFSpec@{%
        \the\EPSFDirectorytoks@\the\EPSFNametoks@}%
     \global\edef\EPSFSpec@{\EPSFSpec@}%
     \egroup}

 %%% Is trimming done percentage-wise??
 \newif\ifPCT@ 

 \def\PCTtrueErrTst@{}%
 \def\PCTfalseErrTst@{}%

 %%%% Trimming figure dimensions (source)
   %%
  %%% \TrimSide@ 
   %% #1 is one of \TT@,...,\RT@;  #2 is arg of \TrimTop,...
 \def\TrimSide@#1#2{%
    \IN@0pct@#2@%
    \ifIN@
       \PCT@true\PCTtrueErrTst@
       \SPLIT@0pct@#2@%
       \advance#1 by \the\Initialtoks@ pt
    \else
       \PCTfalseErrTst@
       \advance#1 by #2
    \fi}
 
 \def\TrimTop#1{\TrimSide@{\TT@}{#1}}%
 \def\TrimLeft#1{\TrimSide@{\LT@}{#1}}%
 \def\TrimBottom#1{\TrimSide@{\BT@}{#1}}%
 \def\TrimRight#1{\TrimSide@{\RT@}{#1}}%

 \def\TrimBoundingBox#1{%
   \TrimTop{#1}%
   \TrimLeft{#1}%
   \TrimBottom{#1}%
   \TrimRight{#1}%
       }

 %%% Convert percentage trim to dimension trim on source (if nec)
  %% #1 is percent trim (pt is unit); #2 is source width or height
 \def\PCTtoSideTrimDims@{%
    \ifPCT@
      \Rescale\LT@\Wd@{100pt}%\showthe\LT@
      \Rescale\RT@\Wd@{100pt}%
      \Rescale\TT@\Ht@{100pt}%
      \Rescale\BT@\Ht@{100pt}%
    \fi}

 \def\TrimFigDims@{%
    \PCTtoSideTrimDims@
    \advance\Wd@ by -\LT@ 
    \advance\Wd@ by -\RT@ \RT@=\z@
    \advance\Ht@ by -\TT@ \TT@=\z@
    \advance\Ht@ by -\BT@ 
    }
  

 %%% \CalculateFigScale@
  %
  \def\ForceWidth#1{\ForcedDim@true
       \ForcedDim@@#1\ForcedHeight@false}
  
  \def\ForceHeight#1{\ForcedDim@true
       \ForcedDim@@=#1\ForcedHeight@true}

  \def\ForceOn{\ForceOn@true}
  \def\ForceOff{\ForceOn@false\ForcedDim@false}
  
  \def\CalculateFigScale@{%
            %Have default \FigScale or read \FigScale
     \ifForcedDim@\FigScale=1000pt% %% start afresh
           \ifForcedHeight@
                \Rescale\FigScale\ForcedDim@@\Ht@
           \else
                \Rescale\FigScale\ForcedDim@@\Wd@
           \fi
     \fi
     \Real{\FigScale}%
     \edef\FigSc@leReal{\the\Realtoks}%
     }
   
  \def\ScaleFigDims@{\TheScale=\FigScale
      \ifForcedDim@
           \ifForcedHeight@ \Ht@=\ForcedDim@@  \Scale\Wd@
           \else \Wd@=\ForcedDim@@ \Scale\Ht@
           \fi
      \else \Scale\Wd@\Scale\Ht@        
      \fi
      \ifForceOn@\relax\else\global\ForcedDim@false\fi
      \Scale\LT@\Scale\BT@  %%%\Scale\Wd@\Scale\Ht@
      \Scale\XShift@\Scale\YShift@
      }
      
  %%% \ShowDisplacementBoxes
   %%  shows (prints) corrected scaled and positioned
   %%  bounding boxes; for diagnostics
  %%% \HideDisplacementBoxes makes them invisible again
   %%
 \def\HideDisplacementBoxes{\global\def\FrameSpider##1{\null}}
 \def\ShowDisplacementBoxes{\global\def\FrameSpider##1{##1}}
 \let\HideFigureFrames\HideDisplacementBoxes %% some synonyms
 \let\ShowFigureFrames\ShowDisplacementBoxes
 \ShowDisplacementBoxes
 
  %%% \hSlide#1, \vSlide#1
   %%
 \def\hSlide#1{\advance\XSlide@ by #1}
 \def\vSlide#1{\advance\YSlide@ by #1}
 
  %%% \SetInkShift@, \InkShift@#1
   %%
  \def\SetInkShift@{%
            \advance\XShift@ by -\LT@
            \advance\XShift@ by \XSlide@
            \advance\YShift@ by -\BT@
            \advance\YShift@ by -\YSlide@
             }
   %
  \def\InkShift@#1{\Shifted@{\Scrunched{#1}}}
 
  %%% \CleanRegisters@
   %
  \def\CleanRegisters@{%
      \globaldefs=1\relax
        \XShift@=\z@\YShift@=\z@\XSlide@=\z@\YSlide@=\z@
        \TT@=\z@\LT@=\z@\BT@=\z@\RT@=\z@
      \globaldefs=0
      \global\PCT@false}

 \def\ReportEPSF@{%
  \ms@g{ *** Box composed for the EPS file \the\EPSFNametoks@}%
  \edef\temp@{\the\EPSFDirectorytoks@}%
  \ifx\temp@\LastEPSFDirectory@\else
    \WrLog@{--- change EPSF directory ==> \temp@}%
    \global\let\LastEPSFDirectory@\temp@
  \fi
  \def\temp@{}%
  \ifPCfmt@ \edef\temp@{\space \pct@@\space EPSP binary}%
  \fi
  %\show\temp@
  \WrLog@\EX@{\the\EPSFNametoks@\temp@}%
  }

 \gdef\LastEPSFDirectory@{}%% initialize

 %%% Special syntax for several drivers. The macros 
  %% \SetTexturesEPSFSpecial  %% Textures 
  %% \SetUnixCoopEPSFSpecial %% dvi2ps early unix 
  %% \SetNiponUnixEPSFSpecial %% dvi2ps j-version 
  %% \SetBechtolsheimDVI2PSEPSFSpecial and 
  %% \SetBechtolsheimDVITPSEPSFSpecial %% by S.P.Bechtolsheim
  %% \SetLisEPSFSpecial %% dvi2ps by Tony Lis
  %% \SetepsfEPSFSpecial %% ******* will work for many:
  %% ----- dvips, the Y&Y drivers, DirectTeX, CMacTeX, OzTeX>=1.7
  %% \SetRokickiEPSFSpecial %% dvips by Tom Rokicki
  %% \SetDVIPSoneEPSFSpecial %% DVIPSONE of Y&Y same as
  %% \SetDVIWindowEPSFSpecial %% DVIWindow of Y&Y 
  %% \SetDirectTeXEPSFSpecial %% DirectTeX by Wilfried Ricken
  %% \SetCMacTeXEPSFSpecial %% CMacTeX by Tom Kiffe
  %% \SetOzTeXEPSFSpecial %% OzTeX (>=1.7) by Andrew Trevorrow
  %% -----
  %% \SetOldOzTeXEPSFSpecial %% OzTeX 1.42--1.6x 
  %% \SetPSprintEPSFSpecial %% PSprint by Andrew Trevorrow
  %%  --- also for OzTeX versions <= 1.41 !!
  %% \SetArborEPSFSpecial %% ArborTeX DVILASER/PS
  %% \SetClarkEPSFSpecial %% dvitops by James Clark
  %% \SetBeebeEPSFSpecial %% DVIALW by N. Beebe
  %% \SetNorthlakeEPSFSpecial %% Northlake Software
  %% \SetStandardEPSFSpecial %% Nonexistant: Placebo used
  %%
  %% These many drivers are supported roughly
  %% by (re-)defining the macro \EPSFSpecial#1#2, where
  %% #1 = EPS file pathname (use \\ for the letter backslash)
  %% #2 = scale in mils 
  %% Be wary of using strange characters in pathnames!
 
 %% Textures, Blue Sky Research, Barry Smith
 \def\SetTexturesEPSFSpecial{\PSOriginfalse%\PSOrigintrue
  \gdef\EPSFSpecial##1##2{\relax
    \edef\specialtemp{##2}%
    \SPLIT@0.@\specialtemp.@\relax
    \special{illustration ##1 scaled
                        \the\Initialtoks@}}}
 
  %% Unix : dvi2ps by:  Mark Senn, Stephan  Bechtolsheim,  
   % Bob  Brown, Richard, Furuta, James Schaad, Robert  Wells, 
   % Norm Hutchinson, Neal Holt, Scott Jones, Howard Trickey.
   % Introduced by B. Horn <bkph@ai.mit.edu>
  \def\SetUnixCoopEPSFSpecial{\PSOrigintrue % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{psfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  %% Unix : dvi2ps j-version 1.91j
   % Introduced by Takashi Matsushita matsu@tmuds1.phys.metro-u.ac.jp
   % Takafumi Sakurai, sakurai@math.s.chiba-u.ac.jp
   % Kazuhiro Kazama,  kazama@square.ntt.jp
  \def\SetNiponUnixEPSFSpecial{\PSOriginfalse % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{epsfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  %% Unix : dvi2ps j-version 1.91j
   % Introduced by Takashi Matsushita matsu@tmuds1.phys.metro-u.ac.jp
   % Takafumi Sakurai, sakurai@math.s.chiba-u.ac.jp
   % Kazuhiro Kazama,  kazama@square.ntt.jp
  \def\SetNiponUnixEPSFSpecial{\PSOriginfalse % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      {\dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\Aux@\dimen6 
      \Real{\dimen6}% 
      \XShift@=0bp\YShift@=\the\Realtoks bp
      \Shifted@{\special{epsfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}}}
%%%      \Shifted@{\special{illustration ##1}}}}}



  %% dvi2ps and dvitps by S.P. Bechtolsheim,
   % Introduced by B. Horn <bkph@ai.mit.edu> and Carl.M.Jones, 
   % testing by R. Evans <Robert@cm.cardiff.ac.uk>
   % and George Denk <denk@mathematik.tu-muenchen.de>
   % Note that a prolog file psfig.pro
   % specific to the driver should be available.
  \def\SetBechtolsheimEPSFSpecial@{%% tool macro only
   \PSOrigintrue
   \special{\DriverTag@ Include0 "psfig.pro"}%
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4} %% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%% convert dimen to real
      \special{\DriverTag@ Literal "10 10 0 0 10 10 startTexFig
           \the\mag\space 1000 div 
           dup 3.25 neg mul 1 index .25 neg mul translate %% correction line
           \Aux@\space mul dup scale "}%
      \special{\DriverTag@ Include1 "##1"}%
      \special{\DriverTag@ Literal "endTexFig "}%
        }}

  \def\SetBechtolsheimDVITPSEPSFSpecial{\def\DriverTag@{dvitps: }%
      \SetBechtolsheimEPSFSpecial@}

  \def\SetBechtolsheimDVI2PSEPSFSSpecial{\def\DriverTag@{DVI2PS: }%
      \SetBechtolsheimEPSFSpecial@}

  %% dvi2ps by Tony Lis,
   % implantations? ; dates?; availability?
   % Introduced by B. Horn <bkph@ai.mit.edu>
  \def\SetLisEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{pstext="10 10 0 0 10 10 startTexFig\space
           \the\mag\space 1000 div \Aux@\space mul 
           \the\mag\space 1000 div \Aux@\space mul scale"}%
      \special{psfile=##1}%
      \special{pstext=endTexFig}%
        }}

  %% dvips by Tom Rokicki; free driver in portable C 
   % Introduced by W.D. Neumann <neumann@mps.ohio-state.edu>
  \def\SetRokickiEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 10\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{psfile="##1"\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  \def\SetInlineRokickiEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{ps::[begin] 10 10 0 0 10 10 startTexFig\space
           \the\mag\space 1000 div \Aux@\space mul 
           \the\mag\space 1000 div \Aux@\space mul scale}%
      \special{ps: plotfile ##1}%
      \special{ps::[end] endTexFig}%
        }}

  %%% Rokicki's epsf.tex syntax.
   %% Is followed by DirectTeX (W. Ricken)
   %% CMacTeX (Tom Kiffe), Y&Y's drivers, and OzTeX >= 1.7
  \def\SetepsfEPSFSpecial{\PSOriginfalse 
   \gdef\epsfclipstring{}%
   %% make { clip} if you want to clip (non-portably)
   %% down to the PostScript bounding box. (OzTeX refuses this)
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert mil scale to dimen
      \divide\dimen4 by 100 %% 1000ths to 10ths
      \Real{\dimen4}%% dimens 0,2 used here
      \edef\theScaleholder@{\the\Realtoks}%
      \dimen6=\the\URXtoks@ pt %% not bp!!!!!
      \advance\dimen6 by -\the\LLXtoks@ pt
      %% \dimen6 now (bbox width) * points
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}%% dimens 0,2 used here
      \edef\Temp{\the\Realtoks}%
      %% \Temp now holds decimal real no
      %% 10 times desired "real width" (rwi) in bp
      \dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}% 
      \special{PSfile=##1\space
        llx=\the\LLXtoks@\space lly=\the\LLYtoks@\space
        urx=\the\URXtoks@\space ury=\the\URYtoks@\space 
        rwi=\Temp\space rhi=\the\Realtoks
        \epsfclipstring}%
        }}
    %% Rmk: Under epsf.tex only multiples of (1/10) bp available widths
    %% but considerably more accuracy available here.

  \let\SetDirectTeXEPSFSpecial=\SetepsfEPSFSpecial
  \let\SetCMacTeXEPSFSpecial=\SetepsfEPSFSpecial
  \let\SetOzTeXEPSFSpecial=\SetepsfEPSFSpecial
 %%% \SetOzTeXEPSFSpecial only for OzTeX in versions 1.7 and later

 %%% Old OzTeX versions (1.42 to 1.6), by Andrew Trevorrow.
 %%% For versions <= 1.41 use \SetPSprintEPSFSpecial below.
 %%  Nota: Version 1.42--1.5 may give
 %%  spurious "offpage" error notices on printing.
 \def\SetOldOzTeXEPSFSpecial{\PSOrigintrue
 \gdef\EPSFSpecial##1##2{%
 \dimen4=##2pt%% convert real to dimen
 \divide\dimen4 by 1000\relax
 \Real{\dimen4}%% dimens 0,2 used here
 \edef\Aux@{\the\Realtoks}%% convert dimen to real
 \special{epsf=\string"##1\string"\space scale=\Aux@}%
 }} 

 %% PSprint, by AndrewTrevorrow for VaX VMS
 %% and OzTeX versions <= 1.41  
  % tested 2-91 by Max Calviani <ISICA@ASTRPD.infn.it>
  \def\SetPSprintEPSFSpecial{\PSOriginfalse % artifice; see below
   \gdef\EPSFSpecial##1##2{%note order
     \special{##1\space 
       ##2 1000 div \the\mag\space 1000 div mul
       ##2 1000 div \the\mag\space 1000 div mul scale
       \the\LLXtoks@\space neg \the\LLYtoks@\space neg translate
       }}}

 %% DVILASER/PS driver originally written by David Fuchs
  % marketed and supported by ArborTeXt  535 W. William St.
  % Suite 300, Ann Arbor, MI 48103, U.S.A
  % (313) 996-3566 (313) 996-3573
  % help@arbortext.com, Andrew Dobrowolski
 \def\SetArborEPSFSpecial{\PSOriginfalse % check!
   \gdef\EPSFSpecial##1##2{%
     \edef\specialthis{##2}%
     \SPLIT@0.@\specialthis.@\relax % suppress decimals (nec!)
     \special{ps: epsfile ##1\space \the\Initialtoks@}}}

 %% dvitops, (c) James Clark <jjc@jclark.uucp>
  % public domain; distributed by UK TeX Archive
  % computers: unix, msdos, vms, primos and vm/cms,
  % introduced by S. Ratz <spqr@uk.ac.southampton.ecs>
 \def\SetClarkEPSFSpecial{\PSOriginfalse % please test!
   \gdef\EPSFSpecial##1##2{%
     \Rescale {\Wd@@}{##2pt}{1000pt}%
     \Rescale {\Ht@@}{##2pt}{1000pt}%
     \special{dvitops: import 
           ##1\space\the\Wd@@\space\the\Ht@@}}}

 %% DVIPSONE, for PC compatibles
  % Y&Y, 106 Indian Hill, Carlisle MA 01741, USA
  % (508) 371-3286
  % (introduced by B. Horn <bkph@ai.mit.edu>)
  \let\SetDVIPSONEEPSFSpecial\SetepsfEPSFSpecial
  \let\SetDVIPSoneEPSFSpecial\SetepsfEPSFSpecial
  \let\SetDVIWindowEPSFSpecial\SetepsfEPSFSpecial

 %% DVIALW by N. Beebe, public domain 
  % DVI Driver Distribution, Center for Scientific Computing,
  % Department of Mathematics, 220 South Physics Building,
  % University of Utah, Salt Lake City, UT 84112, USA
  % (introduced by B. Horn <bkph@ai.mit.edu>)
  % Proposed standard; see TUGboat article 1993.
  \def\SetBeebeEPSFSpecial{%please test!
   \PSOriginfalse% 
   \gdef\EPSFSpecial##1##2{\relax
    \special{language "PS",
      literal "##2 1000 div ##2 1000 div scale",
      position = "bottom left",
      include "##1"}}}
  \let\SetDVIALWEPSFSpecial\SetBeebeEPSFSpecial

 %% Northlake software
  \def\SetNorthlakeEPSFSpecial{\PSOrigintrue
   \gdef\EPSFSpecial##1##2{%
     \edef\specialthis{##2}%
     \SPLIT@0.@\specialthis.@\relax % suppress decimals (nec!)
     \special{insert ##1,magnification=\the\Initialtoks@}}}

 %%% MultiSpecial for Textures PLUS drivers supporting epsf.tex
 %%% syntax of Rokicki (one of many dvips syntaxes)
  %% *experimental* 2-95
  \def\SetEPSFMultiSpecial{\PSOriginfalse
   \csname newwrite\endcsname \EPSFile@@
   \immediate\openout\EPSFile@@ illustration
   \immediate\write\EPSFile@@ {\P@S@}%
   \immediate\write\EPSFile@@ {\pct@@ Remark: Tranquilizer for OzTeX...}%
   \immediate\closeout\EPSFile@@
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert mil scale to dimen
      \divide\dimen4 by 100 %% 1000ths to 10ths
      \Real{\dimen4}%% dimens 0,2 used here
      \edef\theScaleholder@{\the\Realtoks}%
      \dimen6=\the\URXtoks@ pt %% not bp!!!!!
      \advance\dimen6 by -\the\LLXtoks@ pt
      %% \dimen6 now (bbox width) * points
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}%% dimens 0,2 used here
      \edef\Temp{\the\Realtoks}%
      %% \Temp now holds decimal real no
      %% 10 times desired "real width" (rwi) in bp
      \dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}% 
      \special{PSfile=##1\space
        llx=\the\LLXtoks@\space lly=\the\LLYtoks@\space
        urx=\the\URXtoks@\space ury=\the\URYtoks@\space 
        rwi=\Temp\space rhi=\the\Realtoks}%
    \edef\specialtemp{##2}%
    \SPLIT@0.@\specialtemp.@\relax
    \special{illustration ##1 scaled \the\Initialtoks@}%
    }}

 \def\SetStandardEPSFSpecial{%
   \gdef\EPSFSpecial##1##2{%
     \ms@g{}
     \ms@g{%
       !!! Sorry! There is still no standard for \noexpand
       \special EPSF integration !!!}%
     \ms@g{%
      --- So you will have to identify your driver using a command}%
     \ms@g{%
      --- of the form \string\Set...EPSFSpecial, in order to get}%
     \ms@g{%
      --- your graphics to print.}%  
     \ms@g{%
      Update your BoxedEPSF and read the documentation.}%
     \ms@g{}
     \gdef\EPSFSpecial####1####2{}
     }}

  \SetStandardEPSFSpecial %% currently gives warning
 
 \let\wlog\wlog@ld %%restore logging 

 \catcode`\:=\C@tColon
 \catcode`\;=\C@tSemicolon
 \catcode`\?=\C@tQmark
 \catcode`\!=\C@tEmark
 \catcode`\"=\C@tDqt

 \catcode`\@=\EPSFCatAt

  \endinput

%%%%%%%%%%%% ASCII Character test
 %
 %       Upper case letters: ABCDEFGHIJKLMNOPQRSTUVWXYZ
 %       Lower case letters: abcdefghijklmnopqrstuvwxyz
 %                                   Digits: 0123456789
 % Square, curly, angle braces, parentheses: [] {} <> ()
 %           Backslash, slash, vertical bar: \ / |
 %                              Punctuation: . ? ! , : ;
 %          Underscore, hyphen, equals sign: _ - =
 %                Quotes--right left double: ' ` "
 %"at", "number" "dollar", "percent", "and": @ # $ % &
 %           "hat", "star", "plus", "tilde": ^ * + ~
 %
 %%%%%%%%%%%%%%%%%%%%%%%%
 %
 % Une seule erreur de transmission peut empoisoner un programme!
 %
 % A single transmission error can poison a whole program.
 %
 %%%%%%%%%%%%%%%%%%%%%%%%

murphy.tex

\documentclass[letterpaper,12pt]{article}
\usepackage[hideboxes]{boxedeps}

\usepackage{latexsym}
%alter this command for graphicx 
\newcommand{\pic}[2]{\BoxedEPSF{#1 scaled #2}}

\def\CA{{\cal A}}
\def\CC{{\cal C}}
\def\CS{{\cal S}}

\def\Xor{\pic{Xor.ART} {500}}
\def\Yor{\pic{Yor.ART} {500}}
\def\Ior{\pic{Ior.ART} {500}}
\def\Sigmaior{\pic{Sigmaior.ART} {500}}
\def\Pij{\pic{Pij.ART} {500}}
\def\Wij{\pic{Wij.ART} {500}}


\def\Idor{\pic{Idor.ART} {500}}
\def\Rcurlor{\pic{Rcurlor.ART} {500}}

\def\Murphy{\pic{Murphy.ART} {500}}
\def\Murphysum{\pic{Murphysum.ART} {500}}
\def\Murphycurl{\pic{Murphycurl.ART} {500}}
\def\Murphysumless{\pic{Murphysumless.ART} {500}}

\def\Gnplus{\pic{Gnplus.ART} {500}}
\def\Gshift{\pic{Gshift.ART} {500}}
\def\Gcross{\pic{Gcross.ART} {500}}

\def\Aid{\pic{Aid.ART} {500}}
\def\Aa{\pic{Aa.ART} {500}}
\def\Aaa{\pic{Aaa.ART} {500}}
\def\Aainv{\pic{Aainv.ART} {500}}
\def\Annulus{\pic{Annulus.ART} {500}}
\def\Aeh{\pic{Aeh.ART} {500}}
\def\Ahe{\pic{Ahe.ART} {500}}
\def\Wn{\pic{Wn.ART} {500}}
\def\Vn{\pic{Vn.ART} {500}}
\def\WC{\pic{WC.ART} {500}}

\def\XC{\pic{XC.ART} {500}}
\def\XCYC{\pic{XCYC.ART} {500}}
\def\XYC{\pic{XYC.ART} {500}}
\def\TC{\pic{TC.ART} {500}}
\def\PhiX{\pic{PhiX.ART} {500}}
\def\PsiD{\pic{PsiD.ART} {500}}
\def\PsiX{\pic{PsiX.ART} {500}}
\def\PsiXY{\pic{PsiXY.ART} {500}}
\def\PsiXPsiY{\pic{PsiXPsiY.ART} {500}}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}

\newenvironment{remark}{\par\smallskip%
\noindent\textbf{Remark.}\  }%
{\par\smallskip}

\newenvironment{Def}{\par\smallskip%
\noindent\textbf{Definition.}\  }%
{\par\smallskip}

\newenvironment{proof}[1][{}]{\par\smallskip%
\noindent\textit{Proof #1: }\  }
{\hfill$\Box$\par\smallskip}


\begin{document}
\begin{center}
      {\Large\bf Skein theory and the Murphy operators}

\bigskip
HUGH R. MORTON 

\medskip
 {\it Department of Mathematical Sciences, University of Liverpool,\\
     Peach St, Liverpool, L69 7ZL, England. }\\
{\tt morton@liv.ac.uk}
\end{center}
\vskip 1cm

\begin{abstract}
The Murphy operators in the Hecke algebra
$H_n$ of type $A$ are explicit commuting
elements whose sum generates the centre.
 They can be represented by simple tangles in the Homfly skein theory
version of $H_n$.
 In this paper I present a single tangle which represents their sum,
and which is obviously central. As a consequence
 it is possible to identify a natural basis for the Homfly skein of the
annulus, $\CC$.

Symmetric functions of the Murphy operators are also central in $H_n$.
I define geometrically a  homomorphism
from
$\CC$ to the centre of each algebra $H_n$, and
find an element in $\CC$, independent of $n$, 
whose image  is the $m$th power sum of the
Murphy operators. Generating function
techniques are used to describe images of
other elements of
$\CC$ in terms of the Murphy operators, and to
demonstrate  relations among other natural
skein elements.

\phantom{some stuff to separate the keywords from the abstract}

\noindent {\em Keywords}: skein theory; Murphy operators; power sums;
symmetric functions; annulus; Hecke algebras. 
\end{abstract}


\section*{Introduction.}

The Hecke algebra $H_n$ of type $A$ is  a deformed version of
the group algebra ${\bf C}[S_n]$ of the symmetric
group. It has a simple
skein theory model in terms of $n$-tangles and
the Homfly skein relations.


Certain sums of transpositions,
\[m(j)=\sum_{i=1}^{j-1} (i\,j) \in
{\bf C}[S_n],\ j=2,\ldots,n,\]
 known as Jucys-Murphy elements,
appeared in the work of Jucys \cite{Jucys} in
1974, and later in work of Murphy \cite{Murphy}. 
These elements all commute, and every symmetric
polynomial in them can be shown to lie in the
{\em centre} of the algebra ${\bf C}[S_n]$. For
example
$m(2)=(1\,2), m(3)=(1\,3)+(2\,3)$ and
\[m(2)m(3)=(1\,3\,2)+(1\,2\,3)=m(3)m(2).\]



Dipper and James \cite{Dipper} used a simple deformation of the
transpositions  to define analogous
elements $M(j)\in H_n$, which they called the
{\em Murphy operators}. These elements again all
commute, and symmetric polynomials in them belong
to the centre of $H_n$. Dipper and James
 showed that for generic values of
the deformation parameter these account for the
whole of the centre. 
Katriel, Abdessalam and Chakrabarti \cite{Katriel}
noted that in fact any central element can be
expressed as a polynomial in just the sum
$M=\sum_{j=2}^n M(j)$ of the Murphy operators.
This is a stronger result than for the
non-deformed algebra ${\bf C}[S_n]$, although
the centre has the same dimension
in each case, given by $\pi(n)$, the number of
partitions of $n$.

In this paper  I present  a skein theory
version of the Murphy operators and  their sum, finishing with an
elegant representation of their power sums and other symmetric
polynomials.  Starting in section 2 with a choice of
$n$-string braids corresponding to the
transpositions I  exhibit a  braid $T(j)$
representing each of the individual Murphy
operators $M(j)$, following Ram \cite{Ram}, and then
  a very natural simple
$n$-tangle $T^{(n)}$ which  represents their sum $M$, up to
a linear combination with the identity element in
$H_n$ in each case. From the tangle viewpoint it becomes immediately
clear that $M$ is central. 


The Homfly skein of the
annulus, $\CC$, has been exploited for many
years, for example in the work of Jun Murakami,
and myself and Short, in constructing and analysing
Homfly-based  invariants  by the use
of satellites. These invariants cover the same ground
as the quantum $sl(N)$ invariants for all choices
of module and have been developed variously in
algebraic and skein theoretic ways, for example
by Wenzl \cite{Wenzl}, Aiston and myself \cite{AistonMorton} and
 Blanchet
\cite{Blanchet}. 

 In section 3 I  show how the tangle view of the
Murphy operators gives a direct means for identifying the best
linear basis for $\CC$ when
studying these quantum invariants and establishing
its multiplicative properties. I then introduce a rather overlooked
homomorphism from $\CC$ to the centre of $H_n$. Using generating
function methods I show how to realise other symmetric functions of
the Murphy operators as images of explicit elements of $\CC$. The
principal result is the identification of an element $P_m\in\CC$,
independent of $n$, whose image determines the $m$th power sum of the
Murphy operators in $H_n$.

Section 4 gives the final details of the results of section 3, based
on a further underused skein, $\CA$, whose unexpected algebraic
properties allow for some satisfyingly clean proofs.

\section{The skein models.}

The skein theory model of $H_n$ is
based on the framed Homfly skein relations, in
their simplest form
\begin{eqnarray*}
\Xor\  -\ \Yor& =&(s-s^{-1})\ \Ior\\
\mbox{and }\  \Rcurlor &=&v^{-1}\ \Idor.
\end{eqnarray*}

The Homfly skein ${\cal S}(F)$ of a planar
surface $F$, with some designated input and
output boundary points, is defined as linear
combinations of oriented tangles in $F$, modulo
these two local relations, and Reidemeister moves
II and III. The coefficient ring can be taken
as $\Lambda={\bf Z}[v^{\pm1}, s^{\pm1}]$ with
powers of $s^k-s^{-k}$ in the denominators.

Every skein admits a {\em mirror map},
$\overline{\phantom{w}}:\CS(F)\to\CS(F)$ induced by switching all
crossings in a tangle, coupled with inverting $v$ and $s$ in $\Lambda$.

Write $R_n^n(v,s)$ for the skein ${\cal S}(F)$ of
$n$-tangles,  where $F$
is a rectangle with $n$ inputs at the bottom and
$n$ outputs at the top. Composing $n$-tangles
induces a product which makes $R_n^n(v,s)$ into
an algebra. It has a linear basis of $n!$
elements, and is isomorphic to the Hecke algebra
$H_n(z)$. This algebra has a presentation with
generators
$\{\sigma_i\}, i=1,\ldots,n-1$ satisfying the
braid relations
\begin{eqnarray*}
\sigma_i\sigma_j&=&\sigma_j\sigma_i,
\quad |i-j|>1,\\
\sigma_i\sigma_{i+1}\sigma_i&=&\sigma_{i+1}\sigma_i\sigma_{i+1},
\end{eqnarray*} and the quadratic relations
$\sigma_i^2=z\sigma_i+1$.

In $R_n^n$ the generators are the elementary
braids
\[\sigma_i\ =\ \Sigmaior\]
and the parameter $z$ is $s-s^{-1}$, giving the
alternative form $(\sigma_i-s)(\sigma_i+s^{-1})=0$
for the quadratic relations.

In the special case $z=0$ the Hecke algebra
reduces to ${\bf C}[S_n]$, with $\sigma_i$
becoming the transposition
$(i\;i+1)$. For general $i,j$
we can view the transposition $(i\,j)$
 diagrammatically \[(i\,j)\ =\ \Pij\ .\] The corresponding
positive permutation braid \[\omega_{(i\,j)}\ =\ \Wij\] with
all crossings positive realises this
transposition, as in \cite{AistonMorton}. Regarded via
$R_n^n$  as an element of $H_n(z)$, it
becomes the transposition $(i\,j)$ when we set
$z=0$.

\begin{Def} The {\em Murphy operator} $M(j)\in R^n_n,\ j=2,\ldots,n$ is
defined as
\[M(j)=\sum_{i=1}^{j-1}\omega_{(i\,j)}.\]
\end{Def}
These elements certainly project to the
Jucys-Murphy elements $m(j)$ in ${\bf C}[S_n]$.
 With a
bit of algebraic work they  can be shown to
commute, and their sum
\[M=\sum_{j=2}^n M(j)\quad
=\sum_{i<j}\omega_{(i\,j)}
\] can be shown to lie in the centre of $R^n_n$. These facts are
immediately obvious from the skein representatives $T(j)$ and
$T^{(n)}$, discussed shortly. 

\subsection{Variants of the Hecke algebras.}

A simple adjustment of the skein relations, as in \cite{AistonMorton},
allows for a skein model of $H_n$ whose parameters can be readily
adapted to match any of the different appearances of the algebra.

Extend the coefficient ring to include an invertible parameter $x$,
and define the skein $R^n_n(x,v,s)$ by linear combinations of oriented
$n$-tangles modulo the relations
\begin{eqnarray*}
x^{-1}\Xor\  -\ x\Yor& =&(s-s^{-1})\ \Ior\\
\mbox{and }\  \Rcurlor &=&xv^{-1}\ \Idor.
\end{eqnarray*}

There is a natural algebra homomorphism $R_n^n(v,s) \to R^n_n(x,v,s)$
 induced by replacing each tangle $T$
in the skein 
$R_n^n(v,s)$ by $x^{wr(T)}T$ in $R^n_n(x,v,s)$, where $wr(T)$ is the
writhe of $T$.

We then get corresponding versions of the Murphy elements in our new
skein; for example the new version of $T(j)$ is represented by
$x^{2j-2}T(j)$ in $R^n_n(x,v,s)$.

The parameters in $R^n_n(x,v,s)$ can be adjusted to match the exact
version of $H_n$ under study. Taking $x=s,q=s^2, v=1$ we
get the common algebraist's version $H_n(q)$ of \cite{Dipper}, while
the endomorphism rings arising from the fundamental representation of
$sl(N)_q$ and its standard $R$-matrix correspond to the choice
$x=s^{-1/N}, v=s^{-N}$ with $s=e^{h/2}$.

The choice $x=v$ gives the Homfly skein relation which is invariant
under Reidemeister move I, and $x=q^r, s=q, v=a$ gives the version
used by Kawagoe in \cite{Kawagoe}.

 The elements
$M(j)$ above do indeed
  become the Murphy operators used by 
Dipper and James under the isomorphism of
$R^n_n(v,s)$ with the algebraic version $H_n(q)$ of the Hecke algebra
where $x=s,q=s^2, v=1$.

\section{Geometric views of the Murphy operators.}
The Murphy operators $M(j)$ and their sum $M$ have simple skein
representatives as shown here,  using $R^n_n(v,s)$ as the model for
the Hecke algebra $H_n$.

\begin{theorem} \label{Murphy} {\rm (Ram)}. The Murphy operator
$M(j)$ can be represented in $H_n$ by a single
braid $T(j)$, up to linear combination with the
identity.
\end{theorem}

\begin{theorem} \label{Sum}  {\rm (Morton)}. The sum $M$ of the Murphy operators can be
represented in $H_n$ by a single tangle $T^{(n)}$, again up to linear
combination with the identity.

\end{theorem}


\begin{proof} [ of theorem \ref{Murphy}] In \cite{Ram} Ram notes  that $M(j)$ can
{\em almost} be represented by the single braid
$T(j)$ pictured, where `almost' means that $M(j)$
is a linear combination of $T(j)$ and the
identity.

Set \[T(j)\ =\ \Murphy \] in $H_n$, for each $j\le n$ including the
case
$T(1)=1$. Skein theory shows quickly that $T(j)-1=zM(j)$, giving
\[M(j)=\frac{T(j)-1}{s-s^{-1}}.\]
\end{proof}

The elements $T(j)$ will do equally well in place
of $M(j)$, so long as $z\neq0$, in other words
away from ${\bf C}[S_n]$.

 It is clear
geometrically that the elements $T(j)$ all
commute.  It is not immediately clear
that the {\em sum} of these elements is in the centre
of $H_n$, although their product is the full
twist, a well-known central element.

\begin{proof}  [of theorem \ref{Sum}]
Set \[T^{(n)}=\Murphysum \]in $R_n^n\cong H_n(z)$, with the coefficient
ring extended to include $v^{\pm 1}$.

Apply the skein relation at one crossing to get
\[ \Murphysum\ =\ \Murphysumless \ + \ z\ \Murphycurl.\] Hence
$T^{(n)}=T^{(n-1)}+zv^{-1}T(n)$, using the
standard inclusion of $H_{n-1}$ in $H_n$ to interpret $T^{(n-1)}$ as
an element of $H_n$. Then
\[T^{(n)}-T^{(0)} =zv^{-1}\sum_{j=1}^nT(j),\]
 by induction on $n$. Here $T^{(0)}$ is just a scalar multiple of the
identity, represented by a single disjoint simple loop alongside the
identity braid. Since a trivial loop in any Homfly skein contributes the
scalar $\delta=(v^{-1}-v)/z$ we can rewrite the equation as 
\[T^{(n)}-\frac{v^{-1}-v}{z}=zv^{-1}\sum_{j=1}^n
T(j)=z^2v^{-1}M+nzv^{-1}.\]

Then
\[T^{(n)}=z^2v^{-1}M+\left(nzv^{-1}+\frac{v^{-1}-v}{z}\right)\times
1,\] and hence, again away from $z=0$, we can write $M$ as a linear
combination of
$T^{(n)}$ and $1$.
\end{proof}



From this representation it is 
quite obvious that $T^{(n)}$, and thus $M$,  is in the
centre of
$H_n$.

There is a known set of idempotent elements,
$E_\lambda$, one for each partition $\lambda$ of
$n$, which were originally described
algebraically by Gyoja \cite{Gyoja}. Skein pictures of
these based on the Young diagram for $\lambda$
were given by Aiston and myself \cite{AistonMorton}. We
 showed there, in a skein based counterpart of the results of
\cite{Katriel},  that
$ME_\lambda=m_\lambda E_\lambda$ for an explicit scalar $m_\lambda$
and that
$m_\lambda\ne m_\mu$ when $\lambda\ne\mu$. It
follows at once that
$T^{(n)}E_\lambda=t_\lambda E_\lambda$, where the
scalars $t_\lambda$ are different for each
partition $\lambda$. 

\section{The skein of the annulus.}

The Homfly
skein of the annulus, ${\cal C}$, as discussed in
\cite{Turkey} and originally in \cite{Turaev},  is defined as linear combinations of
diagrams in the annulus, modulo the Homfly skein relations. The
element $X\in\CC$ will be indicated on a diagram as
\[\XC\ .\]

 We shall
make use of  the results of section 2  to study $\CC$, and also to find
simple skein representatives for the power sums of the Murphy
operators in $H_n$ in a way which is independent of
$n$.


The best known relation of $\CC$ with the Hecke algebra $H_n$ is the
closure map
$R_n^n\to {\cal C}$, induced by taking a tangle $T$ to its
closure $\hat {T}$ in the annulus, defined by \[\hat {T}\ =\ \TC\ .\]
This is a linear map, whose image we will call ${\cal C}_n$.

The skein ${\cal C}$ has a product induced by
placing one annulus outside another,  under which $\cal C$ becomes a
commutative algebra;
\[\XYC\ =\ \XCYC\ .\]
Write $A_m \in {\cal C}$ for the closure of
the
$m$-braid \[\sigma_{m-1}\cdots \sigma_2\sigma_1\ =\ \Gcross\ .\]
An explicit  spanning set of
${\cal C}_n$ consists of the monomials in 
$\{A_j\}$ of total weight $n$, where
$A_j$ has weight $j$. Such monomials
correspond bijectively with partitions of $n$.


A very simple skein theory construction
determines a natural linear map $\varphi:{\cal
C}\to{\cal C}$, induced by taking a diagram $X$
in the annulus and linking it once with a simple
loop to get
\[\varphi(X)\ =\ \PhiX\ .\]

It is instructive to look at the eigenvectors of
$\varphi$.  Clearly $\varphi$ carries ${\cal
C}_n$ into itself. What is more, if we take an
element $S\in H_n$ with closure $\hat{S}\in {\cal
C}_n$ and compose it with the central element $T^{(n)}$ then
$ST^{(n)}$ has closure $\varphi(\hat{S})$. 

\begin{theorem} \label{Phi} The eigenvalues of $\varphi|{\cal C}_n$ are
all distinct.
\end{theorem}

\begin{proof}Take
$S=E_\lambda$ to see that the closure of $T^{(n)}E_\lambda$ is
$t_\lambda\hat{E_\lambda}$ and also
$\varphi(\hat{E_\lambda})$. The element
$Q_\lambda=\hat{E_\lambda}\in{\cal C}_n$ is then
an eigenvector of $\varphi$ with eigenvalue
$t_\lambda$. Now there are $\pi(n)$ of these eigenvectors,  (the number
of partitions of $n$), and
the eigenvalues are all distinct, by \cite{AistonMorton}. Since 
${\cal C}_n$ is spanned by $\pi(n)$ elements we
can deduce that the elements $Q_\lambda$ form a
basis for ${\cal C}_n$, and that the eigenspaces of $\varphi$ are all
$1$-dimensional.
\end{proof}

 It follows that {\em any} element of ${\cal C}_n$
which is an eigenvector of $\varphi$ must be a multiple of
some $Q_\lambda$.

The elements $h_i\in {\cal C}$, discussed below, arise
 as
$Q_\lambda$ for the Young diagram $\lambda$ consisting of a
single row with $i$ cells.  Kawagoe \cite{Kawagoe} constructed an
element
$s_\lambda$ of
${\cal C}_n$ for each partition $\lambda$ of $n$,
as a Schur polynomial of the sequence $\{h_i\}$, in Macdonald's context
of symmetric functions, \cite{Macdonald}. He then used skein theory
to show that each
$s_\lambda$ is also an eigenvector of the map
$\varphi$. 
His elements
$s_\lambda$ can now be identified almost at once with the elements
$Q_\lambda$, by theorem \ref{Phi}. 


This gives an immediate proof of the multiplicative properties of the
elements $Q_\lambda$ in the skein of the annulus, since the elements
$s_\lambda$ automatically multiply according to the Littlewood-Richardson
rules for Young diagrams \cite{Macdonald}. Conversely it guarantees
that Kawagoe's elements
$s_\lambda$ are eigenvectors for a  wide range of skein maps on ${\cal
C}$ as well as $\varphi$, using the skein properties of $E_\lambda$
from
\cite{AistonMorton}. 

  Recently Lukac, \cite{Lukac}, 
 has  been able to simplify the proof
that $s_\lambda$ is an eigenvector of $\varphi$, by using algebraic
properties of the skein $\CA$ of the annulus with an input on one
boundary and an output on the other. 
Discussion and use of the skein $\CA$ in section \ref{A} 
leads to simple descriptions for other symmetric functions of the
Murphy operators, derived from a direct interplay between the
skein of the annulus and the centre of the Hecke algebras.

\subsection{Symmetric functions and the skein of the annulus.}

In this section I recall some explicit results about elements in the
Hecke algebras and their closure in $\CC$, and their interpretation in
the context of symmetric functions, following the methods of
Macdonald. This leads to a simple description in terms of generating
functions of some interrelations in $\CC$, and also of the elements to
be used later to represent the power sums of the Murphy operators.

The starting point is the description of the two simplest idempotents
in $H_n$, corresponding to the single row and column Young diagrams.
These are given algebraically in \cite{Jones}; here I  use the
skein version as described in \cite{Turkey} in terms of the positive
permutation braids $\omega_\pi,\ \pi\in S_n$. Define two
quasi-idempotents by
\[a_n=\sum_{\pi\in S_n}s^{l(\pi)}\omega_\pi, \quad b_n =\sum_{\pi\in
S_n}(-s)^{-l(\pi)}\omega_\pi,\] where $l(\pi)=wr(\omega_{\pi})$ is the
writhe of the braid $\omega_{\pi}$. 

The following result is straightforward. 
\begin{lemma} We can write \[a_n=a_{n-1}\gamma_n,\] where
$\gamma_n=1+s\sigma_{n-1}+s^2\sigma_{n-1}\sigma_{n-2}
+\cdots+s^{n-1}\sigma_{n-1}\cdots\sigma_1$. 
\end{lemma} 
We have $\gamma_{n+1}=1+s\sigma_n\gamma_n$, and also an immediate skein
relation
\[ \Gnplus\ =\ \Gshift\ +\ s^{n}\ \Gcross\ ,\] in $R^{n+1}_{n+1}$.

The next lemma is proved in \cite{AistonMorton}.
\begin{lemma} \label{Rowidem} For any braid
$\beta\in B_n$ we have $a_n\beta=\varphi_s(\beta)a_n=\beta a_n$,  where
$\varphi_s(\beta)=s^{wr(\beta)}$.
\end{lemma}
Similar results, with $s$ replaced by $-s^{-1}$, hold for $b_n$.

The element $a_n$ then satisfies
\[a_n^2=\varphi_s(a_n)a_n=\varphi_s(a_{n-1})\varphi_s(\gamma_n)a_{n}.\]

Since $\varphi_s(\gamma_n)=1+s^2+\cdots+s^{2n-2}=s^{n-1}[n]$, we have
an immediate corollary.
\begin{corollary} \label{Idempotent} We can write
\[s^{n-1}[n]h_n=h_{n-1}\gamma_n,\] where $h_n=a_n/\varphi_s(a_n)$ is
the true idempotent.
\end{corollary}

The element $h_n\in H_n$ is the idempotent which corresponds to the
single row with $n$ cells. The single column idempotent is given from
$h_n$ by using $-s^{-1}$ in place of $s$.

With a slight abuse of notation  write $h_n\in {\cal C}$
in place of $\hat{h}_n$ for the closure of this element in ${\cal C}$,
and $e_n\in{\cal C}$ for the closure of the single column
idempotent.

\begin{remark} Aiston uses  the notation $Q_{c_n}$ and
$Q_{d_n}$  in \cite{Aiston}, in place of $e_n$ and $h_n$ which are
used here to suggest
 the terminology and techniques of symmetric functions from
\cite{Macdonald}.
\end{remark}

Write \[H(t)=1+\sum_{n=1}^\infty h_n t^n\] for the generating function
of the elements $\{h_n\}$, regarded as a formal power series with
coefficients in ${\cal C}$, and similarly set
$E(t)=1+\sum_{n=1}^\infty e_n t^n$. In \cite{Aiston} Aiston showed that
\begin{theorem} \[E(-t)H(t)=1,\] as power series in ${\cal C}$.
\end{theorem}

If we regard the elements $h_n$ formally as the $n$th complete
symmetric functions in a large number $N$ of variables $x_1,
\ldots,x_N$, setting
\[H(t)=\prod_{i=1}^N \frac{1}{1-x_it}\ ,\] then $E(t)=\prod(1+x_it)$
and
$e_n$ is the $n$th elementary symmetric function in the variables.

Following Macdonald, the Schur function $s_\lambda(x_1,\ldots,x_n)$ for each
partition $\lambda$
can be expressed as a polynomial in the elements $h_n$ 
 which does not depend on $N$ for large enough
$N$, and is therefore determined formally by the series $H(t)$ and
$\lambda$ as an element $s_\lambda$ of ${\cal C}$. This is the element
$s_\lambda$ used by Kawagoe, which is identified  with the
idempotent closures $Q_\lambda$ by Lukac in \cite{Lukac}.

The $\pi(n)$ elements $\{s_\lambda\}$ with $|\lambda|=n$ form a linear
basis for ${\cal C}_n$, and can be expressed in terms of the monomials
of weight $n$ in $\{h_r\}$ by the classical Jacobi-Trudy formulae
\cite{Macdonald}. 

Indeed the skein ${\cal C}^+$, defined as $\cup{\cal C}_n$, is spanned
by all monomials in $\{h_r\}$, and can be interpreted as the ring of
symmetric polynomials in an unlimited number of variables $\{x_i\}$.


Monomials in the geometrically simpler closed braid elements
$\{A_m\}$ also span
${\cal C}_n$, and it is  thus interesting to relate these directly to
$\{h_r\}$. An attractive formula connecting these two generating sets
can be derived from corollary \ref{Idempotent}. 

\begin{theorem} \label{AH}
Write $A(t)=1+z\sum_{m=1}^\infty A_mt^m$, with $z=s-s^{-1}$.

Then \[A(t)=\frac{H(st)}{H(s^{-1}t)}=H(st)E(-s^{-1}t).\]
\end{theorem} 
The proof will be given in section \ref{A}.

\begin{lemma} The elements $\{h_n\}$ are invariant under
the mirror map switching crossings and inverting $s$ and $v$.
\end{lemma}
\begin{proof}In $H_n$ we have
$\sigma_ia_n=\varphi_s(\sigma_i)a_n=sa_n$. Then
$\overline{\sigma_ia_n}=\sigma_i^{-1}\overline{a}_n=s^{-1}\overline{a}_n$
giving $\sigma_i\overline{a}_n=s\overline{a}_n$. It follows that
$\beta\overline{a}_n=\varphi_s(\beta)\overline{a}_n$ for any $\beta$
and so $a_n\overline{a}_n=\varphi_s(a_n)\overline{a}_n$.

On the other hand $a_n\overline{a}_n=a_n\varphi_s(\overline{a}_n)$ by
lemma \ref{Rowidem}, so that
\[h_n=\frac{a_n}{\varphi_s(a_n)}
=\frac{\overline{a}_n}{\varphi_s(\overline{a}_n)}=\overline{h}_n.\]
\end{proof}
\begin{corollary} The inverse series to $A(t)=1+z\sum A_mt^m$ is
$\overline{A}(t)=1-z\sum \overline{A}_mt^m$, where $\overline {A}_m$
is the  negative closed braid corresponding to $A_m$.
\end{corollary}
\begin{proof} Apply the mirror map in theorem \ref{AH} to get
\[\overline{A}(t)={\overline{H(st)}}\ /\ {\overline{H(s^{-1}t)}}
={H(s^{-1}t)}\ /\ {H(st)}=A(t)^{-1}.\]
\end{proof}

In our later description of the power sums of Murphy operators we use
the element
$P_m
\in {\cal C}$ which can be interpreted formally as the $m$th power sum,
$P_m=\sum x_i^m$, of the variables. This is determined unambiguously as
a polynomial in
$\{h_r\}$, and hence an element of ${\cal C}$, by Newton's power series
equation
\[\sum_{m=1}^\infty \frac{P_m}{m}t^m=\ln H(t).\]
Aiston showed in \cite{Adams} that $[m]P_m$ is the sum of $m$ closed
$m$-string braids given  by switching the first
$i$ crossings of the closed braid $A_m$ from positive to negative, for
$i=0,\ldots,m-1$.  Her proof depends on some quantum group translations
to identify
$s_\lambda$ and $Q_\lambda$, and to write $A_m$ in terms of
$s_\lambda$. In another article I give a direct skein theory proof of
her result, using theorem \ref{AH}.

\subsection{Symmetric functions of the Murphy operators.}
The  main result in this paper, besides theorem \ref{Sum} and its
consequences, is an expression for the
$m$th power sum of the Murphy operators in $H_n$ in terms of the
element
$P_m\in {\cal C}$.

For this I use a previously unremarked relation between the
Hecke algebras and the skein of the annulus.
This relation takes the form of a very natural homomorphism $\psi_n$
from
${\cal C}
$ to the centre of each algebra $H_n$.  

 The diagram
\[D\ =\ \PsiD\] determines a  map $\psi_n:{\cal C}\to H_n$,
induced by placing $X\in \CC$  around the circle in
$D$ and the identity of $H_n$ on the arc, to get
\[\psi_n(X)\ =\ \PsiX\ \in H_n.\]
 It is clear that 
\[\psi_n(XY)\ =\ \PsiXY\ =\ \PsiXPsiY\ =\psi_n(X)\psi_n(Y),\]
so that
$\psi_n$ is an algebra homomorphism.
Moreover the elements $\psi_n(X)$ obviously all lie in the
centre of $H_n$, and in fact it follows from theorem \ref{Power}
below that the image of $\psi_n$ consists af all symmetric polynomials
in the Murphy operators and so, by \cite{Dipper}, makes up the whole of
the centre in the generic case.

We already know from theorem \ref{Sum} that the sum $\sum T(j)$ is
essentially $T^{(n)}$, and we can write $T^{(n)}=\psi(X)$ with $X=h_1$,
represented by a single string around the core of the annulus. We may
then look for elements
$X_2$ with
$\psi(X_2)=\sum T(j)^2$ and more generally $\psi(X_m)=\sum
T(j)^m$.

These can certainly be found for each fixed
$n$, using for  example a suitable polynomial in $h_1$ which depends on
$n$. Much more surprising is the result of theorem \ref{Power},
that there exists an element
$X_m$ {\em independent of $n$} with the property
that
\[\psi_n(X_m)=\sum_{j=1}^n T(j)^m,\] up to a
multiple of the identity. Consequently any other
symmetric function of the Murphy operators can be
achieved as $\psi(X)$ where the choice of $X$ is
essentially independent of $n$.

In fact, we can take $X_m=P_m$, to get the following explicit result,
proved in section \ref{A}.
\begin{theorem} \label{Power} For any $n$ we have
\[\psi_n(P_m) -\psi_0(P_m) = (s^m-s^{-m})v^{-m}\sum_{j=1}^n T(j)^m.\]
\end{theorem}


This is particularly satisfactory in that $P_m$ itself arises as a
power sum, although not of any identifiable objects in the ring $\cal
C$ itself.

 Since any element
$X\in {\cal C}^+$ can be written as a polynomial in
$\{P_m\}$ we can then express $\psi(X)$ as a symmetric function of the
Murphy operators $T(j)$.  It is useful to be able to write the
elements 
 $\psi(h_i)$ in terms of the Murphy operators; theorem \ref{Power}
leads to a  compact expression using  formal power series with
coefficients in the centre of the Hecke algebra. 

I shall do this in terms of
the {\em Murphy series} $HM(t)$ of the Hecke algebra,
whose coefficients, lying in the centre of $H_n$,  are defined to be
the complete symmetric functions of the elements
$T(j)$. Thus the Murphy series in $H_n$ can be
written explicitly as \[HM(t)=\prod_{j=1}^n
(1-T(j)t)^{-1}.\] The series has a formal inverse, $EM(-t)$, where the
coefficients of
$EM(t)=\prod(1+T(j)t)$ are the elementary symmetric functions of the
elements $T(j)$, also in the centre of
$H_n$.

\begin{theorem}

\[\psi_n(H(t))=\psi_0(H(t))\frac{HM(sv^{-1}t)}{HM(s^{-1}v^{-1}t)}.\]


\end{theorem}

\begin{proof} It is enough to establish that the logarithms of the
two sides are equal.  Now  \[\ln
(\psi_n(H(t)))=\psi_n(\ln(H(t)))=\sum_{m=1}^\infty
\frac{\psi_n(P_m)}{m} t^m,\] while \[\ln (HM(t))=\sum_{m=1}^\infty
\frac{\sum T(j)^m}{m} t^m.\] The coefficients of $t^m$ in $\ln
(\psi_n(H(t)))$ and in
\begin{eqnarray*}
&&\ln\left(\psi_0(H(t))\frac{HM(sv^{-1}t)}{HM(s^{-1}v^{-1}t)}\right)\\
&=&\ln\left(\psi_0(H(t))\right)+
\ln\left(HM(sv^{-1}t)\right)-\ln\left(HM(s^{-1}v^{-1}t)\right)
\end{eqnarray*}
are equal, by theorem
\ref{Power}.
\end{proof}


\section{The annulus with two boundary points.} \label{A}

To prove theorems \ref{AH} and \ref{Power} I introduce a third skein 
$\CS(F)$, whose underlying surface is the annulus with a single input
on one boundary component and corresponding output on the other, as
shown,

\begin{center}
\Annulus\ .
\end{center}

Write $\CA$ for this skein, whose elements are represented as linear
combinations of oriented tangles in the annulus consisting of a single
arc and a number of closed curves.  As for the skein $\CC$ of the
annulus there is a product on $\CA$ induced by placing one annulus
outside the other.  The identity element $e\in \CA$ is represented by
the tangle \[ \Aid\ .\]
Write $a\in \CA$ for the element represented by \[ \Aa\ .\] From this
we can construct $a^m$ for each $m\in {\bf Z}$, giving for example
\[ a^2\ =\ \Aaa\ ,\quad a^{-1}\ =\ \Aainv\ .\]

There are {\em two} bilinear products, $l:\CC\times\CA\to\CA$ and
$r:\CA\times\CC\to\CA$ induced by placing an element of $\CC$
respectively under or over an element of $\CA$. For example, $h_1\in
\CC$, represented by a single counterclockwise loop, gives
\[l(h_1,e)\ =\ \Ahe\ ,\quad r(e,h_1)\ =\ \Aeh\ .\]

From the skein relation in $\CA$ we have
$l(h_1,e)-r(e,h_1)=(s-s^{-1})a$.

\subsection{Algebraic properties of $\CA$.}
Kawagoe \cite{Kawagoe} and other authors have used a skein which is
linearly isomorphic to $\CA$, based on
the annulus with input and output on the {\em same} component. The
significant advantage of $\CA$ lies in its algebraic properties.

Both $l$ and  $r$ are algebra homomorphisms, since it is clear
diagrammatically that $l(c_1,a_1)l(c_2,a_2)=l(c_1c_2,a_1a_2)$, while
$r$ behaves similarly.


\begin{theorem} The algebra $\CA$ is commutative.
\end{theorem}

\begin{proof}
Unlike the case of $\CC$ this is not immediately clear. Using standard
skein theory techniques we can represent any element of $\CA$ as a
linear combination of tangles consisting of a totally descending arc
lying over a number of closed curves. Each such tangle represents
$l(c_m,a^m)=l(c_m,e)a^m$ for some $m$ and some $c_m\in \CC$. The
general element of $\CA$ can then be written as a Laurent polynomial
\[\sum_{m\in {\bf Z}}l(c_m,e)a^m\]
in $a$, with coefficients in the commutative subalgebra
$l(\CC,e)\subset \CA$. Since $a$ commutes with $l(\CC,e)$ it follows
that any two elements of $\CA$ commute.
\end{proof}

The subalgebras $l(\CC,e)$ and $r(e,\CC)$ are both isomorphic to
$\CC$, but they are not equal. Their difference determines a sort of
commutator map $[\ ,e]:\CC\to\CA$, defined by $[c,e]=l(c,e)-r(e,c)$.

We have already noted that $[h_1,e]=(s-s^{-1})a$. 
We can deduce
theorem \ref{Power}
about the power sums of Murphy operators from the next theorem.
\begin{theorem}\label{Powercomm} For $m\ge 1$ we have
$[P_m,e]=(s^m-s^{-m})a^m$.
\end{theorem}

\subsection{Skein interaction between $H_n$, $\CA$ and $\CC$.}
We can make use of wiring diagrams, as described in
\cite{AistonMorton}, to induce linear maps between skeins. A {\em
wiring diagram} $W$ is an inclusion of one surface $F$ into another
$F'$, along with a fixed diagram of curves in $F'$ which avoid $F$, and
connect any distinguished input and output points on the boundaries of
both, respecting orientation.  A wiring $W$ induces a linear map,
denoted $\CS(W):\CS(F)\to \CS(F')$, using $W$ to extend any diagram in
$F$ to a diagram in $F'$.

To avoid over-elaborate notation  I shall write $W$ in place of
$\CS(W)$ where there is no risk of confusion.

The diagram \[W_n\ =\ \Wn\] with $n$ strings running around the
annulus then induces a linear map $W_n:R^{n+1}_{n+1}\to \CA$, by
inserting an $(n+1)$-tangle in the empty box.

When an element $S\in H_n$ is included in $H_{n+1}$ in the standard
way, using their skein versions, we can see from a diagram that
$W_n(S)=W_{n-1}(S)a$.
Then $W_n(1)=a^n$, where $a\in \CA$ is the element shown above.

\begin{theorem}
The elements $W_n(h_{n+1}), W_n(h_n)$ and $l(h_n,e)$ in $\CA$ satisfy
the linear relation
\[[n+1]W_n(h_{n+1})=s^{-1}[n]W_n(h_n)+l(h_n,e).\]
\end{theorem}
\begin{proof}
The relation \[ \Gnplus\ =\ \Gshift\ +\ s^{n}\ \Gcross\] gives
$W_n(h_n\gamma_{n+1})=W_n(\gamma_n h_n)+s^nW_n(h_n
\sigma_n\cdots\sigma_1)$,  after taking $\gamma_n$ from the second
diagram around the strings of the wiring.

Now $\gamma_n h_n=s^{n-1}[n]h_n$ and
$W_n(h_n\sigma_n\cdots\sigma_1)=l(h_n,e)$. Combined with the equation
$s^n[n+1]h_{n+1}=h_n\gamma_{n+1}$ from corollary \ref{Idempotent} the
result follows at once.
\end{proof}

Now write $Y_n=[n+1]W_n(h_{n+1})$ and set $Y(t)=\sum_{n=0}^\infty
Y_nt^n$.
\begin{corollary}\label{HY} As power series with coefficients in $\CA$
we have
\[l(H(t),e)=(e-s^{-1}at)Y(t).\]
\end{corollary}
\begin{proof} Since $W_n(h_n)=W_{n-1}(h_n)a=aW_{n-1}(h_n)$ the relation
can be written $Y_n=s^{-1}aY_{n-1}+l(h_n,e)$. This gives
$Y(t)=s^{-1}atY(t)+l(H(t),e)$, and hence the result.
\end{proof}

 The mirror map, switching crossings and inverting $s$
and $v$, when applied to
$l(X,e)$ for any $X\in \CC$ gives $r(e,\overline X)$. Now
$\overline{H(t)}=H(t)$ and so 
$\overline{Y(t)}=Y(t)$, giving
\[r(e,H(t))=(e-sat)Y(t).\]

Subtracting this from the equation in corollary \ref{HY} then gives
$[H(t),e]=(s-s^{-1})atY(t)$, a result used by Lukac in \cite{Lukac}.

\begin{proof}[of theorem \ref{Powercomm}]
Take logarithms of the equations and then subtract. Then
\begin{eqnarray*}
\ln(e-s^{-1}at)-\ln(e-sat)&=&\ln\left(l(H(t),e)\right)
-\ln\left(r(e,H(t))\right)\\
&=& l(\ln(H(t)),e)-r(e,\ln(H(t)))\\
&=&\sum_{m=1}^\infty
[\frac{P_m}{m}t^m,e].
\end{eqnarray*}
Now $\ln(e-s^{-1}at)=-\sum_{m=1}^\infty s^{-m}a^mt^m/m$, and it only
remains to compare the coefficients of $t^m$ to see that
$[P_m,e]=(s^m-s^{-m})a^m$.
\end{proof}

\begin{proof}[of theorem \ref{Power}] Use the  wiring \[V_n\ =\ \Vn\]
with $n-1$ strings passing through the annulus to carry
$\CA$ into
$R^n_n$. Then for any $X$ in $\CC$ we have $V_n(l(X,e))=\psi_n(X)$ and
$V_n(r(e,X))=\psi_{n-1}(X)$ with the standard inclusion.
We can also see that $V_n(a)=v^{-1}T(n)$, and extend this to
$V_n(a^m)=v^{-m}T(n)^m$. Then
$\psi_n(P_m)-\psi_{n-1}(P_m)=(s^m-s^{-m})v^{-m}T(n)^m$.

Induction on $n$ completes the proof that
\[\psi_n(P_m)-\psi_0(P_m)=(s^m-s^{-m})v^{-m}\sum_{j=1}^n T(j)^m.\]
\end{proof}


To conclude this paper I give the proof of theorem \ref{AH}, using one
final wiring, this time to induce a map from $\CA$ to $\CC$.

Write
\[C\ =\ \WC\] for the wiring diagram shown, which provides a form of
closure for an element of $\CA$, and determines a linear map
$C:\CA\to\CC$. It is clear that $C(a^m) =A_{m+1}$ for every $m\ge 0$.
We can also see readily that  \[C(F\ r(e,X))=C(r(F,X))=C(F)X\] for
every $F\in \CA$ and $X\in \CC$.

\begin{proof} [of theorem \ref{AH}] The relation
$(e-sat)Y(t)=r\left(e,H(t)\right)$ gives
\[Y(t)=\left(\sum_{m=0}^\infty s^ma^mt^m\right)r\left(e,H(t)\right).\]
Then
\begin{eqnarray*} C\left(Y(t)\right)&=&C\left(\sum_{m=0}^\infty
s^ma^mt^m\right)H(t)\\
&=&\left(\sum_{m=0}^\infty s^mA_{m+1}t^m\right)H(t),
\end{eqnarray*}
giving
\[zst  C\left(Y(t)\right)=\left(A(st)-1\right)H(t).\]

Now $C\left(W_n(h_{n+1})\right)$ is the closure of the $(n+1)$-tangle
$\sigma_1\sigma_2\cdots\sigma_nh_{n+1}=s^nh_{n+1}$. Then
$C(Y_n)=[n+1]s^nh_{n+1}\in\CC$. It follows that
\begin{eqnarray*}
zst 
C\left(Y(t)\right)&=&\sum_{n=0}^\infty(s^{n+1}-s^{-(n+1)})s^{n+1}h_{n+1}t^{n+1}\\
&=&\sum_{n=0}^\infty(s^{2n+2}-1)h_{n+1}t^{n+1}\\
&=&H(s^2t)-H(t).
\end{eqnarray*}
This gives $H(s^2t)-H(t)=A(st)H(t)-H(t)$, and thus
$H(st)=A(t)H(s^{-1}t)$, to complete the proof.
\end{proof}
\medskip
\noindent
Version 1.2, February 2001.

\begin{thebibliography}{99}

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algebras and quantum group invariants}.  PhD dissertation, University
of Liverpool, 1996.

\bibitem{Adams} A.K. Aiston,  {Adams operators and knot decorations}. 
      Liverpool University preprint 1997,  q-alg
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\bibitem{Blanchet} C. Blanchet, {Hecke algebras, modular categories
and 3-manifolds quantum invariants}. {\it Topology} 39 (2000), 
193-223.

\bibitem{Dipper}  R. Dipper and G.D. James, {Blocks and
idempotents of Hecke algebras of general linear groups}. {\it
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\bibitem{Gyoja} A. Gyoja, {A q-analogue of {Y}oung symmetrisers}. {\it
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\bibitem{Jones} V.F.R. Jones, {Hecke algebra representations of braid
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\bibitem{Jucys} A. Jucys, 
{Factorization of {Y}oung's projection operators for symmetric groups}.
{\it Litovsk. Fiz. Sb.} 11 (1971), 1-10. 

\bibitem{Katriel}  J. Katriel, B. Abdesselam and A. Chakrabarti,
{The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes
the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$.} {\it
J.~Math.~Phys.\/} 36 (1995), 5139-5158.

\bibitem{Kawagoe} K. Kawagoe, {On the skeins in the annulus and
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$3$-manifolds}. {\it J. Knot Theory Ramifications} 7 (1998),
187--203. 

\bibitem{Lukac} S. Lukac, {} PhD dissertation, University of
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\bibitem{Macdonald} I.G. Macdonald, {Symmetric functions and {H}all
polynomials}. Clarendon Press, Oxford, 2nd edition (1995).

\bibitem{Turkey}  H.R. Morton,
{ Invariants of links and 3-manifolds from skein
theory and from quantum groups}. In  {`Topics
in knot theory'}, the Proceedings of the NATO Summer
Institute in Erzurum 1992, NATO ASI Series C 399, ed.~M.Bozh\" uy\" uk.  Kluwer
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\bibitem{AistonMorton} H.R.Morton and A.K.Aiston, 
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\bibitem{Murphy} G.E. Murphy, 
{A new construction of {Y}oung's seminormal representation of the
symmetric groups}.  {\it J. Algebra} 69 (1981),  287-297. 

\bibitem{Ram} A. Ram,
{Seminormal representations of {W}eyl groups and {I}wahori-{H}ecke
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99-133. 

\bibitem{Turaev} V.G. Turaev, {The {C}onway and {K}auffman modules 
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Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) }  
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\bibitem{Wenzl} H. Wenzl, {Representations of braid groups and the
quantum Yang-Baxter equation}. {\it Pacific J.~Math.\/} 145 (1990),
153-180.

\end{thebibliography}
\end{document}





math0102098.extracted_bib
\begin{thebibliography}{99}

\bibitem{Aiston} A.K. Aiston, {Skein theoretic idempotents of {H}ecke
algebras and quantum group invariants}.  PhD dissertation, University
of Liverpool, 1996.

\bibitem{Adams} A.K. Aiston,  {Adams operators and knot decorations}. 
      Liverpool University preprint 1997,  q-alg
9711015. 


\bibitem{Blanchet} C. Blanchet, {Hecke algebras, modular categories
and 3-manifolds quantum invariants}. {\it Topology} 39 (2000), 
193-223.

\bibitem{Dipper}  R. Dipper and G.D. James, {Blocks and
idempotents of Hecke algebras of general linear groups}. {\it
Proc.~London Math.~Soc.\/} 54 (1987), 57-82.

\bibitem{Gyoja} A. Gyoja, {A q-analogue of {Y}oung symmetrisers}. {\it
Osaka J. Math.} 23 (1986), 841-852.

\bibitem{Jones} V.F.R. Jones, {Hecke algebra representations of braid
groups and link polynomials}. {\it Ann. Math.} 126 (1987), 335-388.

\bibitem{Jucys} A. Jucys, 
{Factorization of {Y}oung's projection operators for symmetric groups}.
{\it Litovsk. Fiz. Sb.} 11 (1971), 1-10. 

\bibitem{Katriel}  J. Katriel, B. Abdesselam and A. Chakrabarti,
{The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes
the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$.} {\it
J.~Math.~Phys.\/} 36 (1995), 5139-5158.

\bibitem{Kawagoe} K. Kawagoe, {On the skeins in the annulus and
applications to invariants of
$3$-manifolds}. {\it J. Knot Theory Ramifications} 7 (1998),
187--203. 

\bibitem{Lukac} S. Lukac, {} PhD dissertation, University of
Liverpool, 2001.

\bibitem{Macdonald} I.G. Macdonald, {Symmetric functions and {H}all
polynomials}. Clarendon Press, Oxford, 2nd edition (1995).

\bibitem{Turkey}  H.R. Morton,
{ Invariants of links and 3-manifolds from skein
theory and from quantum groups}. In  {`Topics
in knot theory'}, the Proceedings of the NATO Summer
Institute in Erzurum 1992, NATO ASI Series C 399, ed.~M.Bozh\" uy\" uk.  Kluwer
1993, 107-156.

\bibitem{AistonMorton} H.R.Morton and A.K.Aiston, 
      {Idempotents of {H}ecke algebras of type {A}}.  {\it J. Knot
Theory Ramifications} 7 (1998), 463-487.

\bibitem{Murphy} G.E. Murphy, 
{A new construction of {Y}oung's seminormal representation of the
symmetric groups}.  {\it J. Algebra} 69 (1981),  287-297. 

\bibitem{Ram} A. Ram,
{Seminormal representations of {W}eyl groups and {I}wahori-{H}ecke
algebras}.  {\it Proc. London Math. Soc.}  75 (1997),
99-133. 

\bibitem{Turaev} V.G. Turaev, {The {C}onway and {K}auffman modules 
of a solid torus}. {\it Zap. Nauchn. Sem. 
Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) }  
167 (1988), {\it Issled. Topol.} 6, 79-89.

\bibitem{Wenzl} H. Wenzl, {Representations of braid groups and the
quantum Yang-Baxter equation}. {\it Pacific J.~Math.\/} 145 (1990),
153-180.

\end{thebibliography}