Two-row nilpotent orbits of cyclic quivers
Anthony Henderson
We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free.
pst-node.tex
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%% This uses the header file `pst-node.pro'.
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%%
%% COPYRIGHT 1993, 1994, 1999 by Timothy Van Zandt, tvz@nwu.edu.
%%
%% This program can be redistributed and/or modified under the terms
%% of the LaTeX Project Public License Distributed from CTAN
%% archives in directory macros/latex/base/lppl.txt.
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\def\ncarc{\pst@object{ncarc}}
\def\ncarc@i{\check@arrow{\ncarc@ii}}
\def\ncarc@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
yB yA sub xB xA sub \tx@Atan dup
\psk@arcangleA\space add /AngleA ED
\psk@arcangleB\space sub 180 add /AngleB ED
\psk@ncurvB\space \psk@ncurvA\space
\tx@NCCurve}}
\def\pcarc{\pst@object{pcarc}}
\def\pcarc@i{\pc@object\ncarc@ii}
\def\tx@NCAngles{NCAngles }
\def\ncangles{\pst@object{ncangles}}
\def\ncangles@i{\check@arrow{\ncangles@ii}}
\def\ncangles@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCAngles}}
\def\ncangles@iii{%
tx@Dict begin \psline@iii pop end
/AngleA \psk@angleA def
/AngleB \psk@angleB def
/ArmA \psk@armA def
/ArmB \psk@armB def
/ArmTypeA \psk@armtypeA def
/ArmTypeB \psk@armtypeB def }
\def\pcangles{\pst@object{pcangles}}
\def\pcangles@i{\pc@object\ncangles@ii}
\def\tx@NCAngle{NCAngle }
\def\ncangle{\pst@object{ncangle}}
\def\ncangle@i{\check@arrow{\ncangle@ii}}
\def\ncangle@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCAngle}}
\def\pcangle{\pst@object{pcangle}}
\def\pcangle@i{\pc@object\ncangle@ii}
\def\tx@NCBar{NCBar }
\def\ncbar{\pst@object{ncbar}}
\def\ncbar@i{\check@arrow{\ncbar@ii}}
\def\ncbar@ii#1#2{\nc@object{Open}{#1}{#2}{1.5}{%
\ncangles@iii /AngleB \psk@angleA def \tx@NCBar}}
\def\pcbar{\pst@object{pcbar}}
\def\pcbar@i{\pc@object\ncbar@ii}
\def\tx@NCDiag{NCDiag }
\def\ncdiag{\pst@object{ncdiag}}
\def\ncdiag@i{\check@arrow{\ncdiag@ii}}
\def\ncdiag@ii#1#2{%
\nc@object{Open}{#1}{#2}{1.5}{\ncangles@iii \tx@NCDiag}}
\def\pcdiag{\pst@object{pcdiag}}
\def\pcdiag@i{\pc@object\ncdiag@ii}
\def\tx@NCDiagg{NCDiagg }
\def\ncdiagg{\pst@object{ncdiagg}}
\def\ncdiagg@i{\check@arrow{\ncdiagg@ii}}
\def\ncdiagg@ii#1#2{%
\nc@object{Open}{#1}{#2}{.5}{\ncangles@iii \tx@NCDiagg}}
\def\pcdiagg{\pst@object{pcdiagg}}
\def\pcdiagg@i{\pc@object\ncdiagg@ii}
\def\tx@NCLoop{NCLoop }
\def\psset@loopsize#1{\pst@getlength{#1}\psk@loopsize}
\psset@loopsize{1cm}
\def\ncloop{\pst@object{ncloop}}
\def\ncloop@i{\check@arrow{\ncloop@ii}}
\def\ncloop@ii#1#2{%
\nc@object{Open}{#1}{#2}{2.5}%
{\ncangles@iii /loopsize \psk@loopsize def \tx@NCLoop}}
\def\pcloop{\pst@object{pcloop}}
\def\pcloop@i{\pc@object\ncloop@ii}
\def\tx@NCCircle{NCCircle }
\def\nccircle{\pst@object{nccircle}}
\def\nccircle@i{\check@arrow{\nccircle@ii}}
\def\nccircle@ii#1#2{%
\pssetlength\pst@dima{#2}%
\nc@object{Open}{#1}{#1}{.5}{%
/AngleA \psk@angleA def
/r \pst@number\pst@dima def
\tx@NCCircle \psarc@v end}}
\def\tx@NCBox{NCBox }
\def\ncbox{\pst@object{ncbox}}
\def\ncbox@i{\check@arrow{\ncbox@ii}}
\def\ncbox@ii#1#2{%
\def\pst@linetype{2}%
\nc@object{Closed}{#1}{#2}{.5}{%
tx@Dict begin \psline@iii pop end
\psk@boxheight \psk@boxdepth
\tx@NCBox}}
\def\pcbox{\pst@object{pcbox}}
\def\pcbox@i{\pc@object\ncbox@ii}
\def\tx@NCArcBox{NCArcBox }
\def\psset@boxheight#1{\pst@getlength{#1}\psk@boxheight}
\def\psset@boxdepth#1{\pst@getlength{#1}\psk@boxdepth}
\def\psset@boxsize#1{%
\psset@boxheight{#1}%
\let\psk@boxdepth\psk@boxheight}
\psset@boxsize{.4cm}
\def\ncarcbox{\pst@object{ncarcbox}}
\def\ncarcbox@i{\check@arrow{\ncarcbox@ii}}
\def\ncarcbox@ii#1#2{%
\def\pst@linetype{1}%
\nc@object{Closed}{#1}{#2}{.5}{%
\psk@arcangleA \psk@boxheight \psk@boxdepth \pst@number\pslinearc
\tx@NCArcBox}}
\def\pcarcbox{\pst@object{pcarcbox}}
\def\pcarcbox@i{\pc@object\ncarcbox@ii}
\def\tx@Tfan{Tfan }
% Changed according pst-beta.bug December 3, 1993
% nrot=:<angle> does not work when : is active.
\begingroup
\catcode`\:=13
\gdef\pst@activerot{\def:{\string:}}
\endgroup
\def\psset@nrot#1{%
\begingroup
\pst@activerot
\pst@expandafter{\@ifnextchar:{\psset@@nrot}{\psset@@rot}}{#1}\@nil
\global\let\pst@tempg\psk@rot
\endgroup
\let\psk@nrot\pst@tempg}
\def\psset@@nrot:#1\@nil{%
\psset@@rot#1\@nil
\edef\psk@rot{NAngle \ifx\psk@rot\@empty\else\psk@rot add \fi}}
\psset@nrot{0}
\def\tx@LPutCoor{LPutCoor }
\def\tx@LPut{LPut }
\def\psset@npos#1{%
\def\pst@tempa{#1}%
\ifx\pst@tempa\@empty
\def\psk@npos{\npos@default}%
\else
\pst@checknum{#1}\psk@npos
\fi}
\psset@npos{}
\def\ncput{\pst@object{ncput}}
\def\ncput@i{\pst@killglue\pst@makebox{\ncput@ii}}
\def\ncput@ii{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\pst@makesmall\pst@hbox
\pst@rotate\psk@nrot\pst@hbox
\ncput@iii
\endgroup
\pst@shortput}
\def\ncput@iii{%
\leavevmode
\hbox{%
\pst@Verb{%
\pst@nodedict
/t \psk@npos def
\tx@LPut
end
\tx@PutBegin}%
\box\pst@hbox
\pst@Verb{\tx@PutEnd}}}
\def\naput{\pst@object{naput}}
\def\naput@i{\pst@killglue\pst@makebox{\naput@ii{NAngle 90 add}}}
\def\naput@ii#1{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\def\psk@refangle{#1 }%
\let\psk@rot\psk@nrot
\uput@vii
{exch pop add a \tx@PtoC h1 add exch w1 add exch }%
{tx@Dict /NCLW known { NCLW add } if }%
\ncput@iii
\endgroup
\pst@shortput}
\def\nbput{\pst@object{nbput}}
\def\nbput@i{\pst@killglue\pst@makebox{\naput@ii{NAngle 90 sub}}}
\def\psset@tpos#1{%
\pst@checknum{#1}\psk@tpos
\ifdim\psk@tpos \p@<\z@
\def\psk@tpos{.5}%
% DG/SR modification begin - Sep. 23, 1998 - Patch 7
%\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@epha
\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@ehpa
% DG/SR modification end
\else
\ifdim\psk@tpos \p@>\p@
\def\psk@tpos{.5}%
% DG/SR modification begin - Sep. 23, 1998 - Patch 7
%\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@epha
\@pstrickserr{Bad `tpos' value: `#1'. Must be 0<tpos<1}\@ehpa
% DG/SR modification end
\fi
\fi}
\psset@tpos{.5}
\def\tvput{\pst@object{tvput}}
\def\tvput@i{\pst@makebox{\psput@tput{H}{1}}}
\def\tlput{\pst@object{tlput}}
\def\tlput@i{\pst@makebox{\psput@tput{H}{true}}}
\def\trput{\pst@object{trput}}
\def\trput@i{\pst@makebox{\psput@tput{H}{false}}}
\def\thput{\pst@object{thput}}
\def\thput@i{\pst@makebox{\psput@tput{V}{1}}}
\def\taput{\pst@object{taput}}
\def\taput@i{\pst@makebox{\psput@tput{V}{true}}}
\def\tbput{\pst@object{tbput}}
\def\tbput@i{\pst@makebox{\psput@tput{V}{false}}}
\def\tx@HPutAdjust{HPutAdjust }
\def\tx@VPutAdjust{VPutAdjust }
\def\psput@tput#1#2{%
\begingroup
\use@par
\pst@tputmakesmall
\leavevmode
\hbox{%
\pst@Verb{%
\pst@nodedict
/t \psk@tpos \pst@tposflip def
tx@NodeDict /HPutPos known
{ #1PutPos }
{ CP /Y ED /X ED /NAngle 0 def /NCLW 0 def }
ifelse
/Sin NAngle sin def
/Cos NAngle cos def
/s \pst@number\pslabelsep NCLW add def
/l \pst@number\pst@dima def
/r \pst@number\pst@dimb def
/h \pst@number\pst@dimc def
/d \pst@number\pst@dimd def
% DG/SR modification begin - Sep. 26, 1997 - Patch 4
%\ifnum1=0#2\else
\ifnum1=0#2 \else
% DG/SR modification end
/flag #2 def
\csname tx@#1PutAdjust\endcsname
\fi
\tx@LPutCoor
end
\tx@PutBegin}%
\box\pst@hbox
\pst@Verb{\tx@PutEnd}}%
\endgroup
\pst@shortput}
\def\pst@tposflip{}
\def\pst@tputmakesmall{%
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\pst@dimg=\psk@href\pst@dimg
\pst@dimb\pst@dima
\advance\pst@dima\pst@dimg % leftsize
\advance\pst@dimb-\pst@dimg % rightsize
\pst@dimd=\psk@vref\relax
\pst@dimc=\ht\pst@hbox
\advance\pst@dimc-\pst@dimd % height
\advance\pst@dimd\dp\pst@hbox % depth
\setbox\pst@hbox=\hbox to\z@{%
\kern-\pst@dima\vbox to\z@{\vss\box\pst@hbox\vskip-\pst@dimd}\hss}}
\def\MakeShortNab#1#2{%
  \def\pst@shortput@nab{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \let\pst@tempg\naput
    \else
      \ifx#2\next
        \let\pst@tempg\nbput
      \else
        \ifx\@sptoken\next
          \let\pst@tempg\pst@shortput
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortNab{^}{_}
\def\MakeShortTablr#1#2#3#4{%
  \def\pst@shortput@tablr{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \let\pst@tempg\taput
    \else
      \ifx#2\next
        \let\pst@tempg\tbput
      \else
        \ifx#3\next
          \let\pst@tempg\tlput
        \else
          \ifx#4\next
            \let\pst@tempg\trput
          \else
            \ifx\@sptoken\next
              \let\pst@tempg\pst@shortput
            \fi
          \fi
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortTablr{^}{_}{<}{>}
\def\MakeShortTab#1#2{%
  \def\pst@shortput@tab{%
    \def\pst@tempg{\next}%
    \ifx#1\next
      \def\pst@tempg{%
        \@nameuse{%
          t\ifodd\psk@treemode\ifpstreeflip b\else a\fi
          \else\ifpstreeflip r\else l\fi\fi put}}%
    \else
      \ifx#2\next
        \def\pst@tempg{%
          \@nameuse{%
            t\ifodd\psk@treemode\ifpstreeflip a\else b\fi
            \else\ifpstreeflip l\else r\fi\fi put}}%
      \else
        \ifx\@sptoken\next
          \let\pst@tempg\pst@shortput
        \fi
      \fi
    \fi
    \pst@tempg}}
\MakeShortTab{^}{_}
\def\psset@shortput#1{%
\def\pst@tempg{#1}%
\ifx\pst@tempg\@none
\let\pst@shortput\ignorespaces
\else
\@ifundefined{pst@shortput@#1}%
{\@pstrickserr{Bad short put: `#1'}\@ehpa}%
{\edef\pst@shortput{\noexpand\afterassignment\expandafter\noexpand
\csname pst@shortput@#1\endcsname\noexpand\let\noexpand\next}}%
\fi}
\psset@shortput{none}
\def\lput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\lput@i}{\lput@ii}}}
\def\lput@i[#1]{\addto@par{ref=#1}\lput@ii}
\def\lput@ii{\@ifnextchar({\lput@iv}{\lput@iii}}
\def\lput@iii#1{\addto@par{nrot=#1}\@ifnextchar({\lput@iv}{\ncput@i}}
\def\lput@iv(#1){\addto@par{npos=#1}\ncput@i}
\def\mput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\mput@i}{\ncput@i}}}
\def\mput@i[#1]{\addto@par{ref=#1}\ncput@i}
\def\Lput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Lput@ii}{\Lput@i}}}
\def\Lput@i#1{\addto@par{labelsep=#1}\Lput@ii}
\def\Lput@ii[#1]{\addto@par{ref={#1}}\@ifnextchar({\Lput@iv}{\Lput@iii}}
\def\Lput@iii#1{\addto@par{nrot={#1}}\@ifnextchar({\Lput@iv}{\Lput@v}}
\def\Lput@iv(#1){\addto@par{npos=#1}\Lput@v}
\def\Lput@v{\pst@killglue\pst@makebox{\Lput@vi}}
\def\Lput@vi{%
\begingroup
\use@par
\if@star\pst@starbox\fi
\Rput@vi
\pst@makesmall\pst@hbox
\pst@rotate\psk@nrot\pst@hbox
\ncput@iii
\endgroup
\pst@shortput}
\def\Mput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Mput@ii}{\Mput@i}}}
\def\Mput@i#1{\addto@par{labelsep=#1}\Mput@ii}
\def\Mput@ii[#1]{\addto@par{ref={#1}}\Lput@v}
\def\aput@#1{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\aput@i#1}{\aput@ii#1}}}
\def\aput@i#1[#2]{\addto@par{labelsep=#2}\aput@ii#1}
\def\aput@ii#1{\@ifnextchar({\aput@iv#1}{\aput@iii#1}}
\def\aput@iii#1#2{\addto@par{nrot=#2}\@ifnextchar({\aput@iv#1}{#1}}
\def\aput@iv#1(#2){\addto@par{npos=#2}#1}
\def\aput{\aput@\naput@i}
\def\bput{\aput@\nbput@i}
\def\Aput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Aput@i}{\naput@i}}}
\def\Aput@i[#1]{\addto@par{labelsep=#1}\naput@i}
\def\Bput{\def\pst@par{}\pst@ifstar{\@ifnextchar[{\Bput@i}{\nbput@i}}}
\def\Bput@i[#1]{\addto@par{labelsep=#1}\nbput@i}
\def\node@coor#1;#2\@nil{%
\pst@getnode{#1}\pst@tempg
\edef\pst@coor{%
\pst@nodedict
tx@NodeDict \pst@tempg known
{ \pst@tempg load \tx@GetCenter }
{ 0 0 }
ifelse
end }}
\def\Node@coor[#1]#2;#3\@nil{%
\begingroup
\psset{#1}%
\@ifnextchar\bgroup{\Node@@@coor}{\Node@@coor}#2\@nil
\endgroup
\let\pst@coor\pst@tempg}
\def\Node@@coor#1\@nil{%
\pst@getnode{#1}\pst@tempg
\xdef\pst@tempg{%
\pst@nodedict
tx@NodeDict \pst@tempg known
{ \psk@nodesepA \psk@angleA
\pst@tempg load \psk@nodeseptypeA \tx@GetEdge
\psk@offsetA \psk@angleA \tx@AddOffset
\pst@tempg load \tx@GetCenter
3 -1 roll add 3 1 roll add exch }
{ CP }
ifelse
end }}%
\def\Node@@@coor#1{%
\pst@@getcoor{#1}%
\def\psk@angleA{%
\pst@tempg load \tx@GetCenter \pst@coor
3 -1 roll sub 3 1 roll sub neg \tx@Atan}%
\Node@@coor}
\def\nput{\pst@object{nput}}
\def\nput@i#1#2{\pst@killglue\pst@makebox{\nput@ii{#1}{#2}}}
\def\nput@ii#1#2{%
\begingroup
\use@par
\psset@refangle{#1}%
\let\psk@angleA\psk@refangle
\edef\psk@nodesepA{\pst@number\pslabelsep}%
\def\psk@nodeseptypeA{0 }%
\pslabelsep\z@
\uput@vi
\Node@@coor#2\@nil
\let\pst@coor\pst@tempg
\leavevmode
\psput@special\pst@hbox
\endgroup
\ignorespaces}
\newcount\psrow
\newcount\pscol
\newcount\psmatrixcnt
\newskip\psrowsep
\newskip\pscolsep
\def\psset@colsep#1{\pssetlength\pscolsep{#1}}
\def\psset@rowsep#1{\pssetlength\psrowsep{#1}}
\psset@colsep{1.5cm}
\psset@rowsep{1.5cm}
\newif\ifpsmatrix
% DG/SR modification begin - Nov. 27, 1998 - Patch 8
%\let\mscount\@multicnt
\ifx\mscount\@undefined\let\mscount\@multicnt\fi
% DG/SR modification end
\def\psmatrix{%
  \begingroup
{\ifnum0=`}\fi % Don't want to expand any &.
\@ifnextchar[{\psmatrix@i}{\ifnum0=`{\fi}{}\psmatrix@ii}}
\def\psmatrix@i[#1]{%
\ifnum0=`{\fi}{}%
\psset{#1}%
\psmatrix@ii}
\def\psmatrix@ii{%
\KillGlue
\edef\psm@beginmath{%
\ifmmode$\m@th\ifinner\textstyle\else\displaystyle\fi\fi}%
\edef\psm@endmath{\ifmmode$\fi}%
\let\\\psm@cr
\advance\psmatrixcnt 1
\def\psm@thenode{M-\the\psmatrixcnt-\the\psrow-\the\pscol}%
\tabskip\z@
\psrow1
\pscol\z@
\psset@shortput{tablr}%
\leavevmode
\vbox\bgroup\halign\bgroup&%
\begingroup
\global\advance\pscol 1
\csname psrowhook\romannumeral\psrow\endcsname
\csname pscolhook\romannumeral\pscol\endcsname
\psm@beginnode##\psm@endnode\endgroup
\cr}
\def\endpsmatrix{%
\crcr\egroup\unskip\egroup
\endgroup}
\def\psm@cr{{\ifnum0=`}\fi\@ifnextchar[{\psm@@cr}{\psm@@@cr{}}}
\def\psm@@cr[#1]{\psm@@@cr{\vskip#1\relax}}
\def\psm@@@cr#1{%
\ifnum0=`{\fi}{}\cr
\noalign{%
\global\advance\psrow 1
\global\pscol\z@
\vskip\psrowsep
#1}}
\def\psm@beginnode{%
\@ifnextchar\psm@endnode
{\let\psm@endnode@i\relax\setbox\pst@hbox=\hbox{}}%
{\pst@object{psm@beginnode}}}
\def\psm@beginnode@i{%
\setbox\pst@hbox=\hbox\bgroup
\psm@beginmath
\begingroup
\ignorespaces}
\def\psm@endnode@i{%
\unskip
\endgroup
\psm@endmath
\egroup
\use@par
\@psttrue}
\def\psm@endnode{%
\@pstfalse
\psm@endnode@i
\ifnum\pscol>1 \hskip\pscolsep \fi
\psk@mnodesize
\hfil
\nodealigntrue
\if@pst
\csname mnode@\psk@mnode\endcsname
\else
\csname mnode@\psk@emnode\endcsname
\fi
\psk@mcol
\psk@@mnodesize}
% DG/SR modification begin - Sep. 3, 1999 - Patch 10 - From Michael Sharpe
%\def\psspan#1{\mscount#1\relax\loop\ifnum\mscount>\@ne \sp@n\repeat}
\def\psspan#1{\global\mscount#1\relax\pstloop\ifnum\mscount>\@ne\sp@n\repeat}
\def\pstloop#1\repeat{\gdef\pstiterate{#1\relax\expandafter\pstiterate\fi}%
\pstiterate
\let\pstiterate\relax}
% DG/SR modification end
\def\psset@name#1{\pst@getnode{#1}\psk@name}
\let\psk@name\relax
\def\psset@mcol#1{%
\ifx r#1\relax
\let\psk@mcol\relax
\else
\ifx l#1\relax
\let\psk@mcol\hfill
\else
\let\psk@mcol\hfil
\fi
\fi}
\psset@mcol{c}
\def\psset@mnodesize#1{%
\pssetlength\pst@dimg{#1}%
\ifdim\pst@dimg<\z@
\let\psk@mnodesize\relax
\let\psk@@mnodesize\relax
\else
\edef\psk@mnodesize{\noexpand\hbox to\number\pst@dimg sp\noexpand\bgroup}%
\let\psk@@mnodesize\egroup
\fi}
\psset@mnodesize{-1pt}
\def\mnode@R{\rnode@iii\Rnode@ii{\psm@thenode}}
\def\mnode@r{\rnode@iii\rnode@iv{\psm@thenode}}
\def\mnode@oval{\ovalnode@ii{\psm@thenode}}
\def\mnode@tri{\trinode@ii{\psm@thenode}}
\def\mnode@dia{\dianode@ii{\psm@thenode}}
\def\mnode@C{{\nodealigntrue\cnode@ii(\z@,\z@){\psk@radius}{\psm@thenode}}}
\def\mnode@f{{\nodealigntrue\fnode@ii(\z@,\z@){\psm@thenode}}}
\def\mnode@circle{\circlenode@ii{\psm@thenode}}
\def\mnode@p{\pnode(\z@,\z@){\psm@thenode}}
% DG/SR modification begin - Jul. 22, 1997 - Patch 1
\def\mnode@dot{\dotnode@ii(\z@,\z@){\psm@thenode}}
% DG/SR modification end
\def\mnode@none{\box\pst@hbox}
\def\psset@mnode#1{%
\@ifundefined{mnode@#1}%
{\@pstrickserr{\string\psmatrix\space node `#1' not defined.}\@ehpa}%
{\edef\psk@mnode{#1}}}
\def\psset@emnode#1{%
\@ifundefined{mnode@#1}%
{\@pstrickserr{\string\psmatrix\space node `#1' not defined.}\@ehpa}%
{\edef\psk@emnode{#1}}}
\psset@mnode{R}
\psset@emnode{none}
%%%% FROM pst-coil.tex
\def\nccoil{\pst@object{nccoil}}
\def\nccoil@i{\check@arrow{\nccoil@ii}}
\def\nccoil@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
\tx@NCCoor
tx@Dict begin
% DG/SR modification begin - Mar. 19, 1998 - Patch 5
4 2 roll
% DG/SR modification end
\psk@coilwidth \pscoilheight
\psk@coilarmA \psk@coilarmB
\psk@coilaspect \psk@coilinc
\pst@coildict \tx@Coil end
end}}
\def\nczigzag{\pst@object{nczigzag}}
\def\nczigzag@i{\check@arrow{\nczigzag@ii}}
\def\nczigzag@ii#1#2{\nc@object{Open}{#1}{#2}{.5}{%
\tx@NCCoor
tx@Dict begin
% DG/SR modification begin - Mar. 19, 1998 - Patch 5
4 2 roll
% DG/SR modification end
\pscoilheight
\psk@coilwidth
\psk@coilarmA
\psk@coilarmB
\pst@coildict \tx@ZigZag end
\psline@iii
\tx@Line
end}}
\catcode`\@=\TheAtCode\relax
\endinput
%%
%% END pst-node.tex

pstricks.tex
%% BEGIN: pstricks.tex
%%
%% PostScript macros for Generic TeX: main input file for PSTricks 97.
%% See the PSTricks User's Guide for description.
%% This uses the header file `pstricks.pro'.
%%
\def\fileversion{97 patch 14}
\def\filedate{1999/12/23}
%%
%% COPYRIGHT 1993, 1994, 1999 by Timothy Van Zandt, tvz@nwu.edu.
%%
%% This program can be redistributed and/or modified under the terms
%% of the LaTeX Project Public License Distributed from CTAN
%% archives in directory macros/latex/base/lppl.txt.
%%
\csname PSTricksLoaded\endcsname
\let\PSTricksLoaded\endinput
\edef\PstAtCode{\the\catcode`\@}
\catcode`\@=11\relax
\expandafter\ifx\csname @latexerr\endcsname\relax
\long\def\@ifundefined#1#2#3{\expandafter\ifx\csname
#1\endcsname\relax#2\else#3\fi}
\def\@namedef#1{\expandafter\def\csname #1\endcsname}
\def\@nameuse#1{\csname #1\endcsname}
\def\@eha{%
Your command was ignored.^^J
Type \space I <command> <return> \space to replace
it with another command,^^J
or \space <return> \space to continue without it.}
\def\@spaces{\space\space\space\space}
\def\typeout#1{\immediate\write\@unused{#1}}
\alloc@7\write\chardef\sixt@@n\@unused
\def\@empty{}
\def\@gobble#1{}
\def\@nnil{\@nil}
\def\@ifnextchar#1#2#3{%
\let\@tempe#1\def\@tempa{#2}\def\@tempb{#3}\futurelet\@tempc\@ifnch}
\def\@ifnch{%
\ifx\@tempc\@sptoken
\let\@tempd\@xifnch
\else
\ifx\@tempc\@tempe \let\@tempd\@tempa \else \let\@tempd\@tempb \fi
\fi
\@tempd}
\begingroup
\def\:{\global\let\@sptoken= } \:
\def\:{\@xifnch} \expandafter\gdef\: {\futurelet\@tempc\@ifnch}
\endgroup
\fi
\typeout{`PSTricks' v\fileversion\space\space <\filedate> (tvz)}
\def\@pstrickserr#1#2{%
\begingroup
\newlinechar`\^^J
\edef\pst@tempc{#2}%
\expandafter\errhelp\expandafter{\pst@tempc}%
\typeout{%
PSTricks error. \space See User's Guide for further information.^^J
\@spaces\@spaces\@spaces\@spaces
Type \space H <return> \space for immediate help.}%
\errmessage{#1}%
\endgroup}
\def\@ehpa{%
Your command was ignored. Default value substituted.^^J
Type \space <return> \space to procede.}
\def\@ehpb{%
Your command was ignored. Will recover best I can.^^J
Type \space <return> \space to procede.}
\def\@ehpc{%
You better fix this before proceding.^^J
See the PSTricks User's Guide or ask your system administrator for help.^^J
Type \space X <return> \space to quit.}
\def\pst@misplaced#1{\@pstrickserr{Misplaced \string#1 command}\@ehpb}
\newdimen\pst@dima
\newdimen\pst@dimb
\newdimen\pst@dimc
\newdimen\pst@dimd
\newdimen\pst@dimg
\newdimen\pst@dimh
\newbox\pst@hbox
\newbox\pst@boxg
\newcount\pst@cnta
\newcount\pst@cntb
\newcount\pst@cntc
\newcount\pst@cntd
\newcount\pst@cntg
\newcount\pst@cnth
\newif\if@pst
\newtoks\pst@toks
\newif\if@star
\def\pst@ifstar#1{%
\@ifnextchar*{\@startrue\def\next*{#1}\next}{\@starfalse#1}}
\def\pst@expandafter#1#2{%
\def\next{#1}%
\edef\@tempa{#2}%
\ifx\@tempa\@empty
\@pstrickserr{Unexpected empty argument!}\@ehpb
\def\@tempa{\@empty}%
\fi
\expandafter\next\@tempa}
\def\pst@dimtonum#1#2{\edef#2{\pst@@dimtonum#1}}
\def\pst@@dimtonum#1{\expandafter\pst@@@dimtonum\the#1}
{\catcode`\p=12 \catcode`\t=12 \global\@namedef{pst@@@dimtonum}#1pt{#1}}
\def\pst@pyth#1#2#3{%
\ifdim#1>#2\pst@@pyth#1#2#3\else\pst@@pyth#2#1#3\fi}
\def\pst@@pyth#1#2#3{%
\ifdim4#1>9#2%
#3=#1\advance#3 .2122#2%
\else
#3=.8384#1\advance#3 .5758#2%
\fi}
\def\pst@divide#1#2#3{%
\pst@@divide{#1}{#2}%
\pst@dimtonum\pst@dimg{#3}}
\def\pst@@divide#1#2{%
\pst@dimg=#1\relax
\pst@dimh=#2\relax
\pst@cntg=\pst@dimh
\pst@cnth=67108863
\pst@@@divide\pst@@@divide\pst@@@divide\pst@@@divide
\divide\pst@dimg\pst@cntg}
\def\pst@@@divide{%
\ifnum
\ifnum\pst@dimg<\z@-\fi\pst@dimg<\pst@cnth
\multiply\pst@dimg\sixt@@n
\else
\divide\pst@cntg\sixt@@n
\fi}
\def\pst@configerr#1{%
\@pstrickserr{\string#1 not defined in pstricks.con}\@ehpc}
% % \begin{macrocode}
\def\pstVerb#1{\pst@configerr\pstVerb}
\def\pstverb#1{\pst@configerr\pstverb}
\def\pstverbscale{\pst@configerr\pstverbscale}
\def\pstrotate{\pst@configerr\pstrotate}
\def\pstheader#1{\pst@configerr\pstheader}
\def\pstdriver{\pst@configerr\pstdriver}
\@ifundefined{pstcustomize}%
{\def\pstcustomize{\endinput\let\pstcustomize\relax}}{}
\input pstricks.con
\newif\ifPSTricks
\PSTrickstrue
\def\PSTricksOff{%
\def\pstheader##1{}%
\def\pstverb##1{}%
\def\pstVerb##1{}%
\PSTricksfalse}
\@ifundefined{pst@def}{\def\pst@def#1<#2>{\@namedef{tx@#1}{#2 }}}{}
\@ifundefined{pst@ATH}{\def\pst@ATH<#1>{}}{}
\pstheader{pstricks.pro}
\def\pst@dict{tx@Dict begin }
\def\pst@theheaders{pstricks.pro}
\def\pst@Verb#1{\pstVerb{\pst@dict #1 end}}
\def\tx@Atan{Atan }
\def\tx@Div{Div }
\def\tx@NET{NET }
\def\tx@Pyth{Pyth }
\def\tx@PtoC{PtoC }
\def\tx@PathLength@{PathLength@ }
\def\tx@PathLength{PathLength }
\pst@dimg=\pstunit\relax
\ifdim\pst@dimg=1bp
\def\pst@stp{.996264 dup scale}
\else
\edef\pst@stp{1 \pst@@dimtonum\pst@dimg\space div dup scale}
\fi
\def\tx@STP{STP }
\def\tx@STV{STV }
\def\pst@number#1{\pst@@dimtonum#1\space}
\def\pst@checknum#1#2{%
\edef\next{#1}%
\ifx\next\@empty
\let\pst@num\z@
\else
\expandafter\pst@@checknum\next..\@nil
\fi
\ifnum\pst@num=\z@
\@pstrickserr{Bad number: `#1'. 0 substituted.}\@ehpa
\def#2{0 }%
\else
\edef#2{\ifnum\pst@num=\tw@-\fi\the\pst@cntg.%
\expandafter\@gobble\the\pst@cnth\space}%
\fi}
\def\pst@@checknum{%
\@ifnextchar-%
{\let\pst@num\tw@\expandafter\pst@@@checknum\@gobble}%
{\let\pst@num\@ne\pst@@@checknum}}
\def\pst@@@checknum#1.#2.#3\@nil{%
\afterassignment\pst@@@@checknum\pst@cntg=0#1\relax\@nil
\afterassignment\pst@@@@checknum\pst@cnth=1#2\relax\@nil}
\def\pst@@@@checknum#1\relax\@nil{%
\ifx\@nil#1\@nil\else\let\pst@num\z@\fi}
\def\pst@getnumii#1 #2 #3\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph}
\def\pst@getnumiii#1 #2 #3 #4\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi}
\def\pst@getnumiv#1 #2 #3 #4 #5\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi
\pst@checknum{#4}\pst@tempj}
\def\pst@getdimnum#1 #2 #3\@nil{%
\pssetlength\pst@dimg{#1}%
\pst@checknum{#2}\pst@tempg}
% DG/SR modification begin - Jan. 7, 1998 - Patch 9
% Missing from pstricks.bug 0.93
%\def\pst@getscale#1#2{%
%\pst@expandafter\pst@getnumii{#1 #1} {} {} {}\@nil
%\@psttrue
%\ifdim\pst@tempg\p@=\z@
%\@pstrickserr{Bad scaling argument `#1'}\@ehpa
%\def\pst@tempg{1 }%
%\@pstfalse
%\fi
%\ifdim\pst@temph\p@=\z@
%\if@pst\@pstrickserr{Bad scaling argument `#1'}\@ehpa\fi
%\def\pst@temph{1 }%
%\fi
%\edef#2{\pst@tempg\space \pst@temph\space scale }%
%\ifdim\pst@tempg\p@=\p@ \ifdim\pst@temph\p@=\p@
%\def#2{}%
%\fi\fi}
\def\pst@getscale#1#2{%
\edef\pst@tempg{#1}%
\ifx\pst@tempg\@none
\def#2{}%
\else
\pst@expandafter\pst@getnumii{#1 #1} {} {} {}\@nil
\ifdim\pst@tempg\p@=\z@
\@pstrickserr{Bad scaling argument `#1'}\@ehpa
\def#2{}%
\else
\ifdim\pst@temph\p@=\z@
\@pstrickserr{Bad scaling argument `#1'}\@ehpa
\def#2{}%
\else
\edef#2{\pst@tempg\space \pst@temph\space scale }%
\fi
\fi
\fi}
% DG/SR modification end
\def\pst@getint#1#2{%
\pst@cntg=#1\relax
\edef#2{\the\pst@cntg\space}}
\begingroup
\catcode`\{=12
\catcode`\}=12
\catcode`\[=1
\catcode`\]=2
\gdef\pslbrace[{ ]
\gdef\psrbrace[} ]
\endgroup
\def\@newcolor#1#2{%
\expandafter\edef\csname #1\endcsname{\noexpand\pst@color{#2}}%
\expandafter\edef\csname color@#1\endcsname{#2}%
\ignorespaces}
\def\pst@color#1{%
\def\pst@currentcolor{#1}\pstVerb{#1}\aftergroup\pst@endcolor}
\def\pst@endcolor{\pstVerb{\pst@currentcolor}}
\def\pst@currentcolor{0 setgray}
\def\altcolormode{%
\def\pst@color##1{%
\pstVerb{gsave ##1}\aftergroup\pst@endcolor}%
\def\pst@endcolor{\pstVerb{\pst@grestore}}}
\def\pst@grestore{%
currentpoint
matrix currentmatrix
currentfont
grestore
setfont
setmatrix
moveto}
\def\pst@usecolor#1{\csname color@#1\endcsname\space}
\def\newgray#1#2{%
\pst@checknum{#2}\pst@tempg
\@newcolor{#1}{\pst@tempg setgray}}
\def\newrgbcolor#1#2{%
\pst@expandafter\pst@getnumiii{#2} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi setrgbcolor}}
\def\newhsbcolor#1#2{%
\pst@expandafter\pst@getnumiii{#2} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi sethsbcolor}}
\def\newcmykcolor#1#2{%
\pst@expandafter\pst@getnumiv{#2} {} {} {} {} {}\@nil
\@newcolor{#1}{\pst@tempg \pst@temph \pst@tempi \pst@tempj setcmykcolor}}
\newgray{black}{0}
\newgray{darkgray}{.25}
\newgray{gray}{.5}
\newgray{lightgray}{.75}
\newgray{white}{1}
\newrgbcolor{red}{1 0 0}
\newrgbcolor{green}{0 1 0}
\newrgbcolor{blue}{0 0 1}
\newrgbcolor{yellow}{1 1 0}
\newrgbcolor{cyan}{0 1 1}
\newrgbcolor{magenta}{1 0 1}
\def\psset#1{\@psset#1,\@nil\ignorespaces}
\def\@psset#1,{%
\@@psset#1==\@nil
\@ifnextchar\@nil{\@gobble}{\@psset}}
\def\@@psset#1=#2=#3\@nil{%
\@ifundefined{psset@#1}%
{\@pstrickserr{Graphics parameter `#1' not defined.}\@ehpa}%
{\@nameuse{psset@#1}{#2}}}%
\def\psset@style#1{%
\@ifundefined{pscs@#1}%
{\@pstrickserr{Custom style `#1' undefined}\@ehpa}%
{\@nameuse{pscs@#1}}}
\def\newpsstyle#1#2{\@namedef{pscs@#1}{\psset{#2}}}
\def\@none{none}
\def\pst@getcolor#1#2{%
\@ifundefined{color@#1}%
{\@pstrickserr{Color `#1' not defined}\@eha}%
{\edef#2{#1}}}
\newdimen\psunit \psunit 1cm
\newdimen\psxunit \psxunit 1cm
\newdimen\psyunit \psyunit 1cm
\let\psrunit\psunit
\def\pstunit@off{\let\@psunit\ignorespaces\ignorespaces}
\def\pssetlength#1#2{%
\let\@psunit\psunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\psaddtolength#1#2{%
\let\@psunit\psunit
\afterassignment\pstunit@off
\advance#1 #2\@psunit}
\def\pssetxlength#1#2{%
\let\@psunit\psxunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\pssetylength#1#2{%
\let\@psunit\psyunit
\afterassignment\pstunit@off
#1 #2\@psunit}
\def\psset@unit#1{%
\pssetlength\psunit{#1}%
\psxunit=\psunit
\psyunit=\psunit}
\def\psset@runit#1{\pssetlength\psrunit{#1}}
\def\psset@xunit#1{\pssetxlength\psxunit{#1}}
\def\psset@yunit#1{\pssetylength\psyunit{#1}}
\def\pst@getlength#1#2{%
\pssetlength\pst@dimg{#1}%
\edef#2{\pst@number\pst@dimg}}
\def\pst@@getlength#1#2{%
\pssetlength\pst@dimg{#1}%
\edef#2{\number\pst@dimg sp}}
\def\pst@getcoor#1#2{\pst@@getcoor{#1}\let#2\pst@coor}
\def\pst@coor{0 0 }
\def\pst@getcoors#1#2{%
\def\pst@aftercoors{\addto@pscode{#1 \pst@coors }#2}%
\def\pst@coors{}%
\pst@@getcoors}
\def\pst@@getcoors(#1){%
\pst@@getcoor{#1}%
\edef\pst@coors{\pst@coor\pst@coors}%
\@ifnextchar({\pst@@getcoors}{\pst@aftercoors}}
\def\pst@getangle#1#2{\pst@@getangle{#1}\let#2\pst@angle}
\def\pst@angle{0 }
\def\cartesian@coor#1,#2,#3\@nil{%
\pssetxlength\pst@dimg{#1}%
\pssetylength\pst@dimh{#2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@number\pst@dimh}}
\def\NormalCoor{%
\def\pst@@getcoor##1{\pst@expandafter\cartesian@coor{##1},\relax,\@nil}%
\def\pst@@getangle##1{%
\pst@checknum{##1}\pst@angle
\edef\pst@angle{\pst@angle \pst@angleunit}}%
\def\psput@##1{\pst@@getcoor{##1}\leavevmode\psput@cartesian}}
\NormalCoor
\def\degrees{\@ifnextchar[{\@degrees}{\def\pst@angleunit{}}}
\def\@degrees[#1]{%
\pst@checknum{#1}\pst@tempg
\edef\pst@angleunit{360 \pst@tempg div mul }%
\ignorespaces}
\def\radians{\def\pst@angleunit{57.2956 mul }}
\def\pst@angleunit{}
\def\SpecialCoor{%
\def\pst@@getcoor##1{%
\begingroup
\pst@activecoor
\xdef\pst@tempg{##1}%
\endgroup
\expandafter\special@coor\pst@tempg||\@nil}%
\def\pst@@getangle##1{%
\begingroup
\pst@activecoor
\xdef\pst@tempg{##1}%
\endgroup
\expandafter\special@angle\pst@tempg\@empty)\@nil}%
\def\psput@##1{\pst@@getcoor{##1}\leavevmode\psput@special}}
\begingroup
\catcode`\|=13
\catcode`\;=13
\catcode`\!=13
\gdef\pst@activecoor{%
\def|{\string|}%
\def;{\string;}%
\def!{\string!}}
\endgroup
\def\special@coor#1|#2|#3\@nil{%
\ifx#3|\relax
\mixed@coor{#1}{#2}%
\else
\special@@coor#1;;\@nil
\fi}
\def\special@@coor#1{%
\ifcat#1a\relax
\def\next{\node@coor#1}%
\else
\ifx#1[\relax
\def\next{\Node@coor[}%
\else
\ifx#1!\relax
\def\next{\raw@coor}%
\else
\def\next{\special@@@coor#1}%
\fi
\fi
\fi
\next}
\def\special@@@coor#1;#2;#3\@nil{%
\ifx#3;\relax
\polar@coor{#1}{#2}%
\else
\cartesian@coor#1,\relax,\@nil
\fi}
\def\mixed@coor#1#2{%
\begingroup
% DG/SR modification begin - Oct. 27, 1997 - Patch 6
%\specialcoor@ii#1;;\@nil
%\let\pst@tempa\pst@coor
%\specialcoor@ii#2;;\@nil
\special@@coor#1;;\@nil
\let\pst@tempa\pst@coor
\special@@coor#2;;\@nil
% DG/SR modification end
\xdef\pst@tempg{\pst@tempa pop \pst@coor exch pop }%
\endgroup
\let\pst@coor\pst@tempg}
\def\polar@coor#1#2{%
\pssetlength\pst@dimg{#1}%
\pst@@getangle{#2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@angle \tx@PtoC}}
\def\raw@coor#1;#2\@nil{%
\edef\pst@coor{%
#1 \pst@number\psyunit mul exch \pst@number\psxunit mul exch }}
\def\node@coor#1\@nil{%
\@pstrickserr{You must load `pst-node.tex' to use node coordinates.}\@ehps
\def\pst@coor{0 0 }}
\def\Node@coor{\node@coor}
\def\special@angle#1#2)#3\@nil{%
\ifx!#1\relax
\edef\pst@angle{#2 \pst@angleunit}%
\else
\ifx(#1\relax
\pst@@getcoor{#2}%
\edef\pst@angle{\pst@coor exch \tx@Atan}%
\else
\pst@checknum{#1#2}\pst@angle
\edef\pst@angle{\pst@angle \pst@angleunit}%
\fi
\fi}
\def\Cartesian{%
\def\cartesian@coor##1,##2,##3\@nil{%
\pssetxlength\pst@dimg{##1}%
\pssetylength\pst@dimh{##2}%
\edef\pst@coor{\pst@number\pst@dimg \pst@number\pst@dimh}}%
\@ifnextchar({\Cartesian@}{}}
\def\Cartesian@(#1,#2){%
\pssetxlength\psxunit{#1}%
\pssetylength\psyunit{#2}%
\ignorespaces}
\def\Polar{%
\def\psput@cartesian{\psput@special}%
\def\cartesian@coor##1,##2,##3\@nil{\polar@coor{##1}{##2}}}%
\def\psset@origin#1{%
\pst@@getcoor{#1}%
\edef\psk@origin{\pst@coor \tx@NET }}
\def\psk@origin{}
\newif\ifswapaxes
\def\psset@swapaxes#1{%
\@nameuse{@pst#1}%
\if@pst
\def\psk@swapaxes{-90 rotate -1 1 scale }%
\else
\def\psk@swapaxes{}%
\fi}
\psset@swapaxes{false}
\newif\ifshowpoints
\def\psset@showpoints#1{\@nameuse{showpoints#1}}
\psset@showpoints{false}
\let\pst@setrepeatarrowsflag\relax
\def\psset@border#1{%
\pst@getlength{#1}\psk@border
\pst@setrepeatarrowsflag}
\psset@border{0pt}
\def\psset@bordercolor#1{\pst@getcolor{#1}\psbordercolor}
\psset@bordercolor{white}
\newif\ifpsdoubleline
\def\psset@doubleline#1{%
\@nameuse{psdoubleline#1}%
\pst@setrepeatarrowsflag}
\psset@doubleline{false}
\def\psset@doublesep#1{\def\psdoublesep{#1}}
\psset@doublesep{1.25\pslinewidth}
\def\psset@doublecolor#1{\pst@getcolor{#1}\psdoublecolor}
\psset@doublecolor{white}
\newif\ifpsshadow
\def\psset@shadow#1{%
\@nameuse{psshadow#1}%
\pst@setrepeatarrowsflag}
\psset@shadow{false}
\def\psset@shadowsize#1{\pst@getlength{#1}\psk@shadowsize}
\psset@shadowsize{3pt}
\def\psset@shadowangle#1{\pst@getangle{#1}\psk@shadowangle}
\psset@shadowangle{-45}
\def\psset@shadowcolor#1{\pst@getcolor{#1}\psshadowcolor}
\psset@shadowcolor{darkgray}
\def\pst@repeatarrowsflag{\z@}
\def\pst@setrepeatarrowsflag{%
\edef\pst@repeatarrowsflag{%
\ifdim\psk@border\p@>\z@ 1\else\ifpsdoubleline 1\else
\ifpsshadow 1\else \z@\fi\fi\fi}}
\def\psls@none{}
\newdimen\pslinewidth
\def\psset@linewidth#1{\pssetlength\pslinewidth{#1}}
\psset@linewidth{.8pt}
\def\psset@linecolor#1{\pst@getcolor{#1}\pslinecolor}
\psset@linecolor{black}
\def\psls@solid{0 setlinecap stroke }
\def\pst@missing{%
\z@
\@pstrickserr{Missing number or dimension. 0 substituted}\@ehpa}
\def\psset@dash#1{%
\pst@expandafter\psset@@dash{#1} {\pst@missing} {\pst@missing} {}\@nil
\edef\psk@dash{\pst@number\pst@dimg \pst@number\pst@dimh}}
\def\psset@@dash#1 #2 #3\@nil{%
\pssetlength\pst@dimg{#1}%
\pssetlength\pst@dimh{#2}}
\psset@dash{5pt 3pt}
\newif\ifpsdashadjust
\def\psset@dashadjust#1{\@nameuse{psdashadjust#1}}
\psset@dashadjust{true}
\def\psls@dashed{%
\ifpsdashadjust
\psk@dash \pst@linetype\space \tx@DashLine
\else
[ \psk@dash ] 0 setdash stroke
\fi}
\def\tx@DashLine{DashLine }
\def\psset@dotsep#1{\pst@getlength{#1}\psk@dotsep}
\psset@dotsep{3pt}
\def\psls@dotted{%
\ifpsdashadjust
\psk@dotsep \pst@linetype\space \tx@DotLine
\else
[ 0 \psk@dotsep CLW add ] 0 setdash 1 setlinecap stroke
\fi}
\def\tx@DotLine{DotLine }
\def\psset@linestyle#1{%
\@ifundefined{psls@#1}%
{\@pstrickserr{Line style `#1' not defined}\@eha}%
{\edef\pslinestyle{#1}}}
\psset@linestyle{solid}
\def\psfs@none{}
\def\psset@fillcolor#1{\pst@getcolor{#1}\psfillcolor}
\psset@fillcolor{white}
\def\psfs@solid{\pst@fill{\pst@usecolor\psfillcolor fill}}
\def\psset@hatchwidth#1{\pst@getlength{#1}\psk@hatchwidth}
\psset@hatchwidth{.8pt}
\def\psset@hatchsep#1{\pst@getlength{#1}\psk@hatchsep}
\psset@hatchsep{4pt}
\def\psset@hatchcolor#1{\pst@getcolor{#1}\pshatchcolor}
\psset@hatchcolor{black}
\def\psset@hatchangle#1{\pst@getangle{#1}\psk@hatchangle}
\psset@hatchangle{45}
\def\pst@linefill{%
\psk@hatchangle rotate
\psk@hatchwidth SLW
\pst@usecolor\pshatchcolor
\psk@hatchsep \tx@LineFill}
\def\psfs@vlines{\pst@fill\pst@linefill}
\@namedef{psfs@vlines*}{\psfs@solid \psfs@vlines}
\def\psfs@hlines{\pst@fill{90 rotate \pst@linefill}}
\@namedef{psfs@hlines*}{\psfs@solid \psfs@hlines}
\def\psfs@crosshatch{\psfs@vlines \psfs@hlines}
\@namedef{psfs@crosshatch*}{\psfs@solid \psfs@vlines \psfs@hlines}
\def\tx@LineFill{LineFill }
\def\psset@fillstyle#1{%
\edef\pst@tempg{#1}\def\pst@temph{none}%
\ifx\pst@tempg\pst@temph
\let\psk@fillstyle\relax
\else
\@ifundefined{psfs@#1}%
{\@pstrickserr{Undefined fill style: `#1'}\@eha}%
{\edef\psk@fillstyle{\expandafter\noexpand\csname psfs@#1\endcsname}}%
\fi}
\def\psset@addfillstyle#1{%
\@ifundefined{psfs@#1}%
{\@pstrickserr{Undefined fill style: `#1'}\@eha}%
{\edef\psk@fillstyle{%
\expandafter\noexpand\psk@fillstyle
\expandafter\noexpand\csname psfs@#1\endcsname}}}
\psset@fillstyle{none}
\def\psset@arrows#1{%
\begingroup
\pst@activearrows
\xdef\pst@tempg{#1}%
\endgroup
\expandafter\psset@@arrows\pst@tempg\@empty-\@empty\@nil
\if@pst\else
\@pstrickserr{Bad arrows specification: #1}\@ehpa
\fi}
\def\psset@@arrows#1-#2\@empty#3\@nil{%
\@psttrue
\def\next##1,#1-##2,##3\@nil{\def\pst@tempg{##2}}%
\expandafter\next\pst@arrowtable,#1-#1,\@nil
\@ifundefined{psas@\pst@tempg}%
{\@pstfalse\def\psk@arrowA{}}%
{\let\psk@arrowA\pst@tempg}%
\@ifundefined{psas@#2}%
{\@pstfalse\def\psk@arrowB{}}%
{\def\psk@arrowB{#2}}}
\def\psk@arrowA{}
\def\psk@arrowB{}
\def\pst@arrowtable{,<->,<<->>,>-<,>>-<<,(-),[-]}
\begingroup
\catcode`\<=13
\catcode`\>=13
\catcode`\|=13
\gdef\pst@activearrows{\def<{\string<}\def>{\string>}\def|{\string|}}
\endgroup
\def\tx@BeginArrow{BeginArrow }
\def\tx@EndArrow{EndArrow }
\def\psset@arrowscale#1{\pst@getscale{#1}\psk@arrowscale}
\psset@arrowscale{1}
\def\psset@arrowsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@arrowsize{\pst@number\pst@dimg \pst@tempg}}
\psset@arrowsize{1.5pt 2}
\def\psset@arrowlength#1{\pst@checknum{#1}\psk@arrowlength}
\psset@arrowlength{1.4}
\def\psset@arrowinset#1{\pst@checknum{#1}\psk@arrowinset}%
\psset@arrowinset{.4}
\def\tx@Arrow{Arrow }
\@namedef{psas@>}{%
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@>>}{%
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
0 h T
gsave
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
CP
grestore
CP newpath moveto
2 copy
L
stroke
moveto}
\@namedef{psas@<}{%
true \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@<<}{%
true \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow
CP newpath moveto 0 a neg L stroke 0 h neg T
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\def\psset@tbarsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@tbarsize{\pst@number\pst@dimg \pst@tempg}}
\psset@tbarsize{2pt 5}
\def\tx@Tbar{Tbar }
\@namedef{psas@|}{\psk@tbarsize \tx@Tbar}
\@namedef{psas@|*}{0 CLW -2 div T \psk@tbarsize \tx@Tbar}
\@namedef{psas@>|}{%
\psk@tbarsize \tx@Tbar
0 CLW 2 div T
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\@namedef{psas@>|*}{%
0 CLW -2 div T
\psk@tbarsize \tx@Tbar
0 CLW 2 div T
newpath
false \psk@arrowinset \psk@arrowlength \psk@arrowsize \tx@Arrow}
\edef\pst@arrowtable{\pst@arrowtable,|<*->|*,|<->|}
\def\psset@bracketlength#1{\pst@checknum{#1}\psk@bracketlength}
\psset@bracketlength{.15}
\def\tx@Bracket{Bracket }
\@namedef{psas@]}{\psk@bracketlength \psk@tbarsize \tx@Bracket}
\def\psset@rbracketlength#1{\pst@checknum{#1}\psk@rbracketlength}
\psset@rbracketlength{.15}
\def\tx@RoundBracket{RoundBracket }
\@namedef{psas@)}{\psk@rbracketlength \psk@tbarsize \tx@RoundBracket}
\def\psas@c{1 \psas@@c}
\def\psas@cc{0 CLW 2 div T 1 \psas@@c}
\def\psas@C{2 \psas@@c}
\def\psas@@c{%
setlinecap
0 0 moveto
0 CLW 2 div L
stroke
0 0 moveto}
\def\psas@{}
\psset@arrows{-}
\def\tx@SD{SD }
\def\tx@EndDot{EndDot }
\def\psas@oo{{\pst@usecolor\psfillcolor true} true \psk@dotsize \tx@EndDot}
\def\psas@o{{\pst@usecolor\psfillcolor true} false \psk@dotsize \tx@EndDot}
\@namedef{psas@**}{{false} true \psk@dotsize \tx@EndDot}
\@namedef{psas@*}{{false} false \psk@dotsize \tx@EndDot}
\def\pst@par{}
\def\addto@par#1{%
\ifx\pst@par\@empty
\def\pst@par{#1}%
\else
\expandafter\def\expandafter\pst@par\expandafter{\pst@par,#1}%
\fi}
\def\addbefore@par#1{%
\ifx\pst@par\@empty
\def\pst@par{#1}%
\else
\toks@{#1}%
\pst@toks\expandafter{\pst@par}%
\edef\pst@par{\the\toks@,\the\pst@toks}%
\fi}
\def\use@par{%
\ifx\pst@par\@empty\else
\expandafter\@psset\pst@par,\@nil
\def\pst@par{}%
\fi}
\def\pst@object#1{%
\pst@ifstar{%
\@ifnextchar[%
{\pst@@object{#1}}%
{\def\pst@par{}\@nameuse{#1@i}}}}
\def\pst@@object#1[#2]{%
\def\pst@par{#2}%
\@ifnextchar+{\@nameuse{#1@i}}{\@nameuse{#1@i}}}
\def\newpsobject#1#2#3{%
\@ifundefined{#2@i}%
{\@pstrickserr{Graphics object `#2' not defined}\@eha}{%
\@namedef{#1}{\pst@object{#1}}%
\@namedef{#1@i}{\addbefore@par{#3}\@nameuse{#2@i}}}%
\ignorespaces}
\def\pst@getarrows#1{\@ifnextchar({#1}{\pst@@getarrows{#1}}}
\def\pst@@getarrows#1#2{\addto@par{arrows=#2}#1}
\def\begin@ClosedObj{%
\leavevmode
\pst@killglue
\begingroup
\use@par
\solid@star
\ifpsdoubleline \pst@setdoublesep \fi
\init@pscode}
\def\end@ClosedObj{%
\ifpsshadow \pst@closedshadow \fi
\ifdim\psk@border\p@>\z@ \pst@addborder \fi
\psk@fillstyle
\pst@stroke
\ifpsdoubleline \pst@doublestroke \fi
\ifshowpoints
% DG modification begin - Mar. 4, 1995
%\addto@pscode{Points aload length 2 div cvi /N ED \psdots@iii}%
\pst@OpenShowPoints
% DG modification end
\fi
\use@pscode
\endgroup
\ignorespaces}
\def\begin@OpenObj{%
\begin@ClosedObj
\let\pst@linetype\pst@arrowtype
\pst@addarrowdef}
\def\begin@AltOpenObj{%
\begin@ClosedObj
\def\pst@repeatarrowsflag{\z@}%
\def\pst@linetype{0}}
\def\end@OpenObj{%
\ifpsshadow \pst@openshadow \fi
\ifdim\psk@border\p@>\z@ \pst@addborder \fi
\psk@fillstyle
\pst@stroke
\ifpsdoubleline \pst@doublestroke \fi
\ifnum\pst@repeatarrowsflag>\z@ \pst@repeatarrows \fi
\ifshowpoints \pst@OpenShowPoints \fi
\use@pscode
\endgroup
\ignorespaces}
\def\begin@SpecialObj{%
\leavevmode
\pst@killglue
\begingroup
\use@par
\init@pscode}
\def\end@SpecialObj{%
\use@pscode
\endgroup
\ignorespaces}
\def\pst@code{}%
\def\init@pscode{%
\addto@pscode{%
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}
\def\addto@pscode#1{\xdef\pst@code{\pst@code#1\space}}
\def\use@pscode{%
\pstverb{%
\pst@dict
\tx@STP
\pst@newpath
\psk@origin
\psk@swapaxes
\pst@code
end}%
\gdef\pst@code{}}
\def\pst@newpath{newpath }
\def\pst@@killglue{\unskip\ifdim\lastskip>\z@\expandafter\pst@@killglue\fi}
\def\KillGlue{\let\pst@killglue\pst@@killglue}
\def\DontKillGlue{\let\pst@killglue\relax}
\DontKillGlue
\def\solid@star{%
\if@star
\pslinewidth=\z@
\psdoublelinefalse
\def\pslinestyle{none}%
\def\psk@fillstyle{\psfs@solid}%
\let\psfillcolor\pslinecolor
\fi}
\def\pst@setdoublesep{%
\pst@getlength\psdoublesep\psdoublesep
\pslinewidth=2\pslinewidth
\advance\pslinewidth\psdoublesep\p@
\let\pst@setdoublesep\relax}
\def\tx@Shadow{Shadow }
\def\pst@closedshadow{%
\addto@pscode{%
gsave
\psk@shadowsize \psk@shadowangle \tx@PtoC
\tx@Shadow
\pst@usecolor\psshadowcolor
gsave fill grestore
stroke
grestore
gsave
\pst@usecolor\psfillcolor
gsave fill grestore
stroke
grestore}}
\def\pst@openshadow{%
\addto@pscode{%
gsave
\psk@shadowsize \psk@shadowangle \tx@PtoC
\tx@Shadow
\pst@usecolor\psshadowcolor
\ifx\psk@fillstyle\relax\else
gsave fill grestore
\fi
stroke}%
\pst@repeatarrows
\addto@pscode{grestore}
\ifx\psk@fillstyle\relax\else
\addto@pscode{%
gsave
\pst@usecolor\psfillcolor
gsave fill grestore
stroke
grestore}
\fi}
\def\pst@addborder{%
\addto@pscode{%
gsave
\psk@border 2 mul
CLW add SLW
\pst@usecolor\psbordercolor
stroke
grestore}}
\def\pst@stroke{%
\ifx\pslinestyle\@none\else
\addto@pscode{%
gsave
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor
\@nameuse{psls@\pslinestyle}
grestore}%
\fi}
\def\pst@fill#1{\addto@pscode{gsave #1 grestore}}%
\def\pst@doublestroke{%
\addto@pscode{%
gsave
\psdoublesep SLW
\pst@usecolor\psdoublecolor
stroke
grestore}}
\def\pst@arrowtype{%
\ifx\psk@arrowB\@empty 0 \else -2 \fi
\ifx\psk@arrowA\@empty 0 \else -1 \fi
add}
\def\pst@addarrowdef{%
\addto@pscode{%
/ArrowA {
\ifx\psk@arrowA\@empty
\pst@oplineto
\else
\pst@arrowdef{A}
moveto
\fi
} def
/ArrowB {
\ifx\psk@arrowB\@empty \else \pst@arrowdef{B} \fi
} def}}
\def\pst@arrowdef#1{%
\ifnum\pst@repeatarrowsflag>\z@
/Arrow#1c [ 6 2 roll ] cvx def Arrow#1c
\fi
\tx@BeginArrow
\psk@arrowscale
\@nameuse{psas@\@nameuse{psk@arrow#1}}
\tx@EndArrow}
\def\pst@repeatarrows{%
\addto@pscode{%
gsave
\ifx\psk@arrowA\@empty\else
ArrowAc ArrowA pop pop
\fi
\ifx\psk@arrowB\@empty\else
ArrowBc ArrowB pop pop pop pop
\fi
grestore}}
\def\pst@OpenShowPoints{%
\addto@pscode{%
gsave
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}
newpath
Points aload length 2 div 2 sub cvi /N ED
N 0 ge
{ \ifx\psk@arrowA\@empty
Dot
\else
pop pop
\fi
N { Dot } repeat
\ifx\psk@arrowB\@empty
Dot
\else
pop pop
\fi }
{ N 2 mul { pop } repeat }
ifelse
grestore}}
\def\pscustom{\pst@object{pscustom}}
\long\def\pscustom@i#1{%
\begin@SpecialObj
\solid@star
\let\pst@ifcustom\iftrue
\let\begin@ClosedObj\begin@CustomObj
\let\end@ClosedObj\endgroup
\def\begin@OpenObj{\begin@CustomObj\pst@addarrowdef}%
\let\end@OpenObj\endgroup
\let\begin@AltOpenObj\begin@CustomObj
\def\begin@SpecialObj{%
\begingroup
\pst@misplaced{special graphics object}%
\def\addto@pscode####1{}
\let\end@SpecialObj\endgroup}%
\def\@multips(##1)(##2)##3##4{\pst@misplaced\multips}%
\def\psclip##1{\pst@misplaced\psclip}%
\def\pst@repeatarrowsflag{\z@}%
\let\pst@setrepeatarrowsflag\relax
\showpointsfalse
\let\showpointstrue\relax
\def\pst@linetype{\pslinetype}%
\let\psset@liftpen\psset@@liftpen
\psset@liftpen{\z@}%
\def\pst@cp{/currentpoint load stopped pop }%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\def\pst@optcp##1##2{%
\ifnum##1=\z@\def##2{/currentpoint load stopped { 0 0 } if }\fi}%
\let\caddto@pscode\addto@pscode
\def\cuse@par##1{{\use@par##1}}%
\the\pst@customdefs
\setbox\pst@hbox=\hbox{#1}%
\psk@fillstyle
\pst@stroke
\end@SpecialObj}
\def\begin@CustomObj{%
\begingroup
\use@par
\addto@pscode{%
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}
\def\pst@oplineto{moveto }
\def\pst@cp{}
\def\pst@optcp#1#2{}
\def\psset@liftpen#1{}
\def\psset@@liftpen#1{%
\ifcase#1\relax
\def\psk@liftpen{\z@}%
\def\pst@cp{/currentpoint load stopped pop }%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\or
\def\psk@liftpen{1}%
\def\pst@cp{}%
\def\pst@oplineto{/lineto load stopped { moveto } if }%
\or
\def\psk@liftpen{2}%
\def\pst@cp{}%
\def\pst@oplineto{moveto }%
\fi}
\psset@liftpen{0}
\def\psk@liftpen{-1}
\def\psset@linetype#1{%
\pst@getint{#1}\pslinetype
\ifnum\pst@dimg<-3
\@pstrickserr{linetype must be greater than -3}\@ehpa
\def\pslinetype{0}%
\fi}
\psset@linetype{0}
\def\caddto@pscode#1{%
\@pstrickserr{Command can only be used in \string\pscustom}\@ehpa}
\let\cuse@par\caddto@pscode
\def\tx@MSave{%
/msavemtrx
[ tx@Dict /msavemtrx known { msavemtrx aload pop } if CM ]
def }
\def\tx@MRestore{%
tx@Dict /msavemtrx known { length 0 gt } { false } ifelse
{ /msavematrx [ msavematrx aload pop setmatrix ] def }
if }
\newtoks\pst@customdefs
\pst@customdefs{%
\def\newpath{\addto@pscode{newpath}}%
\def\moveto(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor moveto}}%
\def\closepath{\addto@pscode{closepath}}%
\def\gsave{\begingroup\addto@pscode{gsave}}%
\def\grestore{\endgroup\addto@pscode{grestore}}%
% DG/SR modification begin - May 12, 1997 - Patch 2
%\def\translate(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor moveto}}%
\def\translate(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor translate}}%
% DG/SR modification end
\def\rotate#1{\pst@@getangle{#1}\addto@pscode{\pst@angle rotate}}%
\def\scale#1{\pst@getscale{#1}\pst@tempg\addto@pscode{\pst@tempg}}%
\def\msave{\addto@pscode{\tx@MSave}}%
\def\mrestore{\addto@pscode{\tx@MRestore}}%
\def\swapaxes{\addto@pscode{-90 rotate -1 1 scale}}%
\def\stroke{\pst@object{stroke}}%
\def\fill{\pst@object{fill}}%
\def\openshadow{\pst@object{openshadow}}%
\def\closedshadow{\pst@object{closedshadow}}%
% DG/SR modification begin - Jan. 7, 1998 - Patch 8
%\def\movepath(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor tx@Shadow}}%
\def\movepath(#1){\pst@@getcoor{#1}\addto@pscode{\pst@coor \tx@Shadow}}%
% DG/SR modification end
\def\lineto{\pst@onecoor{lineto}}%
\def\rlineto{\pst@onecoor{rlineto}}%
\def\curveto{\pst@threecoor{curveto}}%
\def\rcurveto{\pst@threecoor{rcurveto}}%
\def\code#1{\addto@pscode{#1}}%
\def\coor(#1){\pst@@getcoor{#1}\addto@pscode\pst@coor\@ifnextchar({\coor}{}}%
\def\rcoor{\pst@getcoors{}{}}%
\def\dim#1{\pssetlength\pst@dimg{#1}\addto@pscode{\pst@number\pst@dimg}}%
\def\setcolor#1{%
\@ifundefined{color@#1}{}{\addto@pscode{\use@color{#1}}}}%
\def\arrows#1{{\psset@arrows{#1}\pst@addarrowdef}}%
\let\file\pst@rawfile
} % END \pst@customdefs
\def\closedshadow@i{\cuse@par\pst@closedshadow}
\def\openshadow@i{\cuse@par\pst@openshadow}
\def\stroke@i{\cuse@par\pst@stroke}%
\def\fill@i{\cuse@par\psk@fillstyle}%
\def\pst@onecoor#1(#2){%
\pst@@getcoor{#2}%
\addto@pscode{\pst@coor #1}}
\def\pst@threecoor#1(#2)#3(#4)#5(#6){%
\begingroup
\pst@getcoor{#2}\pst@tempa
\pst@getcoor{#4}\pst@tempb
% DG/SR modification begin - Aug.  4, 1999 - Patch 11
%\pst@getcoor{#6}\pst@tembc
\pst@getcoor{#6}\pst@tempc
% DG/SR modification end
\addto@pscode{\pst@tempa \pst@tempb \pst@tempc #1}%
\endgroup}
\def\pst@rawfile#1{%
\begingroup
\def\do##1{\catcode`##1=12\relax}"
\dospecials
\catcode`\%=14
\pst@@rawfile{#1}%
\endgroup}
\def\pst@@rawfile#1{%
\immediate\openin1 #1
\ifeof1
\@pstrickserr{File `#1' not found}\@ehpa
\else
\immediate\read1 to \pst@tempg
\loop
\ifeof1 \@pstfalse\else\@psttrue\fi
\if@pst
\addto@pscode\pst@tempg
\immediate\read1 to \pst@tempg
\repeat
\fi
\immediate\closein1\relax}
\def\tx@NArray{NArray }
\def\tx@NArray{NArray }
\def\tx@Line{Line }
\def\tx@Arcto{Arcto }
\def\tx@CheckClosed{CheckClosed }
\def\tx@Polygon{Polygon }
\def\psset@gangle#1{\pst@getangle{#1}\psk@gangle}
\psset@gangle{0}
\def\tx@Diamond{Diamond }
\def\psdiamond{\pst@object{psdiamond}}
\def\psdiamond@i(#1){%
\@ifnextchar({\psdiamond@ii(#1)}{\psdiamond@ii(0,0)(#1)}}
\def\psdiamond@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\addto@pscode{%
\psline@iii
pop
\psk@dimen
\pst@tempb
\psk@gangle
\pst@tempa
\tx@Diamond}%
\def\pst@linetype{4}%
\end@ClosedObj}
\def\tx@Triangle{Triangle }
\def\pstriangle{\pst@object{pstriangle}}
\def\pstriangle@i(#1){%
\@ifnextchar({\pstriangle@ii(#1)}{\pstriangle@ii(0,0)(#1)}}
\def\pstriangle@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\addto@pscode{%
\psline@iii
pop
\psk@dimen
\pst@tempb
\psk@gangle
\pst@tempa
\tx@Triangle}%
\def\pst@linetype{2}%
\end@ClosedObj}
\def\tx@CCA{CCA }
\def\tx@CCA{CCA }
\def\tx@CC{CC }
\def\tx@IC{IC }
\def\tx@BOC{BOC }
\def\tx@NC{NC }
\def\tx@EOC{EOC }
\def\tx@BAC{BAC }
\def\tx@NAC{NAC }
\def\tx@EAC{EAC }
\def\tx@OpenCurve{OpenCurve }
\def\tx@AltCurve{AltCurve }
\def\tx@ClosedCurve{ClosedCurve }
\def\psset@curvature#1{%
\edef\pst@tempg{#1 }%
\expandafter\psset@@curvature\pst@tempg * * * \@nil}
\def\psset@@curvature#1 #2 #3 #4\@nil{%
\pst@checknum{#1}\pst@tempg
\pst@checknum{#2}\pst@temph
\pst@checknum{#3}\pst@tempi
\edef\psk@curvature{\pst@tempg \pst@temph \pst@tempi}}
\psset@curvature{1 .1 0}
\def\pscurve{\pst@object{pscurve}}
\def\pscurve@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\pscurve@ii}}
\def\pscurve@ii{%
\addto@pscode{%
\pst@cp
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@OpenCurve}%
\end@OpenObj}
\def\psecurve{\pst@object{psecurve}}
\def\psecurve@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psecurve@ii}}
\def\psecurve@ii{%
\addto@pscode{%
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@AltCurve}%
\end@OpenObj}
\def\psccurve{\pst@object{psccurve}}
\def\psccurve@i{%
\begin@ClosedObj
\pst@getcoors[\psccurve@ii}
\def\psccurve@ii{%
\addto@pscode{%
\psk@curvature\space /c ED /b ED /a ED
\ifshowpoints true \else false \fi
\tx@ClosedCurve}%
\def\pst@linetype{1}%
\end@ClosedObj}
\def\psset@dotsize#1{%
\pst@expandafter\pst@getdimnum{#1} 0 {} {}\@nil
\edef\psk@@dotsize{\pst@number\pst@dimg}%
\let\psk@@@dotsize\pst@tempg
\edef\psk@dotsize{%
/DS \psk@@dotsize \psk@@@dotsize CLW mul add 2 div def }}
\psset@dotsize{2pt 2}
\def\psset@dotscale#1{%
\pst@getscale{#1}\psk@dotscale
\ifx\psk@dotscale\@empty
\def\psk@xdotscale{1 }%
\def\psk@ydotscale{1 }%
\else
\let\psk@xdotscale\pst@tempg
\let\psk@ydotscale\pst@temph
\fi}
% DG/SR modification begin - Oct. 17, 1997 - Patch 5
%\psset@dotscale{1}
% DG/SR modification end
\def\pst@Getangle#1#2{%
\pst@getangle{#1}\pst@tempg
\def\pst@temph{0. }%
\ifx\pst@tempg\pst@temph
\def#2{}%
\else
\edef#2{\pst@tempg\space rotate }%
\fi}
\def\psset@dotangle#1{%
\pst@getangle{#1}\psk@@dotangle
\ifdim\psk@@dotangle\p@=\z@
\let\psk@dotangle\@empty
\else
% DG/SR modification begin - Aug. 8, 1997 - Patch 4
%\edef\psk@dotangle{\psk@@dotangle rotate }
\edef\psk@dotangle{\psk@@dotangle rotate }%
% DG/SR modification end
\fi}
\psset@dotangle{0}
\def\pst@getdotsize{%
\pst@dimg=\psk@@@dotsize\pslinewidth
\advance\pst@dimg\psk@@dotsize\p@
\pst@dimh=\psk@ydotscale\pst@dimg
\pst@dimg=\psk@xdotscale\pst@dimg
\divide\pst@dimh 2
\divide\pst@dimg 2\relax}
% DG/SR modification begin - Oct. 17, 1997 - Patch 5
\psset@dotscale{1}
% DG/SR modification end
\def\psdot{\pst@object{psdot}}
\def\psdot@i{\@ifnextchar({\psdot@ii}{\psdot@ii(\z@,\z@)}}
\def\psdot@ii(#1){%
\begin@SpecialObj
\pst@@getcoor{#1}%
\addto@pscode{%
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}%
\pst@coor Dot}%
\end@SpecialObj}
\def\psdots{\pst@object{psdots}}
\def\psdots@i{%
\begin@SpecialObj
\pst@getcoors[\psdots@ii}
\def\psdots@ii{%
\addto@pscode{false \tx@NArray \psdots@iii}%
\end@SpecialObj}
\def\psdots@iii{%
\psk@dotsize
\@nameuse{psds@\psk@dotstyle}
newpath
n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat}
% DG: dead code (to suppress until \psset@dotstyle) ? - Aug. 4, 1997
\def\tx@SQ{SQ }
\def\tx@ST{ST }
\def\tx@SP{SP }
\def\pst@gdot#1{/Dot { gsave T \psk@dotangle \psk@dotscale #1 grestore } def }
\@namedef{psds@*}{\pst@gdot{0 0 DS \tx@SD}}
\@namedef{psds@o}{%
/r2 DS CLW sub def
\pst@gdot{0 0 DS \tx@SD \pst@usecolor\psfillcolor 0 0 r2 \tx@SD}}
\@namedef{psds@square*}{%
/r1 DS .886 mul def
\pst@gdot{r1 \tx@SQ}}
\@namedef{psds@square}{%
/r1 DS .886 mul def /r2 r1 CLW sub def
\pst@gdot{r1 \tx@SQ \pst@usecolor\psfillcolor r2 \tx@SQ}}
\@namedef{psds@triangle*}{%
/y1 DS .778 mul neg def /x1 y1 1.732 mul neg def
\pst@gdot{x1 y1 \tx@ST}}
\@namedef{psds@triangle}{%
/y1 DS .778 mul neg def /x1 y1 1.732 mul neg def
/y2 y1 CLW add def /x2 y2 1.732 mul neg def
\pst@gdot{x1 y1 \tx@ST \pst@usecolor\psfillcolor x2 y2 \tx@ST}}
\@namedef{psds@pentagon*}{%
/r1 DS 1.149 mul def
\pst@gdot{r1 \tx@SP}}
\@namedef{psds@pentagon}{%
DS .93 mul dup 1.236 mul /r1 ED CLW sub 1.236 mul /r2 ED
\pst@gdot{r1 \tx@SP \pst@usecolor\psfillcolor r2 \tx@SP}}
\@namedef{psds@+}{%
/DS DS 1.253 mul def
\pst@gdot{DS 0 moveto DS neg 0 L stroke 0 DS moveto 0 DS neg L stroke}}
\@namedef{psds@|}{%
\psk@tbarsize CLW mul add 2 div /DS ED
\pst@gdot{0 DS moveto 0 DS neg L stroke}}
% DG: end dead code?
\def\psset@dotstyle#1{%
\@ifundefined{psds@#1}%
{\@pstrickserr{Dot style `#1' not defined}\@eha}%
{\edef\psk@dotstyle{#1}}}
\psset@dotstyle{*}
\def\tx@FontDot{FontDot }
\def\newpsfontdot#1[#2]#3#4{%
\@namedef{psds@#1}{%
/#3 \psk@@dotangle [#2] \tx@FontDot
% DG/SR modification begin - Dec. 12, 1999 - Patch 14
%/Dot { moveto #4 show } bind def }}
/Dot { moveto gsave \psk@dotscale #4 show grestore } bind def }}
% DG/SR modification end
\def\newpsfontdotH#1[#2]#3#4#5{%
\@namedef{psds@#1}{%
/#3 \psk@@dotangle [#2] \tx@FontDot
/Dot {
moveto
\iftrue
% DG/SR modification begin - Dec. 23, 1999 - Patch 14
%gsave \pst@usecolor\psfillcolor #5 show grestore
%\fi
%#4 show
gsave \psk@dotscale \pst@usecolor\psfillcolor #5 show grestore
\fi
gsave \psk@dotscale #4 show grestore
% DG/SR modification end
} bind def }}
\pstheader{pst-dots.pro}
\newpsfontdot{*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(b)}
\newpsfontdotH{o}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(c)}{(b)}
\newpsfontdotH{Bo}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(C)}{(b)}
\newpsfontdotH{triangle}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(t)}{(u)}
\newpsfontdotH{Btriangle}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(T)}{(u)}
\newpsfontdot{triangle*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(u)}
\newpsfontdotH{square}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(s)}{(r)}
\newpsfontdotH{Bsquare}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(S)}{(r)}
\newpsfontdot{square*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(r)}
\newpsfontdotH{pentagon}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(p)}{(q)}
\newpsfontdotH{Bpentagon}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(P)}{(q)}
\newpsfontdot{pentagon*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(q)}
% DG/SR modification begin - Mar. 18, 1997 and Dec. 16, 1999 - Patch 14
%\newpsfontdot{diamond*}%
%[1.9 0.0 0.0 1.9 -0.4598 -0.70775]{Symbol}{<E0>}
%\newpsfontdot{diamond}%
%[2.3 0.0 0.0 2.3 -0.8533 -0.5336]{Symbol}{<A8>}
% D.G. modification begin - Jan. 17, 2000
\newpsfontdotH{diamond}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(d)}{(l)}
\newpsfontdotH{Bdiamond}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(D)}{(l)}
\newpsfontdot{diamond*}%
[1.0 0.0 0.0 1.0 0.0 0.0]{PSTricksDotFont}{(l)}
% DG/SR modification end
\newpsfontdot{oplus}%
[1.44928 0.0 0.0 1.44928 -0.562319 -0.478261]{Symbol}{<C5>}
\newpsfontdot{otimes}%
[1.44928 0.0 0.0 1.44928 -0.562319 -0.475362]{Symbol}{<C4>}
\newpsfontdot{x}%
[1.8 0.0 0.0 1.8 -0.495 -0.4788]{Symbol}{<B4>}
\newpsfontdot{+}%
[2.3 0.0 0.0 2.3 -0.6486 -0.5819]{Times-Roman}{<2B>}
\newpsfontdot{asterisk}%
[2.43309 0.0 0.0 2.43309 -0.609489 -1.14477]{Times-Roman}{<2A>}
\newpsfontdot{B+}%
[2.3 0.0 0.0 2.3 -0.6555 -0.5819]{Times-Bold}{<2B>}
\newpsfontdot{Basterisk}%
[2.29358 0.0 0.0 2.29358 -0.576835 -1.08486]{Times-Bold}{<2A>}
\newpsfontdot{|}%
% DG/SR modification begin - Oct. 27, 1997 - Patch 7
%[1.98413 0.0 0.0 1.98413 -0.258929 -0.712302]{Helvetica}{(|)}
[1.98413 0.0 0.0 1.38 -0.258929 -0.5]{Helvetica}{(|)}
% DG/SR modification end
\newpsfontdot{B|}%
% DG/SR modification begin - Oct. 27, 1997 - Patch 7
%[1.98413 0.0 0.0 1.98413 -0.277778 -0.78302]{Helvetica-Bold}{(|)}
[1.98413 0.0 0.0 1.38 -0.277778 -0.5]{Helvetica-Bold}{(|)}
% DG/SR modification end
\iffalse
\newpsfontdot{*}%
[2.77778 0.0 0.0 2.77778 -0.638889 -0.813889]{Symbol}{<B7>}
\newpsfontdot{o}%
[3.33333 0.0 0.0 3.33333 -0.666667 -1.78167]{Symbol}{<B0>}
\newpsfontdot{Bo}%
[4.69484 0.0 0.0 4.69484 -0.78169 -2.97418]{Times-Bold}{<CA>}
\fi
\newdimen\pslinearc
\def\psset@linearc#1{\pssetlength\pslinearc{#1}}
\psset@linearc{0pt}
\def\psline{\pst@object{psline}}
\def\psline@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psline@ii}}
\def\psline@ii{%
\addto@pscode{\pst@cp \psline@iii \tx@Line}%
\end@OpenObj}
\def\psline@iii{%
\ifdim\pslinearc>\z@
/r \pst@number\pslinearc def
/Lineto { \tx@Arcto } def
\else
/Lineto /lineto load def
\fi
\ifshowpoints true \else false \fi}
\def\qline(#1)(#2){%
\def\pst@par{}%
\begin@SpecialObj
\def\pst@linetype{0}%
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{%
\pst@tempa moveto \pst@coor L
\@nameuse{psls@\pslinestyle}}%
\end@SpecialObj}
\def\pspolygon{\pst@object{pspolygon}}
\def\pspolygon@i{%
\begin@ClosedObj
\def\pst@cp{}%
\pst@getcoors[\pspolygon@ii}
\def\pspolygon@ii{%
\addto@pscode{\psline@iii \tx@Polygon}%
\def\pst@linetype{1}%
\end@ClosedObj}
\def\psset@framearc#1{\pst@checknum{#1}\psk@framearc}
\psset@framearc{0}
\def\psset@cornersize#1{%
\pst@expandafter\psset@@cornersize{#1}\@nil}
\def\psset@@cornersize#1#2\@nil{%
\if #1a\relax
\def\psk@cornersize{\pst@number\pslinearc false }%
\else
\def\psk@cornersize{\psk@framearc true }%
\fi}
\psset@cornersize{relative}
\def\tx@Rect{Rect }
\def\tx@OvalFrame{OvalFrame }
\def\tx@Frame{Frame }
\def\psset@dimen#1{%
\pst@expandafter\psset@@dimen{#1}\@nil}
\def\psset@@dimen#1#2\@nil{%
\if #1o\relax
\def\psk@dimen{.5 }%
\else
\if #1m\relax
\def\psk@dimen{0 }%
\else
\if #1i\relax
\def\psk@dimen{-.5 }%
\fi
\fi
\fi}
\psset@dimen{outer}
\def\psframe{\pst@object{psframe}}
\def\psframe@i(#1){%
\@ifnextchar({\psframe@ii(#1)}{\psframe@ii(0,0)(#1)}}
\def\psframe@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{\psk@cornersize \pst@tempa \pst@coor \psk@dimen \tx@Frame}%
\def\pst@linetype{2}%
\showpointsfalse
\end@ClosedObj}
\def\tx@BezierNArray{BezierNArray }
\def\tx@OpenBezier{OpenBezier }
\def\tx@ClosedBezier{ClosedBezier }
\def\tx@BezierShowPoints{BezierShowPoints }
\def\psbezier{\pst@object{psbezier}}
\def\psbezier@i{%
\pst@getarrows{%
\begin@OpenObj
\pst@getcoors[\psbezier@ii}}
\def\psbezier@ii{%
\addto@pscode{%
% DG/SR modification begin - Apr. 28, 1997 - Patch 1
% \psbezier doesn't work inside \pscustom
%\pst@cp
% DG/SR modification end
\ifshowpoints true \else false \fi
\tx@OpenBezier
\ifshowpoints \tx@BezierShowPoints \fi}%
\end@OpenObj}
\def\pscbezier{\pst@object{pscbezier}}
\def\pscbezier@i{%
\begin@ClosedObj
\pst@getcoors[\pscbezier@ii}
\def\pscbezier@ii{%
\addto@pscode{%
\ifshowpoints true \else false \fi
\tx@ClosedBezier
\ifshowpoints \tx@BezierShowPoints \fi}%
\chardef\pst@linetype=1
\end@ClosedObj}
\def\tx@Parab{Parab }
\def\parabola{\pst@object{parabola}}
\def\parabola@i{\pst@getarrows\parabola@ii}
\def\parabola@ii#1(#2)#3(#4){%
\begin@OpenObj
\pst@getcoor{#2}\pst@tempa
\pst@@getcoor{#4}%
\addto@pscode{\pst@tempa \pst@coor \tx@Parab}%
\end@OpenObj}
\def\psset@gridwidth#1{\pst@getlength{#1}\psk@gridwidth}
\psset@gridwidth{.8pt}
\def\psset@griddots#1{%
\pst@cntg=#1\relax
\edef\psk@griddots{\the\pst@cntg}}
\psset@griddots{0}
\def\psset@gridcolor#1{\pst@getcolor{#1}\psgridcolor}
\psset@gridcolor{black}
\def\psset@subgridwidth#1{\pst@getlength{#1}\psk@subgridwidth}
\psset@subgridwidth{.4pt}
\def\psset@subgridcolor#1{\pst@getcolor{#1}\pssubgridcolor}
\psset@subgridcolor{gray}
\def\psset@subgriddots#1{%
\pst@cntg=#1\relax\edef\psk@subgriddots{\the\pst@cntg}}
\psset@subgriddots{0}
\def\psset@subgriddiv#1{%
\pst@cntg=#1\relax\edef\psk@subgriddiv{\the\pst@cntg}}
\psset@subgriddiv{5}
\def\psset@gridlabels#1{\pst@getlength{#1}\psk@gridlabels}
\psset@gridlabels{10pt}
\def\psset@gridlabelcolor#1{\pst@getcolor{#1}\psgridlabelcolor}
\psset@gridlabelcolor{black}
\def\tx@Grid{Grid }
\def\psgrid{\pst@object{psgrid}}
\def\psgrid@i{\@ifnextchar(%
{\psgrid@ii}{\expandafter\psgrid@iv\pic@coor}}
\def\psgrid@ii(#1){\@ifnextchar(%
{\psgrid@iii(#1)}{\psgrid@iv(0,0)(0,0)(#1)}}
\def\psgrid@iii(#1)(#2){\@ifnextchar(%
{\psgrid@iv(#1)(#2)}{\psgrid@iv(#1)(#1)(#2)}}
\def\psgrid@iv(#1)(#2)(#3){%
\begin@SpecialObj
\pst@getcoor{#1}\pst@tempa
\pst@getcoor{#2}\pst@tempb
\pst@@getcoor{#3}%
\ifnum\psk@subgriddiv>1
\addto@pscode{gsave
\psk@subgridwidth SLW \pst@usecolor\pssubgridcolor
\pst@tempb \pst@coor \pst@tempa
\pst@number\psxunit \pst@number\psyunit
\psk@subgriddiv\space \psk@subgriddots\space
{} 0 \tx@Grid grestore}%
\fi
\addto@pscode{gsave
\psk@gridwidth SLW \pst@usecolor\psgridcolor
\pst@tempb \pst@coor \pst@tempa
\pst@number\psxunit \pst@number\psyunit
1 \psk@griddots\space { \pst@usecolor\psgridlabelcolor }
\psk@gridlabels \tx@Grid grestore}%
\end@SpecialObj}
\newif\ifpsmathbox
\psmathboxtrue
\def\pst@mathflag{\z@}
\newtoks\everypsbox
\let\pst@thisbox\relax
\long\def\pst@makenotverbbox#1#2{%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\setbox\pst@hbox=\hbox{%
\ifcase\pst@mathflag\or$\m@th\textstyle\or$\m@th\displaystyle\fi
{\pst@thisbox\the\everypsbox#2}%
\ifnum\pst@mathflag>\z@$\fi}%
#1}
\def\pst@makeverbbox#1{%
\def\pst@afterbox{#1}%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\afterassignment\pst@beginbox
\setbox\pst@hbox\hbox}
\def\pst@beginbox{%
\ifcase\pst@mathflag\or$\m@th\or$\m@th\displaystyle\fi
\bgroup\aftergroup\pst@endbox
\pst@thisbox
\the\everypsbox}
\def\pst@endbox{%
\ifnum\pst@mathflag>\z@$\fi
\egroup
\pst@afterbox}
\def\pst@makebox{\pst@@makebox}
\def\psverbboxtrue{\def\pst@@makebox{\pst@makeverbbox}}
\def\psverbboxfalse{\def\pst@@makebox{\pst@makenotverbbox}}
\psverbboxfalse
\def\pst@longbox{%
\def\pst@makebox{%
\gdef\pst@makebox{\pst@@makebox}%
\pst@makelongbox}}
\def\pst@makelongbox#1{%
\def\pst@afterbox{#1}%
\edef\pst@mathflag{%
\ifpsmathbox\ifmmode\ifinner 1\else 2\fi\else \z@\fi\else \z@\fi}%
\setbox\pst@hbox\hbox\bgroup
\aftergroup\pst@afterbox
\ifcase\pst@mathflag\or$\m@th\or$\m@th\displaystyle\fi
\begingroup
\pst@thisbox
\the\everypsbox}
\def\pst@endlongbox{%
\endgroup
\ifnum\pst@mathflag>\z@$\fi
\egroup}
\def\pslongbox#1#2{%
\@namedef{#1}{\pst@longbox#2}%
\@namedef{end#1}{\pst@endlongbox}}
\newdimen\psframesep
\def\psset@framesep#1{\pssetlength\psframesep{#1}}
\psset@framesep{3pt}
\newif\ifpsboxsep
\def\psset@boxsep#1{\@nameuse{psboxsep#1}}
\psset@boxsep{true}
\def\pst@useboxpar{%
\use@par
\if@star
\let\pslinecolor\psfillcolor
\solid@star
\let\solid@star\relax
\fi
\ifpsdoubleline \pst@setdoublesep \fi}
\def\psframebox{\pst@object{psframebox}}
\def\psframebox@i{\pst@makebox\psframebox@ii}
\def\psframebox@ii{%
\begingroup
\pst@useboxpar
\pst@dima=\pslinewidth
\advance\pst@dima by \psframesep
\pst@dimc=\wd\pst@hbox\advance\pst@dimc by \pst@dima
\pst@dimb=\dp\pst@hbox\advance\pst@dimb by \pst@dima
\pst@dimd=\ht\pst@hbox\advance\pst@dimd by \pst@dima
\setbox\pst@hbox=\hbox{%
\ifpsboxsep\kern\pst@dima\fi
\begin@ClosedObj
\addto@pscode{%
\psk@cornersize
\pst@number\pst@dima neg
\pst@number\pst@dimb neg
\pst@number\pst@dimc
\pst@number\pst@dimd
.5
\tx@Frame}%
\def\pst@linetype{2}%
\showpointsfalse
\end@ClosedObj
\box\pst@hbox
\ifpsboxsep\kern\pst@dima\fi}%
\ifpsboxsep\dp\pst@hbox=\pst@dimb\ht\pst@hbox=\pst@dimd\fi
\leavevmode\box\pst@hbox
\endgroup}
\def\psdblframebox{\pst@object{psdblframebox}}
\def\psdblframebox@i{\addto@par{doubleline=true}\psframebox@i}
\def\psclip#1{%
\leavevmode
\begingroup
\begin@psclip
\begingroup
\def\use@pscode{%
\pstVerb{%
\pst@dict
/mtrxc CM def
CP CP T
\tx@STV
\psk@origin
\psk@swapaxes
newpath
\pst@code
clip
newpath
mtrxc setmatrix
moveto
0 setgray
end}%
\gdef\pst@code{}}%
\def\@multips(##1)(##2)##3##4{\pst@misplaced\multips}%
\def\nc@object##1##2##3##4{\pst@misplaced{node connection}}%
\hbox to\z@{#1}%
\endgroup
\def\endpsclip{%
\end@psclip
\endgroup}%
\ignorespaces}
\def\endpsclip{\pst@misplaced\endpsclip}
\let\begin@psclip\relax
\def\end@psclip{\pstVerb{currentpoint initclip moveto}}
\def\AltClipMode{%
\def\end@psclip{\pstVerb{\pst@grestore}}%
\def\begin@psclip{\pstVerb{gsave}}}
\def\clipbox{\@ifnextchar[{\clipbox@}{\clipbox@[\z@]}}
% DG modification begin - Apr. 3, 1997
% From paulus@immd5.informatik.uni-erlangen.de (Dietrich Paulus)
%\def\clipbox@[#1]{\pst@makebox\clipbox@@{#1}}
\def\clipbox@[#1]{\pst@makebox{\clipbox@@{#1}}}
% DG modification end
\def\clipbox@@#1{%
\pssetlength\pst@dimg{#1}%
\leavevmode\hbox{%
\begin@psclip
\pst@Verb{%
CM \tx@STV CP T newpath
/a \pst@number\pst@dimg def
/w \pst@number{\wd\pst@hbox}a add def
/d \pst@number{\dp\pst@hbox}a add neg def
/h \pst@number{\ht\pst@hbox}a add def
a neg d moveto
a neg h L
w h L
w d L
closepath
clip
newpath
0 0 moveto
setmatrix}%
\unhbox\pst@hbox
\end@psclip}}
\def\psshadowbox{\pst@object{psshadowbox}}
\def\psshadowbox@i{\pst@makebox\psshadowbox@ii}
\def\psshadowbox@ii{%
\begingroup
\pst@useboxpar
\psshadowtrue
\psboxseptrue
\def\psk@shadowangle{-45 }%
\setbox\pst@hbox=\hbox{\psframebox@ii}%
\pst@dimh=\psk@shadowsize\p@
\pst@dimh=.7071\pst@dimh
\pst@dimg=\dp\pst@hbox
\advance\pst@dimg\pst@dimh
\dp\pst@hbox=\pst@dimg
\pst@dimg=\wd\pst@hbox
\advance\pst@dimg\pst@dimh
\wd\pst@hbox=\pst@dimg
\leavevmode
\box\pst@hbox
\endgroup}
\def\pscirclebox{\pst@object{pscirclebox}}
\def\pscirclebox@i{\pst@makebox\pscirclebox@ii}
\def\pscirclebox@ii{%
\begingroup
\pst@useboxpar
\setbox\pst@hbox=\hbox{%
\pst@nodehook
\pscirclebox@iii
\box\pst@hbox}%
\ifpsboxsep \pscirclebox@sep \fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\pscirclebox@iii{%
\if@star
\pslinewidth\z@
\pstverb{\pst@dict \tx@STP \pst@usecolor\psfillcolor
newpath \pscirclebox@iv \tx@SD end}%
\else
\begin@ClosedObj
\def\pst@linetype{4}\showpointsfalse
\addto@pscode{%
\pscirclebox@iv CLW 2 div add 0 360 arc closepath}%
\end@ClosedObj
\fi}
\def\pscirclebox@iv{%
\pst@number{\wd\pst@hbox}2 div
\pst@number{\ht\pst@hbox}\pst@number{\dp\pst@hbox}add 2 div
2 copy \pst@number{\dp\pst@hbox}sub 4 2 roll
\tx@Pyth \pst@number\psframesep add }
\def\pscirclebox@sep{%
\pst@dimb=\ht\pst@hbox
\advance\pst@dimb\dp\pst@hbox
\divide\pst@dimb 2
\pst@dima=.5\wd\pst@hbox
\pst@pyth\pst@dima\pst@dimb\pst@dimc
\advance\pst@dimc\pslinewidth
\advance\pst@dimc\psframesep
\advance\pst@dimb-\pst@dimc
\setbox\pst@hbox=\hbox to2\pst@dimc{%
\hss
\vbox{\kern-\pst@dimb\box\pst@hbox}%
\hss}%
\advance\pst@dimb-\dp\pst@hbox
\dp\pst@hbox=-\pst@dimb}
\let\pst@nodehook\relax
\def\psCirclebox{\pst@object{psCirclebox}}
\def\psCirclebox@i{\pst@makebox\psCirclebox@ii}
\def\psCirclebox@ii{%
\begingroup
\pst@useboxpar
\pst@dima=\ht\pst@hbox
\advance\pst@dima\dp\pst@hbox
\divide\pst@dima\tw@
\pssetlength\pst@dimb\psk@radius
\setbox\pst@hbox=\hbox{%
\pst@nodehook
\pscircle(.5\wd\pst@hbox,\pst@dima){\pst@dimb}%
\box\pst@hbox}%
\ifpsboxsep \psCirclebox@sep \fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psCirclebox@sep{%
\pst@dimc=\pst@dimb
\advance\pst@dimb-\pst@dima
\advance\pst@dima\pst@dimc
\setbox\pst@hbox=\hbox to\tw@\pst@dimc{%
\hss
\vrule width \z@ depth \pst@dimb height \pst@dima
\box\pst@hbox
\hss}}%
\def\psovalbox{\pst@object{psovalbox}}
\def\psovalbox@i{\pst@makebox{\psovalbox@ii}}
\def\psovalbox@ii{%
\begingroup
\pst@useboxpar
\psovalbox@iii
\ifpsboxsep\psovalbox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psovalbox@iii{%
\psovalbox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
0 360
\pst@number\pst@dimc CLW 2 div sub
\pst@number\pst@dimd CLW 2 div sub
\pst@number\pst@dima
\pst@number\pst@dimb
\tx@Ellipse
closepath}%
\def\pst@linetype{2}%
\end@ClosedObj
\unhbox\pst@hbox}}
\def\psovalbox@iv{%
\pst@dimc=\pslinewidth\advance\pst@dimc\psframesep
\pst@dimd=\ht\pst@hbox\advance\pst@dimd\dp\pst@hbox
\pst@dima=.5\wd\pst@hbox
\pst@dimb=.5\pst@dimd\advance\pst@dimb-\dp\pst@hbox
\pst@dimd=.707\pst@dimd
\advance\pst@dimd\pst@dimc
\advance\pst@dimc.707\wd\pst@hbox}
\def\psovalbox@sep{%
\setbox\pst@hbox\hbox to 2\pst@dimc{\hss\unhbox\pst@hbox\hss}%
\pst@dimg=\pst@dimd
\advance\pst@dimg-\pst@dimb
\dp\pst@hbox=\pst@dimg
\advance\pst@dimd\pst@dimb
\ht\pst@hbox=\pst@dimd}
\def\psdiabox{\pst@object{psdiabox}}
\def\psdiabox@i{\pst@makebox{\psdiabox@ii}}
\def\psdiabox@ii{%
\begingroup
\pst@useboxpar
\psdiabox@iii
\ifpsboxsep\psdiabox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\psdiabox@iv{%
\pst@dimg=.707\pslinewidth
\advance\pst@dimg.707\psframesep
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\pst@dimc=\pst@dima
\advance\pst@dimc\pst@dimg
\pst@dimd=\ht\pst@hbox
\advance\pst@dimd\dp\pst@hbox
\divide\pst@dimd 2
\pst@dimb=\pst@dimd
\advance\pst@dimb-\dp\pst@hbox
\advance\pst@dimd\pst@dimg}
\def\psdiabox@iii{%
\psdiabox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
\psline@iii
pop
.5
\pst@number\pst@dimc 2 mul \pst@number\pst@dimd 2 mul
0
\pst@number\pst@dima \pst@number\pst@dimb
\tx@Diamond}%
\def\pst@linetype{4}%
\end@ClosedObj
\box\pst@hbox}}
\def\psdiabox@sep{%
\setbox\pst@hbox\hbox to 4\pst@dimc{\hss\unhbox\pst@hbox\hss}%
\multiply\pst@dimd 2
\advance\pst@dimd\pst@dimb
\ht\pst@hbox\pst@dimd
\advance\pst@dimd-2\pst@dimb
\dp\pst@hbox\pst@dimd}
\def\psset@trimode#1{\pst@expandafter\psset@@trimode{#1}\@empty\@empty\@nil}
\def\psset@@trimode#1#2#3\@nil{%
\let\pst@tempg#1\relax
\ifx\pst@tempg*%
\let\psk@@trimode\@empty
\let\pst@tempg#2\relax
\else
\let\psk@@trimode\relax
\fi
\edef\psk@trimode{%
\ifx R\pst@tempg 1 \else\ifx D\pst@tempg 2
\else\ifx L\pst@tempg 3 \else 0 \fi\fi\fi}}
\psset@trimode{U}
\def\pstribox{\pst@object{pstribox}}
\def\pstribox@i{\pst@makebox{\pstribox@ii}}
\def\pstribox@ii{%
\begingroup
\pst@useboxpar
\pstribox@iii
\ifpsboxsep\pstribox@sep\fi
\leavevmode
\box\pst@hbox
\endgroup}
\def\pstribox@iii{%
\pstribox@iv
\setbox\pst@hbox=\hbox{%
\begin@ClosedObj
\addto@pscode{%
\psline@iii
pop
.5
\pst@number\pst@dimc \pst@number\pst@dimd
\ifodd\psk@trimode exch \fi
\psk@trimode -90 mul
\pst@number\pst@dima \pst@number\pst@dimb
\tx@Triangle}%
\def\pst@linetype{2}%
\end@ClosedObj
\box\pst@hbox}}
\def\pstribox@iv{%
\pst@dimh=\pslinewidth
\advance\pst@dimh\psframesep
\pst@dimg=\ht\pst@hbox
\advance\pst@dimg-\dp\pst@hbox
\divide\pst@dimg 2
\edef\pst@tempa{\number\pst@dimg sp}% For use by nodes.
\ifodd\psk@trimode
\pst@dimb\pst@dimg
\else
\pst@dima=\wd\pst@hbox
\divide\pst@dima 2
\fi
\ifcase\psk@trimode
\pst@dimb=-\dp\pst@hbox
\advance\pst@dimb-\pst@dimh
\or
\pst@dima=-\pst@dimh
\or
\pst@dimb=\ht\pst@hbox
\advance\pst@dimb\pst@dimh
\or
\pst@dima=\wd\pst@hbox
\advance\pst@dima\pst@dimh
\fi
\pst@dimd=\dp\pst@hbox
\advance\pst@dimd\ht\pst@hbox
\ifx\psk@@trimode\relax
\pst@dimc=\wd\pst@hbox
\advance\pst@dimc\ifodd\psk@trimode 1.447\else 1.789\fi\pst@dimh
\multiply\pst@dimc 2
\advance\pst@dimd\ifodd\psk@trimode 1.789\else 1.447\fi\pst@dimh
\multiply\pst@dimd 2
\else
\ifodd\psk@trimode
\advance\pst@dimd 1.1547\wd\pst@hbox
\advance\pst@dimd 3.4641\pst@dimh
\pst@dimc=.866\pst@dimd
\else
\advance\pst@dimd .866\wd\pst@hbox %.866=(sqrt(3)/2)
\advance\pst@dimd 3\pst@dimh
\pst@dimc=1.1547\pst@dimd % 1.1547=(2/sqrt(3))
\fi
\fi}
\def\pstribox@sep{%
\ifodd\psk@trimode
\advance\pst@dimb.5\pst@dimd
\ht\pst@hbox=\pst@dimb
\advance\pst@dimd-\pst@dimb
\dp\pst@hbox=\pst@dimd
\else
\setbox\pst@hbox\hbox to \pst@dimc{\hss\unhbox\pst@hbox\hss}%
\global\pst@dimg=.5\pst@dimc
\fi
\ifcase\psk@trimode
\dp\pst@hbox-\pst@dimb
\advance\pst@dimd\pst@dimb
\ht\pst@hbox\pst@dimd
\or
\pst@dimg=.5\wd\pst@hbox
\global\advance\pst@dimg-\pst@dima
\setbox\pst@hbox\hbox to \pst@dimc{\kern-\pst@dima\box\pst@hbox\hss}%
\or
\ht\pst@hbox\pst@dimb
\advance\pst@dimd-\pst@dimb
\dp\pst@hbox\pst@dimd
\or
\pst@dimg=\pst@dimc
\advance\pst@dimg-\pst@dima
\global\advance\pst@dimg.5\wd\pst@hbox
\setbox\pst@hbox\hbox to \pst@dimc{%
\hss\box\pst@hbox\kern\psframesep\kern\pslinewidth}%
\fi}
\def\psset@arcsepA#1{\pst@getlength{#1}\psk@arcsepA}
\def\psset@arcsepB#1{\pst@getlength{#1}\psk@arcsepB}
\def\psset@arcsep#1{%
\psset@arcsepA{#1}\let\psk@arcsepB\psk@arcsepA}
\psset@arcsep{0}
\def\tx@ArcArrow{ArcArrow }
\def\psarc{\pst@object{psarc}}
\def\psarc@i{%
\@ifnextchar({\psarc@iii}{\psarc@ii}}
\def\psarc@ii#1{\addto@par{arrows=#1}%
\@ifnextchar({\psarc@iii}{\psarc@iii(0,0)}}
\def\psarc@iii(#1)#2#3#4{%
\begin@OpenObj
\pst@getangle{#3}\pst@tempa
\pst@getangle{#4}\pst@tempb
\pst@@getcoor{#1}%
\pssetlength\pst@dima{#2}%
\addto@pscode{\psarc@iv \psarc@v}%
\gdef\psarc@type{0}%
\showpointsfalse
\end@OpenObj}
\def\psarc@iv{%
\pst@coor /y ED /x ED
/r \pst@number\pst@dima def
/c 57.2957 r \tx@Div def
/angleA
\pst@tempa
\psk@arcsepA c mul 2 div
\ifcase \psarc@type add \or sub \fi
def
/angleB
\pst@tempb
\psk@arcsepB c mul 2 div
\ifcase \psarc@type sub \or add \fi
def
\ifshowpoints\psarc@showpoints\fi
\ifx\psk@arrowA\@empty
\ifnum\psk@liftpen=2
r angleA \tx@PtoC
y add exch x add exch
moveto
\fi
\fi}
\def\psarc@v{%
x y r
angleA
\ifx\psk@arrowA\@empty\else
{ ArrowA CP }
{ \ifcase\psarc@type add \or sub \fi }
\tx@ArcArrow
\fi
angleB
\ifx\psk@arrowB\@empty\else
{ ArrowB }
{ \ifcase\psarc@type sub \or add \fi }
\tx@ArcArrow
\fi
\ifcase\psarc@type arc \or arcn \fi}
\def\psarc@type{0}
\def\psarc@showpoints{%
gsave
newpath
x y moveto
x y r \pst@tempa \pst@tempb
\ifcase\psarc@type arc \or arcn \fi
closepath
CLW 2 div SLW
[ \psk@dash\space ] 0 setdash stroke
grestore }
\def\psarcn{\pst@object{psarcn}}
\def\psarcn@i{\def\psarc@type{1}\psarc@i}
\def\pscircle{\pst@object{pscircle}}
\def\pscircle@i{\@ifnextchar({\pscircle@do}{\pscircle@do(0,0)}}
\def\pscircle@do(#1)#2{%
\if@star
{\use@par\qdisk(#1){#2}}%
\else
\begin@ClosedObj
\pst@@getcoor{#1}%
\pssetlength\pst@dimc{#2}%
\def\pst@linetype{4}%
\addto@pscode{%
\pst@coor
\pst@number\pst@dimc
\psk@dimen CLW mul sub
0 360 arc
closepath}%
\showpointsfalse
\end@ClosedObj
\fi
\ignorespaces}
\def\qdisk(#1)#2{%
\def\pst@par{}%
\begin@SpecialObj
\pst@@getcoor{#1}%
\pssetlength\pst@dimg{#2}%
\addto@pscode{\pst@coor \pst@number\pst@dimg \tx@SD}%
\end@SpecialObj}
\def\psset@radius#1{\pst@@getlength{#1}\psk@radius}
\psset@radius{.25cm}
\def\psCircle{\pst@object{psCircle}}
\def\psCircle@i{\@ifnextchar({\psCircle@ii}{\psCircle@ii(0,0)}}
\def\psCircle@ii(#1){\pscircle@do(#1){\psk@radius}}
\def\pswedge{\pst@object{pswedge}}
\def\pswedge@i{\@ifnextchar({\pswedge@ii}{\pswedge@ii(0,0)}}
\def\pswedge@ii(#1)#2#3#4{%
\begin@ClosedObj
\pssetlength\pst@dimc{#2}
\pst@getangle{#3}\pst@tempa
\pst@getangle{#4}\pst@tempb
\pst@@getcoor{#1}%
\def\pst@linetype{1}%
\addto@pscode{%
\pst@coor
2 copy
moveto
\pst@number\pst@dimc \psk@dimen CLW mul sub % Adjusted radius
\pst@tempa \pst@tempb
arc
closepath}%
\showpointsfalse
\end@ClosedObj}
\def\tx@Ellipse{Ellipse }
\def\psellipse{\pst@object{psellipse}}
\def\psellipse@i(#1){\@ifnextchar(%
{\psellipse@ii(#1)}{\psellipse@ii(0,0)(#1)}}
\def\psellipse@ii(#1)(#2){%
\begin@ClosedObj
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\addto@pscode{%
0 360
\pst@coor
\ifdim\psk@dimen\p@=\z@\else
\psk@dimen CLW mul
dup 4 -1 roll sub neg 3 1 roll sub
\fi
\pst@tempa
\tx@Ellipse
closepath}%
\def\pst@linetype{2}%
\end@ClosedObj}
\def\multips{\@ifnextchar({\def\pst@par{}\multips@ii}{\multips@i}}
\def\multips@i#1{\def\pst@par{rot=#1}\multips@ii}
\def\multips@ii(#1){\@ifnextchar({\multips@iii(#1)}{\multips@iii(\z@,\z@)(#1)}}
\long\def\multips@iii(#1)(#2)#3#4{%
\begingroup
\use@par
\pst@getcoor{#1}\pst@tempa
\pst@@getcoor{#2}%
\pst@cnta=#3\relax
\init@pscode
\addto@pscode{%
\pst@tempa T \the\pst@cnta\space \pslbrace
gsave \ifx\psk@rot\@empty\else\psk@rot rotate \fi}%
\hbox to\z@{%
\def\init@pscode{%
\addto@pscode{%
gsave
\pst@number\pslinewidth SLW
\pst@usecolor\pslinecolor}}%
\def\use@pscode{\addto@pscode{grestore}}%
\def\psclip##1{\pst@misplaced\psclip}%
\def\nc@object##1##2##3##4{\pst@misplaced{node connection}}%
#4}%
\addto@pscode{grestore \pst@coor T \psrbrace repeat}%
\leavevmode
\use@pscode
\endgroup
\ignorespaces}
\def\scalebox#1{\pst@makebox{\@scalebox{#1}}}
\def\@scalebox#1{%
\begingroup
\pst@getscale{#1}\pst@tempa
\let\pst@tempc\pst@tempg
\let\pst@tempd\pst@temph
\@@scalebox
\endgroup}
\def\@@scalebox{%
\leavevmode
\hbox{%
\ifdim\pst@tempd\p@<\z@
\pst@dimg=\pst@tempd\ht\pst@hbox
\pst@dimh=\pst@tempd\dp\pst@hbox
\dp\pst@hbox=-\pst@dimg
\ht\pst@hbox=-\pst@dimh
\else
\ht\pst@hbox=\pst@tempd\ht\pst@hbox
\dp\pst@hbox=\pst@tempd\dp\pst@hbox
\fi
\pst@dima=\pst@tempc\wd\pst@hbox
\ifdim\pst@dima<\z@\kern-\pst@dima\fi
\pst@Verb{CP CP translate \pst@tempa \tx@NET}%
\hbox to \z@{\box\pst@hbox\hss}%
\pst@Verb{%
CP CP translate
1 \pst@tempc div 1 \pst@tempd div scale
\tx@NET}%
\ifdim\pst@dima>\z@\kern\pst@dima\fi}}
\pslongbox{Scalebox}{\scalebox}
\def\scaleboxto(#1,#2){\pst@makebox{\@scaleboxto(#1,#2)}}
\def\@scaleboxto(#1,#2){%
\begingroup
\pssetlength\pst@dima{#1}%
\pssetlength\pst@dimb{#2}%
\ifdim\pst@dima=\z@\else
\pst@divide{\pst@dima}{\wd\pst@hbox}\pst@tempc
\edef\pst@tempc{\pst@tempc\space}%
\fi
\ifdim\pst@dimb=\z@
\ifdim\pst@dima=\z@
\@pstrickserr{%
\string\scaleboxto\space dimensions cannot both be zero}\@ehpa
\def\pst@tempa{}%
\def\pst@tempc{1 }%
\def\pst@tempd{1 }%
\else
\let\pst@tempd\pst@tempc
\fi
\else
\pst@dimc=\ht\pst@hbox
\advance\pst@dimc\dp\pst@hbox
\pst@divide{\pst@dimb}{\pst@dimc}\pst@tempd
\edef\pst@tempd{\pst@tempd\space}%
\ifdim\pst@dima=\z@ \let\pst@tempc\pst@tempd \fi
\fi
\edef\pst@tempa{\pst@tempc \pst@tempd scale }%
\@@scalebox
\endgroup}
\pslongbox{Scaleboxto}{\scaleboxto}
\def\tx@Rot{Rot }
\def\rotateleft{\pst@makebox{\@rotateleft\pst@hbox}}
\def\@rotateleft#1{%
\leavevmode\hbox{\hskip\ht#1\hskip\dp#1\vbox{\vskip\wd#1%
\pst@Verb{90 \tx@Rot}
\vbox to \z@{\vss\hbox to \z@{\box#1\hss}\vskip\z@}%
\pst@Verb{-90 \tx@Rot}}}}
\def\rotateright{\pst@makebox{\@rotateright\pst@hbox}}
\def\@rotateright#1{%
\hbox{\hskip\ht#1\hskip\dp#1\vbox{\vskip\wd#1%
\pst@Verb{-90 \tx@Rot}
\vbox to \z@{\hbox to \z@{\hss\box#1}\vss}%
\pst@Verb{90 \tx@Rot}}}}
\def\rotatedown{\pst@makebox{\@rotatedown\pst@hbox}}
\def\@rotatedown#1{%
\hbox{\hskip\wd#1\vbox{\vskip\ht#1\vskip\dp#1%
\pst@Verb{180 \tx@Rot}%
\vbox to \z@{\hbox to \z@{\box#1\hss}\vss}%
\pst@Verb{-180 \tx@Rot}}}}
\pslongbox{Rotateleft}{\rotateleft}
\pslongbox{Rotateright}{\rotateright}
\pslongbox{Rotatedown}{\rotatedown}
\def\pst@starbox{%
\setbox\pst@hbox\hbox{\psframebox*[boxsep=false]{\unhbox\pst@hbox}}}
\def\pst@@makesmall#1{%
\setbox#1=\hbox to\z@{\hss\vbox to \z@{\vss\box#1\vss}\hss}}
\def\pst@@@makesmall#1{%
\pst@dimh=\psk@xref\wd#1%
\ifx\psk@yref\relax
\pst@dimg=\dp#1%
\else
\pst@dimg=\psk@yref\ht#1%
\advance\pst@dimg\psk@yref\dp#1%
\fi
\setbox#1=\hbox to\z@{%
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tworow2.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%     Two-row Nilpotent Orbits
%      of Cyclic Quivers
%
%     Revised version
%     13 December 2001
%     Further revisions of proof of Lemma 5.4, 6 February 2002
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{amsart}
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\newtheorem{problem}[theorem]{Problem}

\begin{document}
\title{Two-row nilpotent orbits of cyclic quivers}

\author{Anthony Henderson}
\address{School of Mathematics and Statistics,
University of Sydney, NSW 2006, AUSTRALIA}
\email{anthonyh@maths.usyd.edu.au}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We prove that the local intersection cohomology of nilpotent
orbit closures of cyclic quivers is trivial when the two orbits
involved correspond to partitions with at most two rows. This
gives a geometric proof of a result of Graham and Lehrer,
which states that standard modules of the affine Hecke algebra
of $GL_d$ corresponding to nilpotents with at most two Jordan
blocks are multiplicity-free.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Let $\Delta_n$ be the cyclic quiver with $n$ vertices. The isomorphism
classes of nilpotent complex representations of $\Delta_n$ are in
bijection with certain \emph{nilpotent orbits} in a suitable variety
(see \S2 for the definition). The closure of such a nilpotent orbit
is usually singular, and it is an important problem to compute its
local intersection cohomology at the points of another given orbit (see \S3).
If $n=1$, these are the usual nilpotent orbits in
$\mathfrak{gl}_d$, and the problem was solved by Lusztig in \cite{greenuni}.

The main result of this paper (Theorem \ref{mainthm} below) is that
this local intersection cohomology is trivial when the two orbits
involved correspond to partitions with at most two rows
(equivalently, the representations of $\Delta_n$ involved are either
indecomposable or the sum of two indecomposables).

In \S3 we discuss some equivalent formulations of this result. The most
noteworthy concerns the complex representation theory of 
$\widetilde{\H}_d$, the affine Hecke algebra of $GL_d(\C)$
(as defined in \cite{kazhdanlusztig}). This algebra 
has geometrically-defined \emph{standard modules} 
$M_{s,x,q_0}$ indexed by triples $(s,x,q_0)$ where $s\in GL_d(\C)$ 
is semisimple,
$x\in\gl_d(\C)$ is nilpotent, $q_0\in\C^{\times}$, and
\[ \mathrm{Ad}(s)(x)=q_0x. \]
By a well-known result of Ginzburg, Theorem \ref{mainthm} (for $n\geq 2$)
is equivalent to the statement that $M_{s,x,q_0}$ is multiplicity-free
when $q_0$ is a primitive $n$-th root of unity, all eigenvalues of
$s$ are powers of $q_0$, and $x$ has at most two Jordan blocks.
(See \S3 for more on the equivalence.)

This statement was recently proved by Graham and Lehrer
in \cite{grahamlehrer}. Actually, they imposed no conditions
on $s$ and $q_0$, but it is easy to reduce to the above case. Their method
is algebraic, making use of the representation theory
of the affine Temperley-Lieb algebras, and their results apply over more
general ground fields (with induced modules instead of standard modules).
The present paper arose from the author's desire to give
a purely geometric/combinatorial proof
of Theorem \ref{mainthm}, thus reproving Graham and Lehrer's result
(in the complex case).

In \S4 we introduce canonical resolutions of the nilpotent orbit
closures, and derive a dimension formula for the orbits, which appears
to be new (see Lemma \ref{geomlemma}). We also discuss how to determine
the local intersection cohomology, given the Poincar\'e polynomials
of the fibres of these resolutions. Then for the remainder of the paper
we restrict attention to the nilpotent orbits with at most two
rows. Section 5 contains some preliminary Lemmas,
and Section 6 completely describes the closure relations among such
orbits. Section 7 presents the proof of Theorem \ref{mainthm}.

Our notation for partitions and Young diagrams follows
\cite{macdonald}. In particular, if $\lambda$ is a partition, its nonzero parts
are $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{\ell(\lambda)}$,
where $\ell(\lambda)$ is the length; and $\lambda'$ denotes the transpose
partition.

\textit{Acknowledgements.}
I am very grateful to John Graham and Gus Lehrer for telling me
about their result and showing me a draft of \cite{grahamlehrer}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nilpotent Orbits of Cyclic Quivers}
Fix a positive integer $n$. Let $\Delta_n$ be the cyclic
quiver of type $\widetilde{A_{n-1}}$, which has vertex set $I=\Z/n\Z$
and an arrow from $i$ to $i-1$ for all $i\in I$. Let $V=\oplus_{i\in I}V_i$
be an $I$-graded vector space over $\C$. We write $d_i$ for
$\dim V_i$, $\bd$ for the dimension vector
$\textbf{dim}\,V=(d_i)_{i\in I}$, and $d$ for $\dim V=\sum_{i\in I}d_i$.
Let
\[ E_V=\bigoplus_{i\in I}\Hom(V_i,V_{i-1}) \]
be the space of \emph{representations} of $\Delta_n$ on $V$.
Two representations in $E_V$ are isomorphic iff they are in the
same orbit of $G_V=\prod_{i\in I} GL(V_i)$, acting on $E_V$ by conjugation.

We say that $x\in E_V$, and its $G_V$-orbit, are \emph{nilpotent} if
$x$ is nilpotent as an element of $\End(V)$. 
Let $\cN_V$ be the subvariety of nilpotent elements of $E_V$.
The nilpotent $GL(V)$-orbits
in $\End(V)$ are in bijection with $\Lambda(d)$, the set of partitions
of size $d$: let $\O_\lambda$ denote the orbit corresponding to the partition
$\lambda$. The $G_V$-orbits in $\cN_V$ have a similar description, as follows.

The isomorphism classes of indecomposable nilpotent 
representations of $\Delta_n$ are called \emph{segments}. There is
a segment $[i;l)$ for each $i\in I$ and $l\in\Z^{+}$; in our convention
this is the isoclass of indecomposables of length $l$ whose socle is
the simple module corresponding to $i$. The notation is meant to
suggest the multiset $\{i,i+1,\cdots,i+l-1\}$, which (regarded as a
function $I\to\N$) is precisely the dimension vector of an
indecomposable representation in this class. 
Since any representation is a direct
sum of indecomposables, the nilpotent representations are parametrized
by \emph{multisegments} (multisets of segments), which we write with direct
sum notation, e.g.\ $[i_1;l_1)\oplus\cdots\oplus[i_s;l_s)$.

Collecting together all segments $[i;l)$ with the same $i$,
we can think of a multisegment as an $I$-tuple of partitions
$\bl=(\lambda^{(i)})_{i\in I}$. Let $\bL$ be the set of such $I$-tuples,
and for $\bl\in\bL$,
write $\lambda$ for the union $\coprod_{i\in I}\lambda^{(i)}$.
We will identify $\bl$ with the corresponding $I$-labelled Young diagram
of shape $\lambda$. This is the unique $I$-labelled Young diagram
in which the labels from
left to right across a row increase by $1$ at each step,
and $\lambda^{(i)}$ is the subpartition
formed by all rows starting with the label $i$. (Two $I$-labelled
Young diagrams which differ by permuting rows of equal length are
considered the same.)
\begin{example}
Suppose $n=3$. Consider the multisegment
\[ [2;2)\oplus[0;2)\oplus[1;4)\oplus[1;1)\oplus[0;2). \]
The corresponding $\bl$ has $\lambda^{(0)}=(2^{2})$,
$\lambda^{(1)}=(41)$, and $\lambda^{(2)}=(2)$. So
\[ \lambda=(42^{3}1),\text{ and }\bl=
\begin{tableau}
:.1.2.0.1\\
:.0.1\\
:.0.1\\
:.2.0\\
:.1\\
\end{tableau}. \]
\end{example}
The dimension vector of a nilpotent representation in the class corresponding
to $\bl$ is $\bd(\bl)=(d_i(\bl))_{i\in I}$, where $d_i(\bl)$ is 
the number of boxes labelled $i$. 
Consequently, the $G_V$-orbits in $\cN_V$ are in bijection with 
$\bL(\bd)=\{\bl\in\bL\,|\,\bd(\bl)=\bd\}$.
For $\bl\in\bL(\bd)$, write $\O_\bl$ for the corresponding orbit. In other
words,
\[ \O_\bl=\{x\in\cN_V\,|\, \textbf{dim}\, \ker x^{k}=\bd(\bl^{\leq k})\}, \]
where $\bl^{\leq k}$
is obtained from $\bl$ by deleting all but the first $k$ columns.
Clearly the $GL(V)$-orbit containing
$\O_\bl$ is $\O_\lambda$.

We define a partial order $\leq$ on $\bL(\bd)$ by
\[ \bm\leq\bl\Leftrightarrow\O_\bm\subseteq\overline{\O_\bl}. \]
We will be mainly interested in the sub-poset $\bL^{\leq 2}(\bd)$
of $\bL(\bd)$ consisting of all $\bl$ with at most two rows. 
The corresponding nilpotent
representations are either indecomposable or the sum of two
indecomposables; as elements of $\mathrm{End}(V)$, they have at most two
Jordan blocks. Obviously $\bm\in\bL^{\leq 2}(\bd)$, $\bm\leq\bl \Rightarrow
\bl\in\bL^{\leq 2}(\bd)$.

\begin{remark}
All of the above remains true for the linear quiver of type $A_{\infty}$,
if we take $I=\Z$. In this case $\bL^{\leq 2}(\bd)$ has at most $2$ 
elements, of which the maximal one
parametrizes the orbit which is dense in $\cN_V=E_V$. As we will see,
$\bL^{\leq 2}(\bd)$ is more interesting for $\Delta_n$,
but still much simpler to describe than the whole poset $\bL(\bd)$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local Intersection Cohomology}
For $\bl\in\bL(\bd)$, let $\H^{k}IC(\overline{\O_\bl})$ be the
$k$-th intersection cohomology sheaf of the (usually singular)
variety $\overline{\O_\bl}$. As we will see, this vanishes if
$k$ is odd. If $\bm\leq\bl$, let the \emph{local
IC polynomial} $\tilde{K}_{\bl,\bm}(t)\in\N[t]$ be
\[ \tilde{K}_{\bl,\bm}(t)=\sum_{k\geq 0}
\dim\H_{x_\bm}^{2k}IC(\overline{\O_\bl})\, t^{k}, \]
where $\H_{x_\bm}^{2k}IC(\overline{\O_\bl})$ means the stalk at
some point $x_\bm\in\O_\bm$. Trivially we have
$\tilde{K}_{\bl,\bl}(t)=1$ for all $\bl$. 
By basic properties of intersection cohomology, $\tilde{K}_{\bl,\bm}(t)$
has constant term $1$ and satisfies
\begin{equation} \label{degeqn}
\deg \tilde{K}_{\bl,\bm} < (\mathrm{codim}\,_{\overline{\O_\bl}}\,\O_\bm)/2,
\text{ if $\bm<\bl$.}
\end{equation}
If $\bm\nleq\bl$,
we set $\tilde{K}_{\bl,\bm}(t)=0$. The main result of this paper is:
\begin{theorem} \label{mainthm}
For $\bl\geq\bm$ in $\bL^{\leq 2}(\bd)$, $\tilde{K}_{\bl,\bm}(t)=1$.
\end{theorem} \noindent
In other words, for $\bl\geq\bm$ in $\bL^{\leq 2}(\bd)$,
$\overline{\O_\bl}$ is \emph{rationally smooth} at the points of $\O_\bm$
(in general, it is not actually smooth there).

If $n=1$, $\bL(\bd)=\Lambda(d)$ and $\O_\bl=\O_\lambda$.
Lusztig proved in \cite{greenuni} that 
\[ \tilde{K}_{\lambda,\mu}(t)=t^{n(\mu)-n(\lambda)}K_{\lambda,\mu}(t^{-1}), \]
where $K_{\lambda,\mu}(\cdot)$ is the Kostka-Foulkes polynomial
and $n(\lambda)=\sum (k-1)\lambda_k$. So in this case Theorem \ref{mainthm}
is well known.

Several important results involve the polynomials
$\tilde{K}_{\bl,\bm}(t)$, which is why one wants to compute them.
In \cite[\S11]{quivers1} Lusztig showed that
the orbit closures $\overline{\O_\bl}$ could be embedded as open
subvarieties of certain affine Schubert varieties of type $A$,
which explains why $\H^{k}IC(\overline{\O_\bl})=0$ for $k$ odd.
It also means that there is an order-preserving injection 
\[ \bL(\bd)\hookrightarrow \tilde{W}:\bl\mapsto w_\bl, \]
where $\tilde{W}$ is the affine Weyl group of type
$\widetilde{A_{d-1}}$, such that
$\tilde{K}_{\bl,\bm}(t)$
is the affine Kazhdan-Lusztig polynomial $P_{w_\bm,w_\bl}(t)$. 
(This gives a combinatorial algorithm
for computing $\tilde{K}_{\bl,\bm}(t)$ in general, but it is a highly
impractical one.)
So Theorem \ref{mainthm} assserts that
certain very special affine Kazhdan-Lusztig polynomials are trivial. 
One can also interpret $(\tilde{K}_{\bl,\bm}(t))$ as the transition
matrix between two bases of the generic Hall algebra of $\Delta_n$,
an algebra of great importance in the theory of quantum affine 
$\mathfrak{sl}_{n}$ (see for instance \cite[\S3]{ltv}). We
will not say anything further about either of these points of view.

There are also well-known interpretations of (some of) the values
$\tilde{K}_{\bl,\bm}(1)$ in complex representation theory. In the case
$n=1$, $\tilde{K}_{\lambda,\mu}(1)$ is the Kostka number $K_{\lambda,\mu}$,
which gives the multiplicity of the simple representation
$V_{\lambda}$ of the symmetric group $S_d$ in the induced representation
$\Ind_{S_{\mu_1}\times S_{\mu_2}\times\cdots\times S_{\mu_{\ell(\mu)}}}
^{S_d}(\C)$. So in this
case Theorem \ref{mainthm} asserts that an induced representation of the form
$\Ind_{S_{\mu_1}\times S_{\mu_2}}^{S_d}(\C)$
is multiplicity-free, which is again well known.

If on the other hand $n\geq 2$, let $\zeta\in\C^{\times}$ be a primitive
$n$-th root of unity. Let $s\in GL(V)$
be the semisimple element which acts by the scalar $\zeta^{-i}$ on $V_i$. Then
$E_V=\{x\in\End(V)\,|\,\mathrm{Ad}(s)(x)=\zeta x\}$.
Thus for any $x\in \cN_V$, we have an associated standard module
$M_{s,x,\zeta}$ of $\widetilde{\H}_{d}$, the affine Hecke algebra
of $GL_d(\C)$ mentioned in the Introduction. Up to isomorphism,
this depends only on the
$G_V$-orbit of $x$, so we will write it as $M_{\bl}$ for $\bl\in\bL(\bd)$.

Let $L_\bl$ be the quotient of $M_\bl$ defined in 
\cite[Chapter 8]{chrissginzburg}, which is either $0$ or a simple module.
We say that $\bl\in\bL(\bd)$ is \emph{aperiodic} if for all $m$ there exists
some $i\in I$ such that $m$ is not a part of $\lambda^{(i)}$. 
It follows from results of Lusztig that $L_\bl\neq 0$ iff
$\bl$ is aperiodic (see \cite[\S2 and Appendix]{ltv}).
All simple constituents of $M_\bm$ for $\bm\in\bL(\bd)$ are isomorphic
to some such nonzero $L_\bl$, 
and we have the following multiplicity formula due to Ginzburg 
(\cite[Theorem 8.6.15]{chrissginzburg}, \cite[\S2]{ltv}):
\[ [M_\bm:L_\bl]=\tilde{K}_{\bl,\bm}(1). \]
So Theorem \ref{mainthm} implies that if 
$\bm\in\bL^{\leq 2}(\bd)$, $M_\bm$ is multiplicity-free.
In fact, this is equivalent to Theorem \ref{mainthm} (for $n\geq 2$),
since an element of $\bL^{\leq 2}(\bd)$ must be aperiodic, unless $n=2$,
all $d_i$ are equal, and
it is the minimal element of $\bL^{\leq 2}(\bd)$. As mentioned in the
Introduction, this equivalent formulation is part of the main result of 
\cite{grahamlehrer}. Thus Theorem \ref{mainthm}
has already been proved representation-theoretically; we will prove it
geometrically in the course of subsequent sections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resolutions of the Orbit Closures}
The key to our approach is the fact that each orbit closure
$\overline{\O_\bl}$ has an obvious resolution of singularities.
Fix $\bl\in\bL(\bd)$. Let $l=\ell(\lambda')$ be the number of columns of $\bl$.
Recall that $\bl^{\leq k}\in\bL$ is obtained by deleting all but
the first $k$ columns of $\bl$. Define the $I$-graded partial flag variety
\[ \F_\bl=\{0=W^{(0)}\subset W^{(1)}\subset\cdots
\subset W^{(l)}=V\,|\, \textbf{dim}\, W^{(k)}=\bd(\bl^{\leq k})\}, \]
and let
\[ \widetilde{\overline{\O_\bl}}=\{(x,(W^{(k)}))\in\cN_V\times\F_\bl\,|\,
x(W^{(k)})\subseteq W^{(k-1)}, 1 \leq k \leq l\}. \]
In the case $n=1$, this is the cotangent bundle of $\F_\bl=\F_\lambda$.
In general we have:
\begin{lemma} \label{geomlemma}
\begin{enumerate}
\item  $\F_\bl$ is a smooth irreducible projective variety, and
\[ \dim \F_\bl=\sum_{i\in I}\left[\binom{d_i}{2}-
\sum_{k=1}^{l}\binom{d_i(\bl^{\leq k})-d_i(\bl^{\leq k-1})}{2}\right]. \]
\item  The second projection $\widetilde{\overline{\O_\bl}}\to\F_\bl$
is a vector bundle, with fibres of dimension
\[ \sum_{i\in I}\left[\binom{d_i}{2}+
\sum_{k=1}^{l}\binom{d_i(\bl^{\leq k})-d_i(\bl^{\leq k-1})+1}{2}\right]
-\epsilon(\bl), \]
where
\begin{equation*}
\begin{split}
\epsilon(\bl)&=\sum_{\text{rows $R$ of $\bl$}} 
d_{\text{last label of $R$}}(\bl^{\leq |R|})\\
&=\sum_{i\in I}\sum_{p=1}^{\ell(\lambda^{(i)})}
d_{i+\lambda_p^{(i)}-1}(\bl^{\leq \lambda_p^{(i)}}).
\end{split}
\end{equation*}
\item  The first projection $p_\bl:\widetilde{\overline{\O_\bl}}\to\cN_V$
has image $\overline{\O_\bl}$ and is an isomorphism over $\O_\bl$.
Hence $p_\bl$ is a resolution of $\overline{\O_\bl}$, and
\begin{equation*}
\begin{split}
\dim \O_\bl &= \dim \overline{\O_\bl} = 
\dim \widetilde{\overline{\O_\bl}}\\
&= \sum_{i\in I}d_i^{2}-\epsilon(\bl).
\end{split}
\end{equation*}
So if $\bm\leq\bl$, $\mathrm{codim}\,_{\overline{\O_\bl}}\,\O_\bm=
\epsilon(\bm)-\epsilon(\bl)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) is obvious, because $\F_\bl$ is the product of partial flag varieties
in each $V_i$ of the required dimension. 
Clearly $\widetilde{\overline{\O_\bl}}\to\F_\bl$
is a vector bundle, and the fibre over $(W^{(k)})\in\F_\bl$ is
\[ \bigoplus_{i\in I}\{\phi\in\Hom(V_{i+1},V_i)\,|\,
\phi(W_{i+1}^{(k)})\subseteq W_i^{(k-1)}, 1\leq k \leq l\}, \]
which has dimension
\[ \sum_{i\in I}\sum_{k=1}^{l}(d_{i+1}(\bl^{\leq k})-d_{i+1}(\bl^{\leq k-1}))
d_i(\bl^{\leq k-1}). \]
This is the number of ordered pairs of boxes $(b,b')$ in $\bl$ where
the column containing $b'$ is further right than the column containing $b$,
and the label of $b'$ is one more than the label of $b$. Considering the
box $b''$ immediately to the left of $b'$, we see that this is the same
as the number of ordered pairs of boxes $(b,b'')$ with the same label
where the column containing $b''$ is further right than or equal to
the column containing $b$, and where $b''$ is not at the end of a row.
This gives (2). For (3), $p_\bl$ is proper since $\F_\bl$ is projective.
From the definitions,
\[ p_\bl^{-1}(\O_\bl)=\{(x,(W^{(k)}))\in\cN_V\times\F_\bl\,|\,
\ker x^{k}=W^{(k)}, 1 \leq k \leq l\}, \]
which is clearly open in $\widetilde{\overline{\O_\bl}}$ and isomorphic
to $\O_\bl$. The rest follows since we know from (1) and (2) that
$\widetilde{\overline{\O_\bl}}$ is smooth and irreducible.
\end{proof}

Note that $\sum_{i\in I}d_i^{2}=\dim G_V$. So (3) implies that
for $x_\bl\in\O_\bl$, $\dim Z_{G_V}(x_\bl)=\epsilon(\bl)$. (Thus the notation
agrees with \cite[\S3]{ltv}.) Since $ Z_{G_V}(x_\bl)$ is an open subvariety
of the algebra of endomorphisms of $x_\bl$ as a representation of
$\Delta_n$, it is connected. So the only $G_V$-equivariant
simple perverse sheaves on $\cN_V$ are the shifted intersection
cohomology complexes $IC(\overline{\O_\bl})[\dim \O_\bl]$, for
$\bl\in\bL(\bd)$. By the Equivariant Decomposition Theorem 
(see for example \cite[Theorem 8.4.7]{chrissginzburg}),
\begin{equation} \label{decompeqn}
R(p_\bl)_{*}\C[\dim \O_\bl]\cong
\bigoplus_{\substack{\bn\leq\bl\\j\in\Z}}
IC(\overline{\O_\bn})[\dim \O_\bn + j]^{\oplus a_{\bl,\bn,j}}
\end{equation}
for some $a_{\bl,\bn,j}\in\N$. Since the left-hand side is Verdier self-dual,
$a_{\bl,\bn,j}=a_{\bl,\bn,-j}$. Since $p_\bl$ is an isomorphism
over $\O_\bl$, $a_{\bl,\bl,j}$ is $1$ if $j=0$ and $0$ otherwise.

Taking stalk at $x_\bm\in\O_\bm$ of both sides of \eqref{decompeqn}, we get
\[ \sum_{k\geq 0}\dim H^{k}(p_\bl^{-1}(x_\bm))\, t^{k}
= \sum_{\bl\geq\bn\geq\bm}
\left(\sum_{j\in\Z} a_{\bl,\bn,j}\, t^{\epsilon(\bn)-\epsilon(\bl)-j}\right)
\tilde{K}_{\bn,\bm}(t^{2}). \]
Now
\[ p_\bl^{-1}(x_\bm)\cong\{(W^{(k)})\in\F_\bl\,|\,
x_\bm(W^{(k)})\subseteq W^{(k-1)}, 1 \leq k \leq l\}. \]
It is easy to prove (say by induction on $l$) that
$p_\bl^{-1}(x_\bm)$ has a paving by affine spaces, and hence has no
odd cohomologies. (In the cases we use below we will see this another way.)
So in fact $a_{\bl,\bn,j}=0$ if
$j$ is of opposite parity to $\epsilon(\bn)-\epsilon(\bl)$, and
if we write $g_{\bl,\bm}(t)$ for the Poincar\'e polynomial
$\sum_{k\geq 0}\dim H^{2k}(p_\bl^{-1}(x_\bm))\, t^{k}$, we have
\begin{equation} \label{maineqn}
g_{\bl,\bm}(t)
= \sum_{\bl\geq\bn\geq\bm}
\left(\sum_{j\equiv\epsilon(\bn)-\epsilon(\bl)\text{ mod }2} 
a_{\bl,\bn,j}\, t^{(\epsilon(\bn)-\epsilon(\bl)-j)/2}\right)
\tilde{K}_{\bn,\bm}(t).
\end{equation}
Thanks to the degree constraint and others we have mentioned,
knowing $g_{\bl,\bm}(t)$ for all $\bl\geq\bm$ determines
all $a_{\bl,\bm,j}$ and $\tilde{K}_{\bl,\bm}(t)$. In fact:
\begin{lemma} \label{uniqlemma}
If $b_{\bl,\bm,j}\in\C$ and $L_{\bl,\bm}(t)\in\C[t]$ for all
$\bl\geq\bm\in\bL(\bd)$ and $j\in\Z$ satisfy:
\begin{enumerate}
\item for all $\bl\geq\bm$,
\[ g_{\bl,\bm}(t)
= \sum_{\bl\geq\bn\geq\bm}
\left(\sum_{j\equiv\epsilon(\bn)-\epsilon(\bl)\text{ mod }2} 
b_{\bl,\bn,j}\, t^{(\epsilon(\bn)-\epsilon(\bl)-j)/2}\right)
L_{\bn,\bm}(t), \]
\item  $b_{\bl,\bl,j}=\delta_{j,0}$,
\item  $b_{\bl,\bm,j}=b_{\bl,\bm,-j}$,
\item  $b_{\bl,\bm,j}=0$ if $j$ is of opposite parity to
$\epsilon(\bm)-\epsilon(\bl)$,
\item  $L_{\bl,\bl}(t)=1$, and
\item  $\deg L_{\bl,\bm}<(\epsilon(\bm)-\epsilon(\bl))/2$ for $\bm<\bl$,
\end{enumerate}
then $b_{\bl,\bm,j}=a_{\bl,\bm,j}$, $L_{\bl,\bm}(t)=\tilde{K}_{\bl,\bm}(t)$
for all $\bl\geq\bm$, $j\in\Z$.
\end{lemma}
\begin{proof}
We know that $b_{\bl,\bm,j}=a_{\bl,\bm,j}$, 
$L_{\bl,\bm}(t)=\tilde{K}_{\bl,\bm}(t)$ is a solution of (1)--(6).
We now give an algorithm to determine $b_{\bl,\bm,j}$ and $L_{\bl,\bm}(t)$
from (1)--(6), assuming that all $g_{\bl,\bm}(t)$ are known,
thus showing that this
is the unique solution. This algorithm is by induction on
$\epsilon(\bm)-\epsilon(\bl)$. If this is $0$, $\bm=\bl$,
and $b_{\bl,\bl,j}$ and $L_{\bl,\bl}(t)$ are determined by (2) and (5)
respectively. Otherwise, we can apply the induction hypothesis
to $\bl>\bn$ and $\bn>\bm$, and hence assume that all terms in the
right-hand side of (1) have been determined except $\bn=\bl$ and $\bn=\bm$.
So we know the value of
\[ L_{\bl,\bm}(t)+\sum_{j\equiv\epsilon(\bm)-\epsilon(\bl)\text{ mod }2} 
b_{\bl,\bm,j}\, t^{(\epsilon(\bm)-\epsilon(\bl)-j)/2}. \]
Because of (6), this determines $b_{\bl,\bm,j}$ for $j\leq 0$,
$j \equiv\epsilon(\bm)-\epsilon(\bl)\text{ mod }2$, hence for all $j$
by (3) and (4). Thus $L_{\bl,\bm}(t)$ also is determined.
\end{proof} \noindent
The argument of this proof is familiar in other contexts, for instance
in the study of Kazhdan-Lusztig polynomials.
\begin{remark} \label{intervalrem}
Note that if we want to determine a specific polynomial 
$\tilde{K}_{\bl,\bm}(t)$ by this method, we only need to know
$g_{\br,\bn}(t)$ for those $\br$, $\bn$ such that
$\bl\geq\br\geq\bn\geq\bm$; that is, we can restrict attention to the
interval $[\bm,\bl]\subseteq\bL(\bd)$. In particular, in proving
Theorem \ref{mainthm} we can restrict attention to $\bL^{\leq 2}(\bd)$.
In the remaining sections, we will calculate $g_{\bl,\bm}(t)$ for $\bl\geq\bm$
in $\bL^{\leq 2}(\bd)$ and show that it has the required form.
\end{remark}
\begin{remark}
Note that in the case $n=1$, $p_\lambda:\widetilde{\overline{\O_\lambda}}\to
\overline{\O_\lambda}$ is \emph{semismall}; equivalently,
$a_{\lambda,\nu,j}=0$ if $j\neq 0$. For example, $p_{(d)}$ is the famous
Springer resolution of the nilpotent cone. The Poincar\'e polynomial
$g_{\lambda,\mu}(t)$ is the generalized Hall polynomial
$g_{(1^{\lambda_{l}'}),\cdots,(1^{\lambda_{1}'})}^{\mu}(t)$,
the function
$\epsilon(\lambda)$ is $2n(\lambda)+d$, and the nonzero coefficient
$a_{\lambda,\nu,0}$ is the Kostka number $K_{\nu',\lambda'}$
(see \cite[III.6, Example 5]{macdonald}). 
This can be proved geometrically, using
Fourier transform (see \cite[\S5]{myfourier}). 
When $n\geq 2$, Fourier transform does not seem
to be useful for analysing $R(p_\bl)_{*}\C$.
\end{remark}
\begin{remark}
An example when $p_{\bl}$ is not semismall is the case $n=2$,
$\bl=[0;4)\oplus[0;1)$. If $\bm=[0;2)\oplus[0;2)\oplus[0;1)$, the reader
can check that $a_{\bl,\bm,1}=a_{\bl,\bm,-1}=1$.
\end{remark}
\begin{remark}
When $n=2$, it is easy to see that
\[ \epsilon(\bl)=\frac{(d_0-d_1)^{2}+\epsilon(\lambda)}{2}, \]
whence $\dim\O_\bl=\frac{1}{2}\dim\O_\lambda$. This is well known,
since the orbits $\O_\bl$ are the nilpotent orbits of the symmetric
pair $(\gl_d,\gl_{d_0}\oplus\gl_{d_1})$.
\end{remark}
\begin{remark}
When $n\geq 3$, it is not true in general that
$\epsilon(\bl)$ only depends on $\bd$ and $\lambda$.
(Consider $n=3$, $d_0=2$, $d_1=d_2=1$, $\lambda=(21^{2})$.)
However, it is still true that $\bm\leq\bl$, $\mu=\lambda \Rightarrow
\bm=\bl$. This can be seen by interpreting $\overline{\O_\bl}$ as the image
of $p_\bl$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\tabwidth}{2.4ex}
\setlength{\tabheight}{2.4ex}
\section{Preliminary Lemmas}
Now we restrict attention to $\bl\in\bL^{\leq 2}(\bd)$ and the corresponding
nilpotent orbits. It is convenient to return to multisegment notation,
and write elements of $\bL^{\leq 2}(\bd)$ as
$[i_1;l_1)\oplus[i_2;l_2)$ where $l_2$ is possibly zero (by convention
$[i;0)$ is the empty partition). For any $m\in\Z$, write $\{m\}$ for the
unique element of $\{0,1,\cdots,n-1\}\cap (m+n\Z)$. So
\[ \lfloor\frac{m}{n}\rfloor=\frac{m-\{m\}}{n},\
\lceil\frac{m}{n}\rceil=\frac{m+\{-m\}}{n}. \]
\begin{lemma} \label{epsilonlemma}
Let $\bl=[i_1;l_1)\oplus[i_2;l_2)$, where the factors are ordered so that
$l_1\geq l_2$. Then
\begin{equation*}
\begin{split}
\epsilon(\bl)&=\lceil\frac{l_1}{n}\rceil+\lceil\frac{l_2}{n}\rceil
+\lceil\frac{l_2-\{i_1-i_2+l_1-1\}}{n}\rceil\\
&\quad\quad\quad+\lceil\frac{l_2-\{i_1-i_2\}}{n}\rceil.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
This is trivial from the definition of $\epsilon$.
\end{proof}

Fix $\bl=[i_1;l_1)\oplus[i_2;l_2)$, with $l_1\geq l_2\geq 0$. Let
\begin{equation*}
\begin{split} 
\bL^{\leq 2}(\bd)_{\leq\bl}
&=\{\bm\in\bL^{\leq 2}(\bd)\,|\,\bm\leq\bl\},\\
\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}
&=\{\bm=[i_1;m_1)\oplus[i_2;m_2)\,|\,\bm\leq\bl\}.
\end{split}
\end{equation*}
\begin{lemma}
If $l_2\geq 1$, $\bL^{\leq 2}(\bd)_{\leq\bl}=
\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$. If $l_2=0$,
\[ \bL^{\leq 2}(\bd)_{\leq\bl}=\bigcup_{i_2}
\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}, \]
and the union is disjoint except for $\bl$.
\end{lemma}
\begin{proof}
If $l_2 \geq 1$, we have $\textbf{dim}\,\ker x=\{i_1,i_2\}$
for any $x\in\O_\bl$, where $\{i_1,i_2\}$ is considered as a
multiset, and hence $\textbf{dim}\,\ker x\geq\{i_1,i_2\}$
for any $x\in\overline{\O_\bl}$. Thus if $\bm\leq\bl$ and 
$\bm\in\bL^{\leq 2}(\bd)$,
$\bm$ must be $[i_1;m_1)\oplus [i_2;m_2)$ for some
$m_1,m_2\in\N$. If $l_2=0$, we know only that for $x\in\overline{\O_\bl}$, 
$\textbf{dim}\,\ker x\geq\{i_1\}$. But if $\bm<\bl$ and 
$\bm\in\bL^{\leq 2}(\bd)$, $\bm$ must have two rows, so
there is a unique $i_2$ such that $\bm=[i_1;m_1)\oplus [i_2;m_2)$.
\end{proof}
\begin{lemma} \label{condlemma}
Suppose $\bm=[i_1;m_1)\oplus [i_2;m_2)\in\bL^{\leq 2}(\bd)
_{\leq\bl}^{i_1,i_2}$. Then 
\[ m_1+m_2=l_1+l_2,\ l_1\geq m_1\geq l_2, \]
and either
\begin{equation*}
\begin{split}
&\textup{(a)}\ m_1\equiv l_1,\ m_2\equiv l_2 \text{ mod }n; \text{ or}\\
&\textup{(b)}\ m_1\equiv i_2-i_1+l_2,\ m_2\equiv i_1-i_2+l_1 \text{ mod }n.
\end{split}
\end{equation*}
(It is possible that \textup{(a)} and \textup{(b)} both hold.)
\end{lemma}
\begin{proof}
That $m_1+m_2=d=l_1+l_2$ is obvious. Since $\O_\bm\subseteq\overline{\O_\bl}$,
we must have $\O_\mu\subseteq\overline{\O_\lambda}$, which implies that
$l_1\geq m_1\geq l_2$. Finally,
\[ \bd([i_1;m_1)\oplus [i_2;m_2))=\bd([i_1;l_1)\oplus [i_2;l_2)). \]
Subtracting $\bd([i_1;m_1)\oplus [i_2;l_2))$ from both sides, we get
\[ \bd([i_2+l_2;m_2-l_2))=\bd([i_1+m_1;l_1-m_1)). \]
This clearly implies either (a) or (b).
\end{proof}

It will be easier to get a converse to this Lemma after
we describe the fibres $p_\bl^{-1}(x_\bm)$. For $s_1,s_2\in\N$, let
$\B_{(s_1,s_2)}$ be the Springer fibre 
$p_{(s_1+s_2)}^{-1}(x_{(s_1)\oplus(s_2)})$,
a special case of this fibre when $n=1$. (The choices of the vector
space of dimension $s_1+s_2$ and the element $x_{(s_1)\oplus(s_2)}\in
\O_{(s_1)\oplus(s_2)}$ are unimportant.)
\begin{lemma} \label{flaglemma}
Let $\bm=[i_1;m_1)\oplus [i_2;m_2)\neq\bl$,
where $m_1$ and $m_2$ satisfy the conditions in Lemma
\ref{condlemma}. Then $p_\bl^{-1}(x_\bm)\cong\B_{(s_1,s_2)}$, where
\[ s_1=\lfloor\frac{m_1-l_2-\{i_2-i_1\}}{n}\rfloor,\
s_2=\lfloor\frac{m_2-l_2}{n}\rfloor, \]
so that
\[ s_1+s_2=s:=\frac{l_1-l_2-\{i_2-i_1\}-\{i_1-i_2+l_1-l_2\}}{n}. \]
(We declare $\B_{(-1,s_2)}$ to be empty.)
\end{lemma}
\begin{proof}
By definition,
\[ p_\bl^{-1}(x_\bm)\cong\{(W^{(k)})\in\F_\bl\,|\,
x_\bm(W^{(k)})\subseteq W^{(k-1)}, 1 \leq k \leq l_1\}. \]
It is clear that for $1\leq k\leq l_2$, $W^{(k)}$ is forced to be
$\ker x_\bm^{k}$. Hence $p_\bl^{-1}(x_\bm)\cong p_{\bl'}^{-1}(x_{\bm'})$ where
\[ \bl'=[i_1+l_2;l_1-l_2),\ \bm'=[i_1+l_2;m_1-l_2)\oplus
[i_2+l_2;m_2-l_2). \]
Since $\bl\neq\bm$, $m_2-l_2>0$. In this new fibre,
if $i_1\neq i_2$, $W^{(1)}$ is forced to be
$(\ker x_{\bm'})_{i_1+l_2}$. Passing to the quotient space and repeating,
we see that $W^{(k)}$ is uniquely determined for $1\leq k\leq\{i_2-i_1\}$, so
$p_{\bl'}^{-1}(x_{\bm'})\cong p_{\bl''}^{-1}(x_{\bm''})$ where
\begin{equation*}
\begin{split}
\bl''&=[i_2+l_2;l_1-l_2-\{i_2-i_1\}),\\ 
\bm''&=[i_2+l_2;m_1-l_2-\{i_2-i_1\})
\oplus [i_2+l_2;m_2-l_2).
\end{split}
\end{equation*}
(If any lengths become negative here we understand that the corresponding
fibre is empty.)

We now wish to apply the dual argument, so note that
the end box of $\bl''$ is labelled $i_1+l_1-1$, and the end labels
of $\bm''$ are $i_1+l_1-1$ and $i_2+l_2-1$ in some order.
If $i_1+l_1\neq i_2+l_2$,
$W^{(l_1-l_2-\{i_2-i_1\}-1)}$ is forced to be the preimage of
$(\mathrm{coker }x_{\bm''})_{i_2+l_2-1}$. Hence
$W^{(l_1-l_2-\{i_2-i_1\}-k)}$ is unique for 
$1\leq k\leq\{l_1-l_2+i_1-i_2\}$,
so $p_{\bl''}^{-1}(x_{\bm''})\cong
p_{\bl'''}^{-1}(x_{\bm'''})$, where
\[ \bl'''=[i_2+l_2;l_1-l_2-\{i_2-i_1\}-\{l_1-l_2+i_1-i_2\})=[i_2+l_2;ns), \]
and $\bm'''$ is obtained from $\bm''$ by deleting the last
$\{l_1-l_2+i_1-i_2\}$ boxes of the row whose end label is
$i_1+l_1-1$. This is the first row if (a) holds and the second row
if (b) holds; if both hold, there are no boxes to be deleted.
In either case,
$\bm'''=[i_2+l_2;ns_1)\oplus [i_2+l_2;ns_2)$.

So in the end it suffices to show that
\[ p_{[i;n(s_1+s_2))}^{-1}(x_{[i;ns_1)\oplus[i;ns_2)})\cong\B_{(s_1,s_2)}. \]
This is obvious, since each $I$-graded flag in the left-hand side
is uniquely determined by the induced flag in the subspace corresponding to 
$i$.
\end{proof}
\begin{example}
To illustrate this proof, here is an example of the sequence of steps
when $n=3$ (the boxes with bold borders are the ones to be removed):
\[ 
\bl=\begin{tableau}
:\b{2}{2}{:.0.1\\:.1.2\\}.2.0.1.2.0.1.2.0\\:;\\
\end{tableau}\, ,\
\bm=\begin{tableau}
:\b{2}{2}{:.0.1\\:.1.2\\}.2.0.1.2.0\\
:;;.0.1.2\\
\end{tableau} \]

\[ \rightsquigarrow
\bl'= \begin{tableau}
:\b{1}{1}{:.2\\}.0.1.2.0.1.2.0\\
\end{tableau}\, ,\
\bm'=\begin{tableau}
:\b{1}{1}{:.2\\}.0.1.2.0\\
:.0.1.2\\
\end{tableau} \]

\[ \rightsquigarrow
\bl''=\begin{tableau}
:.0.1.2.0.1.2\b{1}{1}{:.0\\}\\
\end{tableau}\, ,\
\bm''=\begin{tableau}
:.0.1.2\b{1}{1}{:.0\\}\\
:.0.1.2\\
\end{tableau} \]

\[ \rightsquigarrow
\bl'''=\begin{tableau}
:.0.1.2.0.1.2\\
\end{tableau}\, ,\
\bm'''=\begin{tableau}
:.0.1.2\\
:.0.1.2\\
\end{tableau}\
\rightsquigarrow\B_{(1,1)}. \]
\end{example}
\begin{corollary} \label{condcor}
Let $\bm=[i_1;m_1)\oplus [i_2;m_2)$. Then 
$\bm\in\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$
if and only if 
\[ m_1+m_2=l_1+l_2,\ l_1\geq m_1\geq l_2+\{i_2-i_1\}, \]
and either \textup{(a)} or \textup{(b)} holds.
\end{corollary}
\begin{proof}
We know from Lemma \ref{geomlemma} that $\overline{\O_\bl}$ is the image of
$p_\bl$, so $\bm\leq\bl$ iff $p_\bl^{-1}(x_\bm)$ is nonempty.
Lemma \ref{flaglemma} shows that for $m_1$, $m_2$ as in Lemma \ref{condlemma},
$p_\bl^{-1}(x_\bm)$ is empty if and only if (a) holds and 
$m_1-l_2<\{i_2-i_1\}$.
\end{proof}
\begin{corollary} \label{flagcor}
For $\bm<\bl$,
$p_\bl^{-1}(x_\bm)$ has no odd cohomologies, and
\[ g_{\bl,\bm}(t)
= \sum_{k=0}^{\min\{s_1,s_2\}}(\binom{s}{k}-\binom{s}{k-1})\,t^{k}, \]
where $s_1$, $s_2$ and $s$ are as in Lemma \ref{flaglemma}.
\end{corollary}
\begin{proof}
Springer fibres such as $\B_{(s_1,s_2)}$ have no odd cohomologies.
The right-hand side is the Green polynomial $Q_{(1^{s})}^{(s_1,s_2)}(t)$,
which is well known to equal $\sum_{k\geq 0}
\dim H^{2k}(\B_{(s_1,s_2)})\,t^{k}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Description of the Posets}
Maintain the notations of \S5. In particular,
$\bl=[i_1;l_1)\oplus[i_2;l_2)$ where $l_1\geq l_2\geq 0$; and
$s$ is as in Lemma \ref{flaglemma}.
Using Corollary \ref{condcor},
we can completely describe the poset
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$
and hence the poset $\bL^{\leq 2}(\bd)$ as a whole.
The arguments are trivial applications of the Lemmas in \S5, and are
mostly left to the reader. There are four cases.

\textbf{Case 1:} $i_1=i_2=i$, $l_1\equiv l_2\text{ mod }n$.\newline
In this case conditions (a) and (b) coincide, and
$s=\frac{l_1-l_2}{n}$. If $l_1-l_2=0$ or $n$,
$\bl$ is the unique element of $\bL^{\leq 2}(\bd)_{\leq\bl}^{i,i}$, 
and conversely.
Otherwise, $[i;l_1-n)\oplus[i;l_2+n)$ is a predecessor of $\bl$,
also in Case 1. So $\bL^{\leq 2}(\bd)_{\leq\bl}^{i,i}$ is a chain:
\begin{equation*}
\begin{CD}
[i;l_1)\oplus [i;l_2) @. \mathbf{(s,0)}\\
@VVV @.\\
[i;l_1-n)\oplus [i;l_2+n) @. \mathbf{(s-1,1)}\\
@VVV @.\\
: @.\\
@VVV @.\\
\qquad
[i;l_1-\lfloor\frac{s}{2}\rfloor n)\oplus [i;l_2+\lfloor\frac{s}{2}\rfloor n)
\qquad
@. \mathbf{(\lceil\frac{s}{2}\rceil,\lfloor\frac{s}{2}\rfloor)}
\end{CD}
\end{equation*}
Here and in subsequent diagrams $\bm\to\bn$ means that $\bn$ is
a predecessor of $\bm$, and the boldface labels are the values 
of $(s_1,s_2)$ as in Lemma \ref{flaglemma} (whose order
is actually not determined in the case that $i_1=i_2$). 
It is easy to see from
Lemma \ref{epsilonlemma} that
$\epsilon([i;l_1-n)\oplus [i;l_2+n))=\epsilon(\bl)-2$.
So the codimension at each link of the chain is $2$. There are
$\lfloor\frac{s}{2}\rfloor$ links.

\textbf{Case 2:} $i_1=i_2=i$, $l_1\not\equiv l_2\text{ mod }n$.\newline
Here conditions (a) and (b) are mutually exclusive, and 
$s=\lfloor\frac{l_1-l_2}{n}\rfloor$. It is easy to see that
$\bl$ is the unique element of 
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i,i}$ if and only if $l_1-l_2<n$;
otherwise, $[i;l_1-\{l_1-l_2\})\oplus[i;l_2+\{l_1-l_2\})$ 
is a predecessor of $\bl$, also in Case 2, and satisfying condition (b).
Again, $\bL^{\leq 2}(\bd)_{\leq\bl}^{i,i}$ is a chain:
\begin{equation*}
\begin{CD}
[i;l_1)\oplus [i;l_2) @. \mathbf{(s,0)}\\
@VVV @.\\
\qquad[i;l_1-\{l_1-l_2\})\oplus[i;l_2+\{l_1-l_2\})\qquad
@. \mathbf{(s,0)}\\
@VVV @.\\
[i;l_1-n)\oplus [i;l_2+n) @. \mathbf{(s-1,1)}\\
@VVV @.\\
[i;l_1-n-\{l_1-l_2\})\oplus[i;\cdots) @. \mathbf{(s-1,1)}\\
@VVV @.\\
: @.\\
@VVV @.\\
[i;l_1-\lfloor\frac{s}{2}\rfloor n-\{l_1-l_2\})\oplus 
[i;\cdots)
@. \mathbf{(\lceil\frac{s}{2}\rceil,\lfloor\frac{s}{2}\rfloor)}
\end{CD}
\end{equation*}
(To make this and later diagrams legible, some row lengths are omitted; recall
that for every element the sum of the lengths is $l_1+l_2$.)
If $s$ is even, the minimal element is actually the same as
the element which might be thought to be directly above it, namely
$[i;l_1-\frac{s}{2}n)\oplus[i;l_2+\frac{s}{2}n)$. So there are
always $s$ links in the chain. This time
the codimension at each link is $1$.

\textbf{Case 3:} $i_1\neq i_2$, $l_1-l_2\equiv i_2-i_1\text{ mod }n$.\newline
In this case conditions (a) and (b) coincide, and
$s=\lfloor\frac{l_1-l_2}{n}\rfloor$. The condition for
$\bl$ to be the only element of
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$ is that $l_1-l_2=\{i_2-i_1\}$. 
Otherwise,
$[i_2;l_1-\{l_1-l_2\})\oplus[i_1;l_2+\{l_1-l_2\})$ is a
predecessor of $\bl$, also in Case 3. The summands here 
are in the right order, i.e.\ $l_1-\{l_1-l_2\}>l_2+\{l_1-l_2\}$.
Once more $\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$ is a chain:
\begin{equation*}
\begin{CD}
[i_1;l_1)\oplus [i_2;l_2) @. \mathbf{(s,0)}\\
@VVV @.\\
\qquad [i_2;l_1-\{l_1-l_2\})\oplus[i_1;l_2+\{l_1-l_2\})
\qquad @. \mathbf{(0,s)}\\
@VVV @.\\
[i_1;l_1-n)\oplus [i_2;l_2+n) @. \mathbf{(s-1,1)}\\
@VVV @.\\
[i_2;l_1-n-\{l_1-l_2\})\oplus[i_1;\cdots) @. \mathbf{(1,s-1)}\\
@VVV @.\\
: @.\\
@VVV @.\\
[i_2;l_1-\lfloor\frac{s}{2}\rfloor n-\{l_1-l_2\})\oplus 
[i_1;\cdots) @. \mathbf{(\lfloor\frac{s}{2}\rfloor,\lceil\frac{s}{2}\rceil)}
\end{CD}
\end{equation*}
If $s$ is even, the minimal element here is also
$[i_1;l_1-\frac{s}{2}n)\oplus[i_2;l_2+\frac{s}{2}n)$, so there
are always $s$ links in the chain. Again the
codimension at each link is $1$.

\textbf{Case 4:} $i_1\neq i_2$, $l_1-l_2\not\equiv i_2-i_1\text{ mod }n$.
\newline
Conditions (a) and (b) are mutually exclusive, and
$s=\lfloor\frac{l_1-l_2}{n}\rfloor$ or $\lfloor\frac{l_1-l_2}{n}\rfloor-1$
depending on whether $\{l_1-l_2\}$ or $\{i_2-i_1\}$ is larger. If
$l_1-l_2<\{i_2-i_1\}$, $\bl$ is the unique element of 
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$, and conversely.
Otherwise, write $A=\{i_2-i_1\}$, $B=\{i_1-i_2+l_1-l_2\}$;
then $[i_1;l_1-B)\oplus [i_2;l_2+B)$
and $[i_2;l_1-A)\oplus[i_1;l_2+A)$
are predecessors of $\bl$, also in Case 4, and satisfying condition (b). 
These coincide iff $l_1-l_2<n+\{i_2-i_1\}$, in which
case their common value is a minimal element; otherwise, the
summands are in the right order. The poset
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$ is as follows:
\[
\psmatrix[colsep=0.15cm,rowsep=1cm]
\mathbf{(s,0)} & {[i_1;l_1)\oplus [i_2;l_2)} &  & \\
\mathbf{(s,0)} & {[i_1;l_1-B)\oplus [i_2;\cdots)} & 
{[i_2;l_1-A)\oplus[i_1;\cdots)} & \mathbf{(0,s)}\\
\mathbf{(s-1,1)} & 
{\begin{array}{c}[i_1;l_1-n)\\\oplus[i_2;\cdots)\end{array}} & 
{\begin{array}{c}[i_2;l_1-A-B)\\\oplus[i_1;\cdots)\end{array}} 
& \mathbf{(0,s)}\\
\mathbf{(s-1,1)} & {\begin{array}{c}[i_1;l_1-n-B)\\
\oplus[i_2;\cdots)\end{array}} &
{\begin{array}{c}[i_2;l_1-n-A)\\
\oplus[i_1;\cdots)\end{array}} & \mathbf{(1,s-1)}\\
 &\phantom{AAAAA}:\phantom{AAAAA} & \phantom{AAAAA}:\phantom{AAAAA}& \\
\mathbf{(\lceil\frac{s}{2}\rceil,\lfloor\frac{s}{2}\rfloor)} &
{\begin{array}{c}[i_1;l_1-\lfloor\frac{s}{2}\rfloor n-B)\\
\oplus [i_2;\cdots)\end{array}} &
{\begin{array}{c}[i_2;l_1-\lfloor\frac{s}{2}\rfloor n-A)\\
\oplus [i_1;\cdots)\end{array}} &
\mathbf{(\lfloor\frac{s}{2}\rfloor,\lceil\frac{s}{2}\rceil)}
\endpsmatrix%\everypsbox{\scriptstyle}
\psset{nodesep=3pt,arrows=->}
\ncline{1,2}{2,2}
\ncline{1,2}{2,3}
\ncline{2,2}{3,2}
\ncline{2,2}{3,3}
\ncline{2,3}{3,2}
\ncline{2,3}{3,3}
\ncline{3,2}{4,2}
\ncline{3,2}{4,3}
\ncline{3,3}{4,2}
\ncline{3,3}{4,3}
\ncline{4,2}{5,2}
\ncline{4,2}{5,3}
\ncline{4,3}{5,2}
\ncline{4,3}{5,3}
\ncline{5,2}{6,2}
\ncline{5,2}{6,3}
\ncline{5,3}{6,2}
\ncline{5,3}{6,3}
%\end{psmatrix}
\]
%\begin{equation*}
%\begin{CD}
%\mathbf{(s,0)} @. [i_1;l_1)\oplus [i_2;l_2) @. @.\\
%@. @VVV @. @.\\
%\mathbf{(s,0)} @. [i_1;l_1-B)\oplus [i_2;l_2+B) @.
%[i_2;l_1-A)\oplus[i_1;\cdots) @. \mathbf{(0,s)}\\
%@. @VVV @VVV @.\\
%\mathbf{(s-1,1)} @. [i_1;l_1-n)\oplus [i_2;l_2+n) @.
%[i_2;l_1-A-B)\oplus[i_1;\cdots) @. \mathbf{(0,s)}\\
%@. @VVV @VVV @.\\
%\mathbf{(s-1,1)} @. \begin{array}{c}[i_1;l_1-n-B)\\
%\oplus[i_2;\cdots)\end{array} @.
%\begin{array}{c}[i_2;l_1-n-A)\\
%\oplus[i_1;\cdots)\end{array} @. \mathbf{(1,s-1)}\\
%@. @VVV @VVV @.\\
%@. : @. : @. \\
%@. @VVV @VVV @.\\
%\mathbf{(\lceil\frac{s}{2}\rceil,\lfloor\frac{s}{2}\rfloor)} @.
%\begin{array}{c}[i_1;l_1-\lfloor\frac{s}{2}\rfloor n-B)\\
%\oplus [i_2;\cdots)\end{array} @.
%\begin{array}{c}[i_2;l_1-\lfloor\frac{s}{2}\rfloor n-A)\\
%\oplus [i_1;\cdots)\end{array} @.
%\mathbf{(\lfloor\frac{s}{2}\rfloor,\lceil\frac{s}{2}\rceil)}
%\end{CD}
%\end{equation*}
If $s$ is even, the bottom two elements coincide and are minimal.
If $s$ is odd, there is one more element below these two, namely
\[ [i_1;l_1-\lceil\frac{s}{2}\rceil n)\oplus
[i_2;l_2+\lceil\frac{s}{2}\rceil n)\qquad
\mathbf{(\lfloor\frac{s}{2}\rfloor,\lceil\frac{s}{2}\rceil)}. \]
So the length of a maximal chain in this poset is $s+1$. Again, all
predecessors have codimension $1$.

\begin{example}
As an example of how these posets
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$ fit together to
make $\bL^{\leq 2}(\bd)$, the following is the case
$n=2$, $d_0=d_1=3$:
\[
\psmatrix[colsep=1cm,rowsep=1cm]
 & {[0;6)} & {[1;6)} & \\
 & {[0;5)\oplus[1;1)} & {[1;5)\oplus[0;1)} & \\
{[0;4)\oplus[0;2)} & {[0;4)\oplus[1;2)} & {[1;4)\oplus[0;2)} & 
{[1;4)\oplus[1;2)} \\
 & {[0;3)\oplus[1;3)} & &
\endpsmatrix%\everypsbox{\scriptstyle}
\psset{nodesep=3pt,arrows=->}
\ncline{1,2}{2,2}
\ncline{1,2}{2,3}
\ncline{1,2}{3,1}
\ncline{1,3}{2,2}
\ncline{1,3}{2,3}
\ncline{1,3}{3,4}
\ncline{2,2}{3,2}
\ncline{2,2}{3,3}
\ncline{2,3}{3,2}
\ncline{2,3}{3,3}
\ncline{3,2}{4,2}
\ncline{3,3}{4,2}
%\end{psmatrix}
\]
%\begin{equation*}
%\begin{CD}
%@. [0;6) @. [1;6) @.\\
%@. @VVV @VVV @.\\
%@. \quad[0;5)\oplus[1;1)\quad @. \quad[1;5)\oplus[0;1)\quad @.\\
%@. @VVV @VVV @.\\
%[0;4)\oplus[0;2) @. [0;4)\oplus[1;2) @. [1;4)\oplus[0;2) @.
%[1;4)\oplus[1;2)\\
%@. @VVV @. @.\\
%@. [0;3)\oplus[1;3) @. @.
%\end{CD}
%\end{equation*}
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{mainthm}}
For $\bl\geq\bm$ in $\bL^{\leq 2}(\bd)$,
define a nonnegative integer $c_{\bl,\bm}$ by the following rule.
If $\bl=\bm$, let $c_{\bl,\bm}=1$. If $\bl>\bm$, then
$\bm\in\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$ for unique $(i_1,i_2)$,
and the interval $[\bm,\bl]$ is contained in 
$\bL^{\leq 2}(\bd)_{\leq\bl}^{i_1,i_2}$.
Define $s_1$, $s_2$, and $s$ as above, and let $k(\bm)=\min\{s_1,s_2\}$.
Say $\bm$ is \emph{special} (relative to $\bl$) if there is no
element $\bn$ of $[\bm,\bl]$ with $k(\bn)=k(\bm)$ except $\bm$ itself.
If $\bm$ is special, let 
$c_{\bl,\bm}=\binom{s}{k(\bm)}-\binom{s}{k(\bm)-1}$.
Otherwise, let $c_{\bl,\bm}=0$. 
We can now prove a stronger
version of Theorem \ref{mainthm}:
\begin{theorem}
If $\bl\geq\bm$ in $\bL^{\leq 2}(\bd)$, then $\tilde{K}_{\bl,\bm}(t)=1$,
$a_{\bl,\bm,j}=0$ if $j\neq 0$, and $a_{\bl,\bm,0}=c_{\bl,\bm}$.
\end{theorem}
\begin{proof}
By inspection of the diagrams in the previous section,
we see that if $\bm$ is special, then $\epsilon(\bm)-\epsilon(\bl)=2k(\bm)$.
Moreover, for all positive $k'\leq k(\bm)$, there is a unique
special $\bn\in[\bm,\bl]$ such that $k(\bn)=k'$. So
Corollary \ref{flagcor} can be rephrased:
\begin{equation*}
g_{\bl,\bm}(t)
= \sum_{\bl\geq\bn\geq\bm}c_{\bl,\bn}\,t^{(\epsilon(\bn)-\epsilon(\bl))/2}.
\end{equation*}
Now we apply Lemma \ref{uniqlemma} (bearing in mind Remark \ref{intervalrem})
with $b_{\bl,\bm,j}=\delta_{j,0}c_{\bl,\bm}$ and $L_{\bl,\bm}(t)=1$. 
The result follows.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}

math0111184.extracted_bib
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\end{thebibliography}