%Paper: hep-th/9408121
%From: Toshiya Kawai <kawai@theory.kek.jp>
%Date: Tue, 23 Aug 1994 14:50:32 +0900
%Date (revised): Wed, 31 Aug 1994 17:13:17 +0900

%
%
%    LaTeX file
%
%
%     The last three pages (p.28 - p.30) must be printed out in landscape mode.
%
%    For instance, try
%
%   dvi2ps -t 27 DVIFILE | lpr ; dvi2ps -f 28 -o landscape DVIFILE | lpr
%
%
\documentstyle[12pt]{article}
\setlength{\textwidth}{16.2cm}
\setlength{\textheight}{21cm}
\addtolength{\oddsidemargin}{-15mm}
\addtolength{\topmargin}{-16mm}
\makeatletter
%\input portland.sty

% This is PORTLAND.STY by H.Partl, TU Wien, as of 16 Dec 1988.
% Definition of commands and environments \portrait and \landscape
% for switching between PORTRAIT and LANDSCAPE printing. To be
% called as document style option with any available style.
%
% Commands:
% \portrait   sets or resets the page layout to the initial values
%             (i.e. the values valid at \begin{document})
% \landscape  sets the page layout such that the horizontal and
%             vertical measures are interchanged with respect to
%             the initial values. The text area will occupy the
%             same position on the page as in \portrait mode.
% Both command issue a \clearpage before changing the layout.
%
% Environments:
% \begin{portrait}  ...  \end{portrait}
% \begin{landscape} ...  \end{landscape}
% Both issue \clearpage at the \begin and at the \end.
%
% Implementation Dependencies:
% You should insert the correct paper height (11in or 197mm or ...)
% into \paperheight for correct positioning.
% If your printer driver allows mixing of protrait and landscape pages
% within one printout, you should insert the corresponding \special
% commands just after the four \clearpage commands.
% If your printer only allows printing of the whole document either in
% protrait or in landscape orientation, you should print the document
% twice (or selected parts of it) - once in portrait and once in landscape -
% and then merge the correct pages.
% If your printer can't print in landscape orientation, then these commands
% can be used only for small pages that fit onto the paper in both direc-
% tions.

\newdimen \paperheight  % needed for calculation of bottom margin.
\paperheight 297mm      % <--- 11in for US paper or 297mm for A4 paper!

\newdimen\phoffset
\newdimen\pvoffset
\newdimen\ptextwidth
\newdimen\ptextheight
\newdimen\ptopmargin
\newdimen\poddsidemargin
\newdimen\pevensidemargin

\newdimen\lhoffset
\newdimen\lvoffset
\newdimen\ltextwidth
\newdimen\ltextheight
\newdimen\ltopmargin
\newdimen\loddsidemargin
\newdimen\levensidemargin

\def\set@portland@values{%
%   portrait values = current values
\phoffset\hoffset
\pvoffset\voffset
\ptextwidth\textwidth
\ptextheight\textheight
\ptopmargin\topmargin
\poddsidemargin\oddsidemargin
\pevensidemargin\evensidemargin
%   landscape margins will be measured from zero point
\lhoffset \z@
\lvoffset \z@
%   landscape textwidth = portrait textheight
\ltextwidth \ptextheight
%   landscape textheight = portrait textwidth
\ltextheight \ptextwidth
%   landscape left margin = portrait bottom margin (same for odd and even)
\loddsidemargin \paperheight
\advance \loddsidemargin by -\ptextheight
\advance \loddsidemargin by -\headsep
\advance \loddsidemargin by -\headheight
\advance \loddsidemargin by -\ptopmargin
\advance \loddsidemargin by -\pvoffset
\advance \loddsidemargin by -2in
\levensidemargin \loddsidemargin
%   landscape top margin (incl. head) = portrait oddside left margin
\ltopmargin \poddsidemargin
\advance \ltopmargin by -\headheight
\advance \ltopmargin by -\headsep
\advance \ltopmargin by \phoffset
}

\def\portrait{\clearpage \message{ \string\portrait }%
    \hoffset\phoffset
    \voffset\pvoffset
    \textwidth\ptextwidth
    \textheight\ptextheight
         \@colht\textheight  \@colroom\textheight \vsize\textheight
         \columnwidth\textwidth \@clubpenalty\clubpenalty
         \if@twocolumn \advance\columnwidth -\columnsep
         \divide\columnwidth\tw@ \hsize\columnwidth \@firstcolumntrue
         \fi
         \hsize\columnwidth \linewidth\hsize
    \topmargin\ptopmargin
    \oddsidemargin\poddsidemargin
    \evensidemargin\pevensidemargin
    }

\def\endportrait{\clearpage \message{ \string\endportrait }}

\def\landscape{\clearpage \message{ \string\landscape }%
    \hoffset\lhoffset
    \voffset\lvoffset
    \textwidth\ltextwidth
    \textheight\ltextheight
         \@colht\textheight  \@colroom\textheight \vsize\textheight
         \columnwidth\textwidth \@clubpenalty\clubpenalty
         \if@twocolumn \advance\columnwidth -\columnsep
         \divide\columnwidth\tw@ \hsize\columnwidth \@firstcolumntrue
         \fi
         \hsize\columnwidth \linewidth\hsize
    \topmargin\ltopmargin
    \oddsidemargin\loddsidemargin
    \evensidemargin\levensidemargin
    }

\def\endlandscape{\clearpage \message{ \string\endlandscape }}

\let\set@document@values \document
\def\document{\set@document@values \set@portland@values}

%\endinput



%\input supertab.sty


% @stylefile{Super tabular
% shortpackagename = {supertab},
% longpackagename  = {supertabular},
% baseformats      = {\LaTeX 2.09},
% version          = {3.6a},
% date             = {1991-02-15},
% author           = {Theo Jurriens,
%                     TAJ@HGRRUG5,
%                     P.O Box 800,
%                     9700 AV Groningen},
% abstract         = {This file provides the supertabular environment, which
%                     is an extension to the standard tabular environment.
%                     Large tabulars are automatically split across pages.
%                     Seperate commands for the table-head and table-tail
%                     are provided, in order to repeat these on each page.}
% infauthor        = {Johannes Braams,
%                     JL_Braams@pttrnl.nl,
%                     PTT Research,
%                     P.O. Box 421,
%                     2260 AK Leidschendam}
% infdate           = {1990-10-16}}
%
%----------------------------------------------------------------------------
%
% supertabular sty
% original idea:  Theo Jurriens 1988
%                 TAJ@hgrrug5        P.O Box 800, 9700 AV Groningen
%
% revised by:     Johannes Braams
%                 JL_Braams@pttrnl.nl   PTT Research Leidschendam (NL)
%
% 15.02.91  - Because of the check for the use of tablefirsthead the
% V 3.6a      combination of an \hline in the head and an \hline as the first
%             thing in the data went wrong. The \futurelet in the definition
%             of \hline found \fi instead of \hline, so no \doublerulesep
%             was added.
%             Also had to modify the way the environments were defined.
%             The blank space (from the CR after the argument of \supertabular)
%             has to be gobbled. This can only be done using a construction
%             like \def\command#1 {...}. So removed the use of \newenvironment
% 04.02.91  - Added the commands \topcaption, \bottomcaption and \tablecaption
% V 3.6       to include a caption within the supertabular environment. The
%             default behaviour is to put the caption before the actual start
%             of the table.
%           - Also added \tablefirsthead and \tablelasttail to let the
%             user specify a different head for the first page of the table
%             and for consecutive pages as well as different tails for first
%             pages and the last one. If these commands are not used, the
%             default behaviour will be to use the value of \tablehead end
%             \tabletail
%           - Removed the need for the \noalign{\global\let\\=\@stabularcr}
%             commands by storing and resetting \@stabularcr
%
% 16.10.90  Added the supertabular* environment that was in an earlier
% V 3.5     version (2.0) by the original author
%           Reintroduced the version numbering
%
% revised by:     Gabriele Kruljac
%                 kruljac@ds0mpi11   Max-Planck-Institute Stuttgart
%
%
% 06.06.89  Correction: now care is taken of probably existing onecolumn
%           head (title or tables ...) in twocolumn sty.
%
% 10.05.89  Correction: the new \\ definition has been added to the
%           begin of each `sub'-tabular
%           Added: algorithm to produce the tabulars in twocolumn style
%
% 06.04.89  Correction: put \global statement in \end{supertabular}
%           into \noalign
%
% 22.02.89  Correction: restore the original meaning of \\ with
%           \end{supertabular}
%
% (Feb 89) The whole algorithm has been changed, so that I can use
%          the most features of a normal tabular:
%          \\  for new line, including  \\[#1]
%          p{...} in the preamble ...
%          Account is taken to \baselinestretch and \arraystretch
%         -I'm not counting the lines because of too much rounding errors
%          but instead I add the (estimated) used space in pt.
%         -The tablehead is taken into this algorithm of proofing, so
%          I really know how much space the head uses.
%         -When no p-arg is given I add a variable \midlineheight to
%          calculate the used space. To calculate \midlineheight I
%          take the \baselineskip, which is active when the supertabular
%          starts (\baselineskip includes the \value of \baselinestretch)
%          and multiply it with \arraystretch.
%         -When a p-arg is given the text will be stored in a box. So
%          I know the height I have to add. Also I reduce the maximum
%          pagesize, so that the last parbox on a page can get up to
%          max 4 lines without producing an overfull vbox.
%         -To do so I had to make some additions to LaTeX's tabular
%          commands. These new commands got a leading `s'.
%
%          Weak points:
%          -When the material of a normal entry (not a p-arg) becomes
%           larger than the estimated \midlineheight, overfull vboxes
%           will be produced at all.
%          -When the last p-arg on a page gets more than 4 lines
%           (probably even more than 3 lines) it will result in an
%           overfull vbox.
%           Also some combinations of \baselinestretch \arraystretch and
%           a large font may lead to one line too much.
%          -if accidentally the last line of the tabular produces
%           a newpage, on the next page the tabletail will be written
%           immediately after the tablehead. Depending on the contents
%           this may result in an error message regarding misplaced
%           \noalign.
%
%           A quick but not very elegant solution: shrink \maxsize by
%           \noalign{\global\maxsize=...pt} after the first \\ of the
%           supertabular.
%
%------------------------------------------------------------------------------
% Added the user-commands \topcaption and \bottomcaption which set the boolean
% @topcaption to determine where to put the tablecaption. The default
% is to put the caption on the top of the table
%
\newif\if@topcaption \@topcaptiontrue
\def\topcaption{\@topcaptiontrue\tablecaption}
\def\bottomcaption{\@topcaptionfalse\tablecaption}
%
% Added the command \tablecaption, with the use of the definition
% of the \caption and \@caption commands from latex.tex.
% This command has to function exactly like \caption does except it
% has to store its argument (and the optional argument) for later
% processing WITHIN the supertabular environment
% JB
%
\long\def\tablecaption{\refstepcounter{table} \@dblarg{\@xtablecaption}}
\long\def\@xtablecaption[#1]#2{%
  \long\def\@process@tablecaption{\@stcaption{table}[#1]{#2}}}
\let\@process@tablecaption\relax

%
% This is a redefinition of LaTeX's \@caption, \@makecaption is
% called within a group so as not to return to \normalsize globally.
% also a fix is made for the `feature' of the \@makecaption of article.sty and
% friends that a caption ALWAYS gets a \vskip 10pt at the top and NONE at the
% bottom. If a user wants to precede his table with a caption this results
% in a collision.
%
\long\def\@stcaption#1[#2]#3{\par%
    \addcontentsline{\csname ext@#1\endcsname}{#1}%
        {\protect\numberline{\csname the#1\endcsname}{\ignorespaces #2}}
  \begingroup
    \@parboxrestore
    \normalsize
    \if@topcaption \vskip -10pt \fi % 'fix'
    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
    \if@topcaption \vskip 10pt \fi % 'fix'
  \endgroup}

                             % \tablehead activates the new tabular \cr
                             % commands
\def\@tablehead{}
\def\tablehead#1{\gdef\@tablehead{#1}}
\def\tablefirsthead#1{\gdef\@table@first@head{#1}}

%+
%     If the user uses an extra amount of tabular-data (like \multicolumn)
%     in \verb=\tabletail= \TeX\ starts looping because of the definition
%     of \verb=\nextline=. So make \verb=\\= act like just a \verb=\cr=
%     inside this tail to prevent the loop.
%     Save and restore the value of \verb=\\=
%-
\def\@tabletail{}
\def\tabletail#1{%
    \gdef\@tabletail{\noalign{\global\let\@savcr=\\\global\let\\=\cr}%
                     #1\noalign{\global\let\\=\@savcr}}}
\def\tablelasttail#1{\gdef\@table@last@tail{#1}}

\newdimen\maxsize            % maximum pagesize
\newdimen\actsize            % actual pagesize
\newdimen\twocolsize         % needed for correct max height if twocolumn
\newdimen\parboxheight       % height plus depth of a parbox-argument
\newdimen\addspace           % stores the value of \\[#1]
\newdimen\midlineheight      % estimated size of a normal line
\newdimen\pargcorrection     % to set page height tolerance if p-arg
\newdimen\computedimens      % computation variable
\newbox\tabparbox

         %%%%  Redefine original LaTeX tabular \cr commands. %%%%
         %%%%  New tabular \cr commands get a leading `s'    %%%%

                             % Insert  \nextline command for counting
\def\@stabularcr{{\ifnum0=`}\fi\@ifstar{\@sxtabularcr}{\@sxtabularcr}}
\def\@sxtabularcr{\@ifnextchar[{\@sargtabularcr}%
                 {\ifnum0=`{\fi}\cr\nextline}}

                             % contents of command unchanged
\def\@sargtabularcr[#1]{\ifnum0=`{\fi}\ifdim #1>\z@
    \unskip\@sxargarraycr{#1}\else \@syargarraycr{#1}\fi}

                             % here copy the value #1 of [ ] of \\
                             % to \addspace
\def\@sxargarraycr#1{\@tempdima #1\advance\@tempdima \dp \@arstrutbox%
    \vrule \@height\z@ \@depth\@tempdima \@width\z@ \cr%
    \noalign{\global\addspace=#1}\nextline}

                             % command will be called when \\[0pt]
\def\@syargarraycr#1{\cr\noalign{\vskip #1\global\addspace=#1}\nextline}

         %%%%  Redefine original LaTeX p-arg commands.       %%%%
         %%%%  New commands get a leading `s'                %%%%

                                      % reduce maximum pagesize to have
                                      % a small tolerance for last entry
\def\@sstartpbox#1{\global\advance\maxsize by -\pargcorrection
                   \global\pargcorrection=0pt
                                      % put text into box to save height
                   \setbox\tabparbox%
                          \vtop\bgroup\hsize#1\@arrayparboxrestore}

\def\@sendpbox{\par\vskip\dp\@arstrutbox\egroup%
               \computedimens=\ht\tabparbox%
               \advance\computedimens by \dp\tabparbox%
               \ifnum\parboxheight<\computedimens
                  \global\parboxheight=\computedimens
               \fi
               \computedimens=0pt
               \box\tabparbox\hfil}

         %%%%  Here start really new supertabular commands   %%%%

                                 % estimate height of normal line
                                 % regarding \array- and \baselinestretch
\def\calmidlineheight{\midlineheight=\arraystretch \baslineskp
                      \global\advance\midlineheight by 1pt
                      \global\pargcorrection=4\midlineheight}

\def\calpage{\global\actsize=\pagetotal  % where am I on the actual page?
             \twocolsize=\textheight            %  added 06.06.89
             \advance\twocolsize by -\@colroom  %        "
             \advance\actsize by \twocolsize    %        "
             \global\advance\actsize by \midlineheight
             \maxsize=\textheight        % start a new page when 90% of
             \multiply \maxsize by 9     % the page are used
             \divide\maxsize by 10
             \ifnum\actsize > \maxsize
                   \clearpage
                   \global\actsize=\pagetotal
             \fi
             \maxsize=\textheight       % now set \maxsize with tolerance
             \global\advance\maxsize by -\midlineheight}   % of one lines
                             % Here is the definition of supertabular
% modified JB (15.2.91)
\def\supertabular#1 {%           % before it was \edef\tableformat,
                                 % but gave error with @{\hspace{..}} !
    \def\tableformat{\string#1} % store preamble
    \global\starfalse % remember this is the normal version

                                 % Check if we have to insert a caption
    \if@topcaption\@process@tablecaption
    \fi

    \def\baslineskp{\baselineskip}
    \calmidlineheight% estimate height of a normal line
    \calpage         % calculate max. pagesize and startpoint

                                 % save old \@tabularcr
    \let\@@tabularcr\@tabularcr%             Added JB 4/2/91
                                % Now insert the definition of \@stabularcr
    \let\@tabularcr\@stabularcr
                                 % save old \\
    \global\let\@oldcr=\\

                                 % activate new parbox algorithm
    \let\@@startpbox=\@sstartpbox
    \let\@@endpbox=\@sendpbox
%
%    Moved the check for the use of \tablefirsthead to befor the start of
%    the tabular environment in order to make the \futurelet inside \hline
%    do its work correctly (15.02.91)
%
    \ifx\@table@first@head\undefined
        \let\@@tablehead=\@tablehead
    \else
        \let\@@tablehead=\@table@first@head
    \fi%                                     Added JB 4/2/91
                                 % start normal tabular environment
    \begin{tabular}{\tableformat}%
    \@@tablehead}%   Added JB 15/2/91

                                 % this is \end{supertabular}
\def\endsupertabular{%
    \ifx\@table@last@tail\undefined%
        \@tabletail%
    \else%
        \@table@last@tail%
    \fi%                                     Added JB 4/2/91
%removed JB                \noalign{\global\let\\=\@oldcr}%
%   \let\@@startpbox=\@startpbox%
%   \let\@@endpbox=\@endpbox%
    \end{tabular}
                                 % restore old \@tabularcr
    \let\@tabularcr\@@tabularcr             % Added JB 4/2/91
                                 % Check if we have to insert a caption
    \if@topcaption
    \else
        \@process@tablecaption
                                 % resore to default behaviour
        \@topcaptiontrue
    \fi
%+
%    Restore the meaning of \verb=\\= to the one it had before the start
%    of this environment. Also re-initialize some control-sequences
%-
    \global\let\\=\@oldcr
    \let\@table@first@head\undefined        % For the next ocurrence
    \let\@table@last@tail\undefined         % of this environment
    \let\@process@tablecaption\relax
}

                             % Here is the definition of supertabular*
\newif\ifstar
\newdimen\tabularwidth
\@namedef{supertabular*}#1#2 {% modified JB (15.2.91)
                                 % before it was \edef\tableformat,
                                 % but gave error with @{\hspace{..}} !
    \def\tableformat{\string#2} % store preamble
    \tabularwidth=#1 % The total width of the tabular
    \global\startrue % remember this is the *-version

                                 % Check if we have to insert a caption
    \if@topcaption\@process@tablecaption\fi

    \def\baslineskp{\baselineskip}
    \calmidlineheight% estimate height of a normal line
    \calpage         % calculate max. pagesize and startpoint


                                 % save old \@tabularcr
    \let\@@tabularcr\@tabularcr%              Added JB 4/2/91
                                % Now insert the definition of \@stabularcr
    \let\@tabularcr\@stabularcr%              Added JB 4/2/91
                                 % save old \\
    \global\let\@oldcr=\\

                                 % activate new parbox algorithm
    \let\@@startpbox=\@sstartpbox
    \let\@@endpbox=\@sendpbox
%
%    The same modification as for \tabular 15.2.91
    \ifx\@table@first@head\undefined
        \let\@@tablehead\@tablehead
    \else
        \let\@@tablehead\@table@first@head
    \fi%                                     Added JB 4/2/91
                                 % start normal tabular environment
    \begin{tabular*}{\tabularwidth}{\tableformat}%
%
%removed JB                \noalign{\global\let\\=\@stabularcr}
    \@@tablehead}%
                                 % this is \end{supertabular*}
\@namedef{endsupertabular*}{%
    \ifx\@table@last@tail\undefined%
        \@tabletail%
    \else%
        \@table@last@tail%
    \fi%                                     Added JB 4/2/91
%removed JB                \noalign{\global\let\\=\@oldcr}%
    \end{tabular*}
                                 % restore old \@tabularcr
    \let\@tabularcr\@@tabularcr
%               \let\@@startpbox=\@startpbox%
%               \let\@@endpbox=\@endpbox%
                                 % Check if we have to insert a caption
    \if@topcaption
    \else
        \@process@tablecaption
                                 % resore to default behaviour
        \@topcaptiontrue
    \fi
%+
%    Restore the meaning of \verb=\\= to the one it had before the start
%    of this environment. Also re-initialize some control-sequences
%-
    \global\let\\=\@oldcr
    \let\@table@first@head\undefined        % For the next ocurrence
    \let\@table@last@tail\undefined         % of this environment
    \let\@process@tablecaption\relax}

\def\nextline{%           %%% algorithm to calculate the pagebreaks %%%
    \noalign{\ifnum\parboxheight=0
                                 % if no p-arg add `normal' line height
                   \advance\actsize by \midlineheight
                                      % \addspace is value #1 of \\[#1]
                   \global\advance\actsize by \addspace
             \else
                                 % if p-arg add height of box and more
                   \global\advance\actsize by \parboxheight
                   \divide\parboxheight by 11\relax
                   \global\advance\actsize by \parboxheight%
                   \global\parboxheight=0pt
             \fi
             \global\addspace=0pt}%
                                      % when page becomes full:
    \ifnum\actsize<\maxsize
%
%    This line is necessary because the tablehead has to be inserted *after*
%    the \if\else\fi-clause. For this purpose \next is used. In the middle
%    of tableprocessing it shoud be an *empty* macro (*not* \relax).
%    (15.2.91)
    \noalign{\global\let\next\@empty}
    \else
                                      % output \tabletail, close tabular
                                      % output all material and start a
                                      % fresh new page
         \@tabletail
         \ifstar%                     % Added 16-10-90
           \end{tabular*}%
         \else%
           \end{tabular}%
         \fi
         \if@twocolumn%                        % added 10.05.89
            \if@firstcolumn%                   %
               \newpage%                       %
            \else%                             %
               \clearpage%                     %
            \fi%                               %
            \twocolsize=\textheight%           % added 06.06.89
            \advance\twocolsize by -\@colroom% %       "
            \global\actsize=\twocolsize%       %       "
         \else                                 %
            \clearpage                         %
            \global\actsize=\midlineheight%
         \fi                                   %
         \let\next\@tablehead%                 % Added 15.2.91
         \ifstar%                              % Added 16-10-90
           \begin{tabular*}{\tabularwidth}{\tableformat}%
         \else%
           \begin{tabular}{\tableformat}%
         \fi%
%removed JB         \noalign{\global\let\\=\@stabularcr}  % added 10.05.89
%removed JB         \@tablehead%
    \fi\next}%                                % Added \next 15.2.91


%\input cite.sty

%%% Save file as: CITE.STY               Source: FILESERV@SHSU.BITNET
%     C I T E . S T Y
%
%     version 3.0  (Oct 1992)
%
%     Compressed, sorted lists of numerical citations: [11-16]
%     see also OVERCITE.STY and DRFTCITE.STY
%
%     Copyright (C) 1989-1992 by Donald Arseneau
%     These macros may be freely transmitted, reproduced, or modified for
%     non-commercial purposes provided that this notice is left intact.
%
%     Instructions follow \endinput.
%  ------------------------------------
%
%  \@citen contains the code that parses the list of names, ignoring
%  spaces after commas, writes the aux file \citation, and formats the
%  number list.  \citen can be used by itself to give citation numbers
%  without the other formatting; e.g., "See also ref.~\citen{junk}."
%
\def\citen{\protect\p@citen}

\def\p@citen#1{%
\edef\@tempa{\@ignspaftercomma,#1, \@end, }% ignore spaces in parameter list
\edef\@tempa{\expandafter\@ignendcommas\@tempa\@end}%
\if@filesw \immediate \write \@auxout {\string \citation {\@tempa}}\fi
\@tempcntb\m@ne    % \@tempcntb tracks higest number
\let\@h@ld\relax   % nothing held from list yet
\let\@citea\@empty % no punctuation preceding first
\let\@celt\over    % not expandable, but identifiable
\def\@cite@list{}% % empty list to start
\@for \@citeb:=\@tempa \do{\@make@cite@list}% make a sorted list of numbers
% After sorted citelist is made, execute it to compress citation ranges.
\@tempcnta\m@ne    % no previous number
\let\@celt\@compress@cite \@cite@list % output number list with compression
\@h@ld}% output anything held over

% For each citation, check if it is defined and if it is a number.
% if a number: insert it in the sorted \@cite@list
% otherwise: output it immediately.
%
\begingroup \catcode`\*=7 % funny catcode for comparisons
\gdef\@make@cite@list{%
 \expandafter\let \expandafter\@B@citeB \csname b@\@citeb \endcsname
 \ifx\@B@citeB\relax % undefined: output ? and warning
    \@citea {\bf ?}\let\@citea\citepunct
    \@warning {Citation `\@citeb' on page \thepage\space undefined}%
 \else %  defined
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\begin{document}
\addtolength{\baselineskip}{.3mm}
\thispagestyle{empty}
\vspace{-1.5cm}

\begin{flushright}
 KEK-TH-409\\
%{KEK preprint} ???\\
August  1994
\end{flushright}
\vspace{5mm}

\begin{center}
  {\large Duality of Orbifoldized Elliptic Genera}
  \footnote{{\noindent To appear in the proceedings of the workshop,
      {\em Quantum Field Theory, Integrable Models and Beyond}, Yukawa
      Institute for Theoretical Physics, Kyoto University, 14-18
      February 1994.  } }\\[13mm]
  {\sc Toshiya Kawai}\\[3mm]
  {\it
    National Laboratory for High Energy Physics (KEK),\\[2mm] Tsukuba,
    Ibaraki 305, Japan} \\[7mm]
  {\sc Sung-Kil Yang}\footnote
  { Supported in part by Grant-in-Aid for Scientific Research on
    Priority Area 231 ``Infinite Analysis'', Japan Ministry of
    Education.}\\[2mm]
  {\it Institute of Physics, University of
    Tsukuba, \\[2mm] Ibaraki 305, Japan}\\[18mm]
\end{center}
\vspace{.5cm}
\begin{center}
{\sc Abstract}
\end{center}
\vspace{.5cm}
\noindent We discuss duality and mirror symmetry phenomena of
Landau-Ginzburg orbifolds considering their elliptic genera.  Under
the duality (or mirror) transform performed by orbifoldizing the
Landau-Ginzburg model via some discrete group of the superpotential we
observe that the roles of the untwisted and twisted sectors are
exchanged. As explicit evidence detailed orbifold data are presented
for $N=2$ minimal models, Arnold's exceptional singularities, $K3$
surfaces constructed from Arnold's singularities and Fermat
hypersurfaces.

%\bigskip
%{\noindent  To appear in the proceedings of the workshop,
%{\em Quantum field theory, integrable models and beyond},
%Yukawa Institute for Theoretical Physics, Kyoto University,
%14-18 February 1994.  }

\bigskip
\begin{flushleft}
{\tt hep-th/9408121}
\end{flushleft}


\newpage



\section{Introduction}



Landau-Ginzburg field theory approach to two-dimensional critical
phenomena uncovers qualitative physical properties behind the exact
algebraic description in terms of conformal field theories
\cite{rZamolodchikov,rLC,rKMS}.  When $N=2$ supersymmetry is
considered even quantitative results can be deduced in the framework
of the Landau-Ginzburg models \cite{rMartinec,rVW,rLVW}. The
non-renormalization theorem for the superpotential of $N=2$ models is
believed to be responsible for this miracle.  $N=2$ Landau-Ginzburg
descriptions have also proved to be efficient in constructing
superstring vacua through orbifoldizing the Landau-Ginzburg models
\cite{rVafa,rIV}. This fact is somewhat mysterious since the
Landau-Ginzburg models do not a priori possess the target space
interpretation while more conventional sigma models have.  There have
been some arguments attempting to clarify the connection between the
Landau-Ginzburg models and the sigma models with Calabi-Yau target
spaces \cite{rGepneri,rGVW,rCecotti}.

Recently a novel scheme has been proposed to understand the
Landau-Ginzburg/Calabi-Yau correspondence \cite{rWitteniii}. One
considers a $U(1)$ gauged Landau-Ginzburg model with the
Fayet-Iliopoulos $D$-term and the theta term.  The model contains
several chiral superfields, one of which, say $P$, plays the role of
an order parameter. The coefficient $r$ of the Fayet-Iliopoulos term
combined with that of the theta term, $\theta$ turns out to be a
complex variable $t=\sqrt{-1}r+\frac{\theta}{2\pi}$ parametrizing the
complexified K\"ahler cone. By tuning $t$ one finds two extremum
regimes. One regime $(r \ll 0)$ represents the Landau-Ginzburg phase
where $p$, the bosonic component of $P$, acquires the vacuum
expectation value $\langle p \rangle \not= 0$. The $U(1)$ symmetry
then breaks down to some discrete group. This discrete group is
employed to orbifoldize the Landau-Ginzburg model.  In the other
regime $(r\gg 0)$ we have $\langle p \rangle = 0$ and the bosonic
components of the chiral superfields other than $P$ are constrained to
take values on a hypersurface in a weighted projective space. Hence
this is the regime of the sigma model. Furthermore it is argued that
one can make an analytic continuation from the Landau-Ginzburg to
Calabi-Yau regimes and vice versa on the complex $t$-plane. This
picture was also confirmed from another independent point of view
\cite{rAGM}\ and has been extended to the $(0,2)$ case
\cite{rWitteniii,rDK}.


One more evidence of the above scheme can be obtained by considering
the elliptic genus \cite{rSW,rWittenii,rAKMW}, {\it i.e.\/} the index
of the right-moving supercharge.  It is expected that the elliptic
genus is independent of the parameter $t$ because of its topological
nature and thus should coincide in both the Calabi-Yau and
Landau-Ginzburg orbifold phases.  Calculations of the elliptic genera
of the Landau-Ginzburg models were initiated in \cite{rWitteni}\ where
the A-type $N=2$ minimal model is taken to explain the essential idea.
This was soon followed by several groups who have extended the
original idea in order to incorporate various $N=2$ models
\cite{rWitteni, rDY, rKYY,rDAY,rHen,rAYS,rBH, rKM,rNW,rK}. In
ref.\cite{rKYY} it was confirmed that the elliptic genus of an
appropriate Landau-Ginzburg orbifold takes the same form as that of
the sigma model with a Calabi-Yau target manifold thus with a good
agreement with the above expectation.






The orbifoldized elliptic genus is also an interesting arena to
consider the mirror symmetry of Calabi-Yau manifolds \cite{rMirror}\
and similar phenomena. In fact the investigation of mirror symmetry
via the elliptic genus has already been taken up in \cite{rBH}.  A
well-known procedure to construct mirror pairs is to orbifoldize the
Landau-Ginzburg model via various symmetry groups of the
superpotential \cite{rGP}.  To compute the elliptic genera of the
resulting Landau-Ginzburg orbifolds necessitates a slightly more
involved formula than for the most frequently studied case. Thus after
introducing {\it generic\/} Landau-Ginzburg orbifolds in sect.2 we
briefly summarize the formulas for their elliptic genera in sect.3.

In sect.4, which is the main part of this contribution, we study
mirror phenomena and their cousins by employing these formulas.  By
picking up typical examples we present detailed data which have
accumulated during our series of analyses of the elliptic genus.
Although these data may be a sort of objects usually to be suppressed
in the literature it is quite impressive to experience duality or
mirror phenomena through explicit data. We thus think it worth
publishing these data in a comprehensible manner. We also believe that
our presentation is in accordance with the editorial spirit of this
proceedings.






\section{$N=2$ Landau-Ginzburg model and its orbifolds}

We consider the Landau-Ginzburg model whose Lagrangian density is
given by
\begin{equation}
  \int d^2\theta d^2\bar\theta\,\sum_{i=1}^NX_i\bar X_i+ \int
  d^2\theta\, W(X_i)+ \int d^2\bar\theta\, \bar W(\bar X_i)\,,
\end{equation}
where the superpotential $W$ is a weighted homogeneous polynomial of
$N$ chiral superfields $X_1, \cdots, X_N$ with weights
$\omega_1, \cdots, \omega_N$,
%
\begin{equation}
  \lambda W(X_1,\cdots,X_N)= W(\lambda^{\omega_1}
  X_1,\cdots,\lambda^{\omega_N} X_N)\,.
\end{equation}
%
We assume that $W$ has an isolated critical point at the origin and
$\omega_i$'s are strictly positive rational numbers such that
$\omega_1,\ldots,\omega_N\le\frac{1}{2}$.  The infrared fixed point
theory is believed to be described by an $N=2$ superconformal field
theory with
\begin{equation}
  \hat c = \sum_{i=1}^N(1-2\omega_i)\,.
\label{centralcharge}
\end{equation}

In general $W$ is invariant under some discrete group $G\subset
GL(N,{\bf C})$ acting on ${}^t(X_1,\ldots,X_N)$ and one can consider
the orbifold theory with respect to $G$.  The resulting theory is
called the Landau-Ginzburg orbifold and will be denoted symbolically
as $W/\!/G$ in the following. We shall restrict ourselves to the case
where $G$ is abelian and their elements take the form
$diag(\bfe{\alpha_1\omega_1},\ldots, \bfe{\alpha_N\omega_N})$ where
$\alpha_i\in {\bf Z}$ and $\bfe{*}=\exp(2\pi\sqrt{-1}*)$.  One
distinguished example of such discrete groups that always exists for
any $W$ is the one generated by
$diag(\bfe{\omega_1},\ldots,\bfe{\omega_N})$.  We shall call this
group as the {\it principal discrete group\/} and denote it by $G_0$. The
Landau-Ginzburg orbifold $W/\!/G_0$ is a fundamental and the most
frequently studied case.




In a favorable situation ({\it i.e.\/} $\hat c \in {\bf Z}$ and $G
\supseteq G_0$) the Landau-Ginzburg orbifold $W/\!/G$ can be
interpreted as an `analytic continuation' of some $N=2$ sigma model
with its target space  smoothed.









\section{Elliptic genus}

Topological properties of a supersymmetric theory in two space-time
dimensions can be succinctly summarized by the elliptic genus
\cite{rSW,rWittenii,rAKMW}.  This quantity was recently refined so as
to incorporate $N=2$ theories and up until now various examples have
been computed \cite{rWitteni, rDY, rKYY,rDAY,rHen,rAYS,rBH,
  rKM,rNW,rK}.



The definition of the $N=2$ elliptic genus is
\cite{rWitteni}\footnote{ Throughout this paper we shall consider
  $(2,2)$ theories only.}
\begin{equation}
  Z(\tau,z)=\mathop{\rm Tr}(-1)^F y^{{(J^{\rm L})}_0}q^{{\cal H}^{\rm
      L}}{\bar q}^{{\cal H}^{\rm R}},\quad y=\bfe{z},\
  q=\bfe{\tau}\quad (\mathop{\rm Im} \tau>0)\,,
  \label{defofegenus}
\end{equation}
where ${(J^{\rm L,R})}_0$ are the left, right $U(1)$ charge operators
and ${\cal H}^{\rm L,R}$ are the left, right Hamiltonians.  We have set
$(-1)^F=\exp[-\pi\sqrt{-1}\{(J^{\rm L})_0-(J^{\rm R})_0\}]$.  As
usual, due to the right supersymmetry $Z(\tau,z)$ is $\bar q$
independent.


The basic properties of the elliptic genus \cite{rKYY}\ are the
modular invariance up to a prefactor
%
\begin{equation}
  Z \left( {a\tau +b \over c\tau +d}, {z \over c\tau +d} \right)=
  \bfe{ \frac{\hat c}{2}\frac{ c z^2}{ c\tau +d}} Z(\tau, z)\,,\quad
  \pmatrix{ a&b\cr c&d\cr} \in SL(2,{\bf Z})\,,
  \label{modular}
\end{equation}
%
and the double quasi-periodicity
%
\begin{equation}
  Z(\tau, z+\lambda \tau +\mu)=(-1)^{\hat c (\lambda +\mu)}
  \bfe{-\frac{\hat c}{2} (\lambda^2 \tau +2\lambda z) } Z(\tau,
  z)\,,\quad \lambda, \mu \in h{\bf Z}\,,
  \label{period}
\end{equation}
%
where $h$ is the least positive integer such that the $U(1)$ charge of
any chiral ring element multiplied by $h$  is an integer.
These two properties together with the `$\chi_y$-genus'
\cite{rHirzebruch}\  determined by
\begin{equation}
  \chi_y=y^{\frac{\hat c}{2}}\lim_{\tau\rightarrow i\infty} Z(\tau,0)\,,
\end{equation}
characterize the elliptic genus.


The elliptic genus of the Landau-Ginzburg model can be computed as
%
\begin{equation}
  Z[W](\tau,z)=\prod_{i=1}^N {\vartheta_1(\tau,(1-\omega_i) z) \over
    \vartheta_1(\tau,\omega_i z)}\,,
  \label{LGgenus}
\end{equation}
%
where
\begin{equation}
  \begin{array}{rcl}
    &\displaystyle \vartheta_1(\tau,z)&= \sqrt{-1} \sum\limits_{n\in{\bf
        Z}}(-1)^n q^{\frac{1}{2}(n-\frac{1}{2})^2}
    y^{n-\frac{1}{2}}\\[2mm] &&\displaystyle = \sqrt{-1}q^{\frac{1}{8}}
    y^{-\frac{1}{2}} \prod_{n=1}^\infty(1-q^n)
    (1-q^{n-1}y)(1-q^ny^{-1})\,,
  \end{array}
\end{equation}
is one of the Jacobi theta functions. Using the theta function
formulae it is easy to check that $Z[W](\tau,z)$ obeys (\ref{modular})
and (\ref{period}) with $\hat c$ given by (\ref{centralcharge}) and
$h$ being the smallest positive integer such that $\omega_ih\in{\bf
  Z}$ for all $1\le i\le N$.



The elliptic genus of the Landau-Ginzburg orbifold $W/\!/G$ is given
by \cite{rKYY,rDAY,rBH}
\begin{equation}
  Z[W/\!/G](\tau,z)=\frac{1}{\vert G\vert}\sum_{{\boldsymbol
      \alpha},{\boldsymbol \beta}\in G} \epsilon({\boldsymbol
    \alpha},{\boldsymbol \beta})\sector{{\boldsymbol
      \beta}}{{\boldsymbol \alpha}}(\tau,z)\,,
\label{eqegenus}
\end{equation}
where
\begin{equation}
  \epsilon({\boldsymbol\alpha},{\boldsymbol \beta})=\prod_{i=1}^N
  (-1)^{\alpha_i+\beta_i+\alpha_i\beta_i}\,,
\label{eqsign}
\end{equation}
and
\begin{eqnarray}
  \sector{{\boldsymbol \beta}}{{\boldsymbol
      \alpha}}(\tau,z)&=&\prod_{i=1}^N
  \bfe{\frac{1-2\omega_i}{2}\alpha_i\beta_i}
  \bfe{\frac{1-2\omega_i}{2}(\alpha_i^2\tau+2\alpha_i
    z)}\nonumber\\[2mm] &&\hspace{3cm}\times
  \frac{\vartheta_1(\tau,(1-\omega_i)(z+\alpha_i\tau+\beta_i))}
  {\vartheta_1(\tau,\omega_i(z+\alpha_i\tau+\beta_i))}\,.
\end{eqnarray}
Here we have made  $\alpha=(\alpha_1,\ldots,\alpha_N)$ represent for the
element $diag(\bfe{\alpha_1\omega_1},\ldots,\bfe{\alpha_N\omega_N})$
of $G$.  Correspondingly the $\chi_y$-genus takes the form
\begin{equation}
  \chi_y[W/\!/G]=y^{\frac{\hat c}{2}}\frac{(-1)^N}{\vert
    G\vert}\sum_{{\boldsymbol \alpha},{\boldsymbol \beta}\in G}
  \prod_{\stackrel{\scriptstyle i}{\omega_i\alpha_i\not\in{\bf Z}}}
  y^{-(\!(\omega_i\alpha_i)\!)} \prod_{\stackrel{\scriptstyle
      i}{\omega_i\alpha_i\in{\bf Z}}} \frac{\sin
    \pi\{(\omega_i-1)z+\omega_i\beta_i\}}
  {\sin\pi(\omega_iz+\omega_i\beta_i)}\,,
\label{chiygenus}
\end{equation}
where $(\!( *)\!)=*-[*]-\frac{1}{2}$.


The orbifoldized elliptic genus $Z[W/\!/G]$ obeys the same modular
transformation property and double quasi-periodicity as those of
$Z[W]$.  In addition, if $\hat c$ is an integer, the conditions for
the double quasi-periodicity $\lambda,\mu\in h{\bf Z}$ are relaxed to
$\lambda,\mu\in {\bf Z}$ and the orbifold theory has a chance to have
correspondence with an $N=2$ sigma model.

The Witten index $Z[W/\!/G](\tau,0)=\chi_{y=1}[W/\!/G]$ reads
\begin{equation}
  \frac{(-1)^N}{\vert G\vert}\sum_{{\boldsymbol \alpha},{\boldsymbol
      \beta}\in G} \prod_{\stackrel{\scriptstyle
      i}{\omega_i\alpha_i\in{\bf Z} \ {\rm and}\
    \omega_i\beta_i\in{\bf Z}}} \left(1-\frac{1}{\omega_i}\right)\,,
\end{equation}
and this reproduces the result of Roan \cite{rRoan}\ which is in turn
the extension of Vafa's formula \cite{rVafa}
\footnote{
Notice that in the formula of $Z[W/\!/G_0](\tau,z)$ given in
ref.\cite{rKYY} we took
$\epsilon(\alpha,\beta)=(-1)^{D(\alpha+\beta+\alpha\beta)}$ where $D$
is an integer such that $Dh\equiv \hat c h\pmod{2}$.  Eq. (\ref{eqsign})
 corresponds to the choice  $D=N$ which is possible
since $\hat c h=Nh-2\sum_i\omega_ih\equiv Nh\pmod{2}$.  If $\hat c$ is
an integer we can instead take $D=\hat c$ which leads to the original
Vafa's formula \cite{rVafa}\ of the Euler characteristic.}.







\section{Self-duality, strange duality and mirror symmetry}

In this section we restrict ourselves to Landau-Ginzburg orbifolds
$W/\!/G$ with either $G=\{id\}$ or $G \supseteq G_0$. We now wish to
present a variety of computational results for the following
phenomena:

{\em There are some cases in which the Landau-Ginzburg orbifold
  $W/\!/G$ has a partner $W^*/\!/G^*$ such that
\begin{itemize}
\item $W^*/\!/G^*$ has the same central charge $\hat c$ as $W/\!/G$.
\item
  \begin{equation}
    Z[W/\!/G](\tau,z)=\pm Z[W^*/\!/G^*](\tau,z)\,,
\label{dualeg}
\end{equation}
or equivalently
\begin{equation}
  \chi_y[W/\!/G]=\pm \chi_y[W^*/\!/G^*]\,.
\label{dualchiy}
\end{equation}
\item By going from $W/\!/G$ to $W^*/\!/G^*$ the roles of the
  untwisted sectors and twisted sectors are interchanged.
\end{itemize}}
Here we have to explain what we mean by untwisted and twisted sectors.
If $G=\{ id\}$ we have only untwisted sectors and no twisted sectors.
If $G \supseteq G_0$, by untwisted sectors we mean the ones with
respect to the subgroup $G_0$. Thus the number of the untwisted
sectors is equal to $\vert G \vert /\vert G_0 \vert$.


Conceptual understanding of these observations is still lacking and it
is certainly true that a mere consideration of elliptic genus does not
suffice and perhaps we have to view things from a broader perspective
(see sect.5 for discussions).  Nevertheless we hope that a relative
ease of computations and their explicitness make these results worth
presenting.



\subsection{Self-duality of  minimal models}

Our first example is the well-known self-duality of $N=2$ minimal
model.  The $N=2$ minimal model is in one to one correspondence with
the Landau-Ginzburg model with its superpotential given by one of
$ADE$ potentials  and its central charge is given by $\hat
c=1-\frac{2}{h}$ where $h$ is the Coxeter number of $ADE$.  If we take
$W^*=W$ and choose $G=\{id\}$ and $G^*=G_0$ then the above situation
is realized as we now see.  As mentioned the Landau-Ginzburg model
$W=W/\!/\{id\}$ has no twisted sectors and its $\chi_y$-genus is, as
well-known, given by
\begin{equation}
  \chi_y[W]=\sum_{i=1}^lt^{m_i-1}\,,
\end{equation}
where $t=y^{\frac{1}{h}}$ and $m_1,\ldots,m_l$ are the exponents of
$ADE$.  The $\chi_y$-genus of the Landau-Ginzburg orbifold
$W/\!/G_0$  can be decomposed  as
\begin{equation}
  \chi_y[W/\!/G_0]=\sum_{\alpha=0}^{h-1}\chi_y^\alpha[W/\!/G_0]\,,
\end{equation}
and each contribution  is  given as follows.
For $A_l$ we have
\begin{equation}
  \chi_y^\alpha[W/\!/G_0]=\left\{
\begin{array}{ll}
        0\,,&\alpha=0\\[1mm]
       -t^{l-\alpha}\,,& \alpha=1,\ldots,l\,.
     \end{array}
\right.
\end{equation}
For $D_l$ and
if $l$ is even, we have
\begin{equation}
 \chi_y^\alpha[W/\!/G_0]=\left\{
\begin{array}{ll}
        0\,,&\alpha=\mbox{even}\\[1mm]
       -t^{2l-3-\alpha}\,,& \alpha=\mbox{odd}, \not = l-1\\[1mm]
        -2t^{l-2}\,,& \alpha=l-1
     \end{array}
\right. \,,
\end{equation}
while if $l$ is odd
\begin{equation}
 \chi_y^\alpha[W/\!/G_0]=\left\{
\begin{array}{ll}
        0\,,&\alpha=\mbox{even}, \not = l-1\\[1mm]
       -t^{2l-3-\alpha}\,,& \alpha=\mbox{odd}\\[1mm]
        -t^{l-2}\,,& \alpha=l-1
     \end{array}
\right. \,.
\end{equation}
For the remaining cases we have
\bigskip

\begin{tabular}{ll}\hline
  &\vphantom{\bigg\vert}$\{-\chi_y^0[W/\!/G_0], -\chi_y^1[W/\!/G_0], \ldots,
  -\chi_y^{h-1}[W/\!/G_0]\}$\\ \hline\\[-2mm]
  $E_6$&$\{ 0,{t^{10}},0,0,{t^7},{t^6},0,{t^4},{t^3},0,0,1\}$ \\
  $E_7$&$\{ 0,{t^{16}},0,0,0,{t^{12}},0,{t^{10}},0,{t^8},0,%
{t^6},0,{t^4},0,0,0,1\}$
\\ $E_8$&$\{ 0,{t^{28}},0,0,0,0,0,{t^{22}},0,0,0,{t^{18}},0,%
{t^{16}},0,0,0,{t^{12}},0,{t^{10}},0,0,0,{t^6},0,0,0,0,0,1\}$\\[-2mm]
\\  \hline
\end{tabular}
\bigskip

To summarize we found that
\begin{equation}
  \chi_y^\alpha[W/\!/G_0]=-\mbox{mult}(\alpha)\, t^{h-1-\alpha}\,,
\end{equation}
where $\mbox{mult}(\alpha)$ is the multiplicity of
$\alpha$ appearing in the set of exponents $\{m_1,\ldots,m_l\}$.
Hence it follows that
\begin{equation}
  \chi_y[W]=-\chi_y[W/\!/G_0]\,,
\end{equation}
and the twisted and the untwisted sectors are interchanged (with minus signs)
between $W$ and $W/\!/G_0$.



\subsection{Arnold's strange duality in terms of  Landau-Ginzburg orbifolds}

Let $W$ be the potential corresponding to one of Arnold's 14
exceptional singularities and let $W^*$ denote its dual in the sense
of strange duality. (See Table 1.)  $W$ and $W^*$ share the same
Coxeter number $h$ and hence $\hat c=1+2/h$.  Take $G=\{ id\}$ and
$G^*=G_0^*$ where $G_0^*$ is the principal discrete group of $W^*$.
Comparing Table 1 and Table 2 we find that
\begin{equation}
  \chi_y[W]=-\chi_y[W^*/\!/G_0^*]\,,
\end{equation}
and the twisted and the untwisted sectors are interchanged (with minus
signs) between $W$ and $W^*/\!/G_0^*$.





\subsection{Landau-Ginzburg orbifolds corresponding to  $K3$ constructed from
  exceptional singularities}

Let $\tilde W(z_1,z_2,z_3)$ be the potential corresponding to one of
Arnold's 14 exceptional singularities and let $\tilde
W^*(z_1,z_2,z_3)$ denote its dual in the sense of strange duality.
Set $W=W(z_1,z_2,z_3,z_4)=\tilde W(z_1,z_2,z_3)+z_4^h$
and similarly for $W^*$.
 Then it is known that one
can construct the $K3$ surface as the resolution of
\begin{equation}
  \{ (z_1,\ldots,z_4)\in {\bf WP}_{\{d_1,d_2,d_3,1\}}^3 \mid
  W(z_1,z_2,z_3,z_4)=0\}\,.
\end{equation}
The Landau-Ginzburg orbifold $W/\!/G_0$ describes the analytic
continuation of the $N=2$ sigma model whose target space is the $K3$
surface.
The $\chi_y$-genus of the $K3$ surface is
\begin{equation}
\chi_y(K3)=2+20y+2y^2\,,
\end{equation}
and we find $\chi_y[W/\!/G_0]=\chi_y(K3)=\chi_y[W^*/\!/G_0^*]$.
 Let us denote the
$\alpha^{\rm th}$ twisted sector contribution to the
$\chi_y[W/\!/G_0]$ by $\chi_y^\alpha[W/\!/G_0]$.  Table 3 shows that
the contributions from the untwisted sector and those from the twisted
sectors are interchanged between $W/\!/G_0$  and  $W^*/\!/G_0^*$. To put
differently,
\begin{equation}
\chi^0_y[W/\!/G_0]+\chi_y^0[W^*/\!/G_0^*]=\chi_y(K3)\,.
\end{equation}
Thus we have seen that the partner of $W/\!/G_0$ is given by
$W^*/\!/G_0^*$.

We remark that subjects related to what has been presented in the
previous and present subsections were earlier discussed by Martinec
\cite{rMartinec}.

\subsection{Mirror symmetry}
Our last example is mirror symmetry considered by Greene and Plesser
\cite{rGP}.  We consider the family of superpotentials given by
\begin{equation}
  W=z_1^d+\cdots+z_d^d\,,\quad d=3,4,5
\end{equation}
and  take $W=W^*$.
We choose $G$ to satisfy
\begin{equation}
  {\bf Z}_d\simeq G_0\subseteq G \subseteq ({\bf Z}_d)^{d-1}.
\end{equation}
Apparently the  number of such $G$'s is $2^{d-2}$ and they are given by:
\bigskip
\smallskip

\begin{center}
  \begin{tabular}{|l|l|l|}\hline
    \multicolumn{3}{|c|}{$d=3$} \\ \hline
     $G$ & \hfil generators \hfil & \phc$\chi(\hat{\cal M}_G)$\\ \hline
    $G_0\simeq {\bf Z}_3$& $(1,1,1)$&$0$\\
     $G_1\simeq ({\bf Z}_3)^2$ & $(1,1,1), (0,1,2)$ &$0$\\ \hline
  \end{tabular}
\end{center}
\smallskip

\begin{center}
  \begin{tabular}{|l|l|l|}\hline
    \multicolumn{3}{|c|}{$d=4$} \\ \hline
     $G$ & \hfil generators \hfil& \phc$\chi(\hat{\cal M}_G)$\\ \hline
     $G_0\simeq{\bf Z}_4$& $(1,1,1,1)$&$24$\\
     $G_1\simeq({\bf Z}_4)^2$ & $(1,1,1,1),(0,0,1,3)$ &$24$\\
     $G_2\simeq({\bf Z}_4)^2$ & $(1,1,1,1),(0,1,1,2)$ &$24$\\
     $G_3\simeq({\bf Z}_4)^3$ & $(1,1,1,1),(0,0,1,3),(0,1,1,2)$ &$24$\\
     \hline
  \end{tabular}
\end{center}
\smallskip

\begin{center}
  \begin{tabular}{|l|l|l|}\hline
    \multicolumn{3}{|c|}{$d=5$} \\ \hline
     $G$& \hfil generators \hfil & \phc$\chi(\hat{\cal M}_G)$\\ \hline
     $G_0\simeq {\bf Z}_5$& $(1,1,1,1,1)$&$-200$\\
     $G_1\simeq ({\bf Z}_5)^2$ & $(1,1,1,1,1), (0,0,0,1,4)$ &$-88$\\
     $G_2\simeq ({\bf Z}_5)^2$ & $(1,1,1,1,1), (0,1,2,3,4)$ &$-40$\\
     $G_3\simeq ({\bf Z}_5)^2$ & $(1,1,1,1,1), (0,1,1,4,4)$ &$8$\\
     $G_4\simeq ({\bf Z}_5)^3$ & $(1,1,1,1,1), (0,1,1,4,4), (0,1,2,3,4)$
&$-8$\\
     $G_5\simeq ({\bf Z}_5)^3$ & $(1,1,1,1,1), (0,1,3,1,0), (0,1,1,0,3)$
&$40$\\
     $G_6\simeq ({\bf Z}_5)^3$ & $(1,1,1,1,1), (0,1,4,0,0), (0,3,0,1,1)$
&$88$\\
     $G_7\simeq ({\bf Z}_5)^4$ & $(1,1,1,1,1), (0,1,2,3,4), (0,1,1,4,4),
      (0,0,0,1,4)$&$200$ \\ \hline
  \end{tabular}
\end{center}


\bigskip
\smallskip

{\noindent If $G=G_k$} then we take  $G^*=G_{2^{d-2}-1-k}$. Note that
$\vert G\vert \vert G^* \vert= d^d$.
The Landau-Ginzburg orbifold $W/\!/G$ corresponds to the sigma model on
$\hat{\cal M}_G$ which is a  resolution of the orbifold
\begin{equation}
  {\cal M}_G=\{ (z_1,\ldots,z_d)\in {\bf CP}^{d-1} : W(z_1,\ldots,z_d)=0\}/
   (G/G_0)\,.
\label{fermat}
\end{equation}
The Euler characteristic of $\hat{\cal M}_G$ is related to the
$\chi_y$-genus by
\begin{equation}
  \chi(\hat{\cal M}_G)=(-1)^d\chi_{y=1}[W/\!/G]\,.
\end{equation}

By examining the data presented below we can confirm that the asserted
situation indeed occurs. However before seeing this let us explain how
to look at tables below.  The elements of $G_i$ are ordered from left
to right then from top to bottom in their tabulations. Notice that the
elements of $G_i$ corresponding to the untwisted sectors take the form
$(0,*,\ldots,*)$. The $\chi_y^\alpha[W/\!/G_i]$ are arrayed in the
same order as for the elements of $G_i$ and should again be read from
left to right then from top to bottom in their tabulations.  Thus for
example the first and second rows of the table of $d=5$, $G_4$
correspond respectively to $1+5y+5y^2+y^3,0,0,0,0$ and
$0,0,2y+2y^2,0,2y+2y^2$ in the table of $\chi_y^\alpha[W/\!/G_4]$.

Now let us consider, as an illustration, the pair of $W/\!/G_1$ and
$W/\!/G_2$ for $d=4$. Both theories have $4$ untwisted sectors and 12
twisted sectors.  The total twisted contribution to
$\chi_y[W/\!/G_1]$ reads $y^2 + 0 + y + 0 +y +y+ y+ y +1 + 0 + y +
0=1+6y+y^2$ while the total untwisted contributions to
$\chi_y[W/\!/G_2]$ reads $(1+5y+y^2)+0+y+0=1+6y+y^2$.  As another
example let us take the pair of $W/\!/G_0$ and $W/\!/G_7$ for $d=5$.
The total twisted contribution to $\chi_y[W/\!/G_0]$ reads
$-y^3-y^2-y-1$. The theory $W/\!/G_7$ has $5^4/5=125$ untwisted
sectors. The first one makes a contribution of $1+y+y^2+y^3$ while
each of the remaining $124$ ones of $0$.

The other cases can be checked similarly. Though we have not worked
out, it is also likely that similar results can be obtained for a
class of mirror pairs considered in \cite{rBeHu}.


%\vspace{2cm}
\newpage

%\input data.tex

\footnotesize
\def\phci{\vphantom{\Big\vert}}
\begin{center}
  (i) $d=3$
\end{center}
\bigskip

%\topcaption{}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$G_0$ }\\ \hline}
%\tabletail{\hline \multicolumn{4}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
 &(0, 0, 0) & (1, 1, 1) & (2, 2, 2)\\
\end{supertabular}
\end{center}
\vspace{1cm}

%\topcaption{}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$G_1$ }\\ \hline}
%\tabletail{\hline \multicolumn{4}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
 &(0, 0, 0) & (0, 1, 2) & (0, 2, 4)\\
 &(1, 1, 1) & (1, 2, 3) & (1, 3, 5)\\
 &(2, 2, 2) & (2, 3, 4) & (2, 4, 6)\\
\end{supertabular}
\end{center}
\vspace{1cm}


%\topcaption{Twisted sector contributions to $\chi_y[W/\!/G_0]$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_0]$
}\\ \hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
$ 1 + y,-y,-1$\\
\end{supertabular}
\end{center}
\vspace{2cm}


%\topcaption{Twisted sector contributions to $\chi_y$ of $\hat {\cal M}_{G_1}$}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$\chi_y^\alpha[W/\!/G_1]$
}\\ \hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
 &$1 + y,0,0,$
  &$ -y,0,0,$
 &$  -1,0,0 $\\
\end{supertabular}
\end{center}
\vspace{2cm}

\newpage


\begin{center}
 (ii)  $d=4$
\end{center}
\bigskip

%\topcaption{}
\tablefirsthead{\hline \multicolumn{5}{|c|}{$G_0$ }\\ \hline}
%\tabletail{\hline \multicolumn{5}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{}
\begin{center}
\begin{supertabular}{|lllll|}
 &(0, 0, 0, 0) & (1, 1, 1, 1) & (2, 2, 2, 2) & (3, 3, 3, 3)\\
\end{supertabular}
\end{center}
\vspace{.5cm}




%\topcaption{}
\tablefirsthead{\hline \multicolumn{5}{|c|}{$G_1$ }\\ \hline}
%\tabletail{\hline \multicolumn{5}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{}
\begin{center}
\begin{supertabular}{|lllll|}
&(0, 0, 0, 0)&(0, 0, 1, 3)&(0, 0, 2, 6)&(0, 0, 3, 9)\\
&(1, 1, 1, 1)&(1, 1, 2, 4)&(1, 1, 3, 7)&(1, 1, 4, 10)\\
&(2, 2, 2, 2)&(2, 2, 3, 5)&(2, 2, 4, 8)&(2, 2, 5, 11)\\
&(3, 3, 3, 3)&(3, 3, 4, 6)&(3, 3, 5, 9)&(3, 3, 6, 12)\\
\end{supertabular}
\end{center}
\vspace{.5cm}



%\topcaption{}
 \tablefirsthead{\hline \multicolumn{5}{|c|}{$G_2$ }\\ \hline}
%\tabletail{\hline \multicolumn{5}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{}
\begin{center}
\begin{supertabular}{|lllll|}
&(0, 0, 0, 0)&(0, 1, 1, 2)&(0, 2, 2, 4)&(0, 3, 3, 6)\\
&(1, 1, 1, 1)&(1, 2, 2, 3)&(1, 3, 3, 5)&(1, 4, 4, 7)\\
&(2, 2, 2, 2)&(2, 3, 3, 4)&(2, 4, 4, 6)&(2, 5, 5, 8)\\
&(3, 3, 3, 3)&(3, 4, 4, 5)&(3, 5, 5, 7)&(3, 6, 6, 9)\\
\end{supertabular}
\end{center}
\vspace{.5cm}

%\topcaption{}
 \tablefirsthead{\hline \multicolumn{5}{|c|}{$G_3$ }\\ \hline}
%\tabletail{\hline \multicolumn{5}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{}
\begin{center}
\begin{supertabular}{|lllll|}
&(0, 0, 0, 0)&(0, 1, 1, 2)&(0, 2, 2, 4)&(0, 3, 3, 6)\\
&(0, 0, 1, 3)&(0, 1, 2, 5)&(0, 2, 3, 7)&(0, 3, 4, 9)\\
&(0, 0, 2, 6)&(0, 1, 3, 8)&(0, 2, 4, 10)&(0, 3, 5, 12)\\
&(0, 0, 3, 9)&(0, 1, 4, 11)&(0, 2, 5, 13)&(0, 3, 6, 15)\\
&(1, 1, 1, 1)&(1, 2, 2, 3)&(1, 3, 3, 5)&(1, 4, 4, 7)\\
&(1, 1, 2, 4)&(1, 2, 3, 6)&(1, 3, 4, 8)&(1, 4, 5, 10)\\
&(1, 1, 3, 7)&(1, 2, 4, 9)&(1, 3, 5, 11)&(1, 4, 6, 13)\\
&(1, 1, 4, 10)&(1, 2, 5, 12)&(1, 3, 6, 14)&(1, 4, 7, 16)\\
&(2, 2, 2, 2)&(2, 3, 3, 4)&(2, 4, 4, 6)&(2, 5, 5, 8)\\
&(2, 2, 3, 5)&(2, 3, 4, 7)&(2, 4, 5, 9)&(2, 5, 6, 11)\\
&(2, 2, 4, 8)&(2, 3, 5, 10)&(2, 4, 6, 12)&(2, 5, 7, 14)\\
&(2, 2, 5, 11)&(2, 3, 6, 13)&(2, 4, 7, 15)&(2, 5, 8, 17)\\
&(3, 3, 3, 3)&(3, 4, 4, 5)&(3, 5, 5, 7)&(3, 6, 6, 9)\\
&(3, 3, 4, 6)&(3, 4, 5, 8)&(3, 5, 6, 10)&(3, 6, 7, 12)\\
&(3, 3, 5, 9)&(3, 4, 6, 11)&(3, 5, 7, 13)&(3, 6, 8, 15)\\
&(3, 3, 6, 12)&(3, 4, 7, 14)&(3, 5, 8, 16)&(3, 6, 9, 18)\\
\end{supertabular}
\end{center}
\newpage



%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_0}$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_0]$ }\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
$ 1 + 19y + {y^2},{y^2},y,1$\\
\end{supertabular}
\end{center}
\vspace{2cm}



%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_0}$}
\tablefirsthead{\hline \multicolumn{5}{|c|}{$\chi_y^\alpha[W/\!/G_1]$ }\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|lllll|}
&$1 + 5y + {y^2},3y,3y,3y,$
&$  {y^2},0,y,0,$
&$  y,y,y,y,$
&$   1,0,y,0$\\
\end{supertabular}
\end{center}
\vspace{2cm}


 %\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_2}$}
\tablefirsthead{\hline \multicolumn{5}{|c|}{$\chi_y^\alpha[W/\!/G_2]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|lllll|}
&$ 1 + 5y + {y^2},0,y,0,$
&$   {y^2},y,y,3y,$
&$   y,0,3y,0,$
&$   1,3y,y,y$\\
\end{supertabular}
\end{center}
\vspace{2cm}



%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_0}$}
\tablefirsthead{\hline \multicolumn{5}{|c|}{$\chi_y^\alpha[W/\!/G_3]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|lllll|}
 & $ 1 + y + {y^2},0,0,0,$
 & $ 0,0,0,0,$
 & $ 0,0,0,0,$
 & $ 0,0,0,0,$\\
 & $ {y^2},y,y,0,$
 & $ 0,y,0,0,$
 & $ y,0,y,0,$
 & $ 0,0,y,0,$\\
 & $ y,0,0,0,$
 & $ y,0,0,y,$
 & $ 0,y,0,y,$
 & $ y,y,0,0,$\\
 & $ 1,0,y,y,$
 & $ 0,0,y,0,$
 & $ y,0,y,0,$
 & $ 0,0,0,y $\\
\end{supertabular}
\end{center}
\newpage


\begin{center}
  (iii) $d=5$
\end{center}
\bigskip

%% quintic %%
%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_0$ }\\ \hline}
%\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{The elements of $G_0$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (1, 1, 1, 1, 1) & (2, 2, 2, 2, 2) & (3, 3, 3, 3,
  3) & (4, 4, 4, 4, 4)\\
\end{supertabular}
\end{center}
\vspace{2cm}

%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_1$ }\\ \hline}
%\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{The elements of $G_1$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 0, 0, 1, 4) & (0, 0, 0, 2, 8) & (0, 0, 0, 3,
  12) & (0, 0, 0, 4, 16)\\ &(1, 1, 1, 1, 1) & (1, 1, 1, 2, 5) & (1, 1,
  1, 3, 9) & (1, 1, 1, 4, 13) & (1, 1, 1, 5, 17)\\ &(2, 2, 2, 2, 2) &
  (2, 2, 2, 3, 6) & (2, 2, 2, 4, 10) & (2, 2, 2, 5, 14) & (2, 2, 2, 6,
  18)\\ &(3, 3, 3, 3, 3) & (3, 3, 3, 4, 7) & (3, 3, 3, 5, 11) & (3, 3,
  3, 6, 15) & (3, 3, 3, 7, 19)\\ &(4, 4, 4, 4, 4) & (4, 4, 4, 5, 8) &
  (4, 4, 4, 6, 12) & (4, 4, 4, 7, 16) & (4, 4, 4, 8, 20)\\
\end{supertabular}
\end{center}
\vspace{2cm}

%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_2$ }\\ \hline}
%\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{The elements of $G_2$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 1, 2, 3, 4) & (0, 2, 4, 6, 8) & (0, 3, 6, 9,
  12) & (0, 4, 8, 12, 16)\\ &(1, 1, 1, 1, 1) & (1, 2, 3, 4, 5) & (1,
  3, 5, 7, 9) & (1, 4, 7, 10, 13) & (1, 5, 9, 13, 17)\\ &(2, 2, 2, 2,
  2) & (2, 3, 4, 5, 6) & (2, 4, 6, 8, 10) & (2, 5, 8, 11, 14) & (2, 6,
  10, 14, 18)\\ &(3, 3, 3, 3, 3) & (3, 4, 5, 6, 7) & (3, 5, 7, 9, 11)
  & (3, 6, 9, 12, 15) & (3, 7, 11, 15, 19)\\ &(4, 4, 4, 4, 4) & (4, 5,
  6, 7, 8) & (4, 6, 8, 10, 12) & (4, 7, 10, 13, 16) & (4, 8, 12, 16,
  20)\\
\end{supertabular}
\end{center}
\vspace{2cm}


%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_3$ }\\ \hline}
%\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{The elements of $G_3$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 1, 1, 4, 4) & (0, 2, 2, 8, 8) & (0, 3, 3, 12,
  12) & (0, 4, 4, 16, 16)\\ &(1, 1, 1, 1, 1) & (1, 2, 2, 5, 5) & (1,
  3, 3, 9, 9) & (1, 4, 4, 13, 13) & (1, 5, 5, 17, 17)\\ &(2, 2, 2, 2,
  2) & (2, 3, 3, 6, 6) & (2, 4, 4, 10, 10) & (2, 5, 5, 14, 14) & (2,
  6, 6, 18, 18)\\ &(3, 3, 3, 3, 3) & (3, 4, 4, 7, 7) & (3, 5, 5, 11,
  11) & (3, 6, 6, 15, 15) & (3, 7, 7, 19, 19)\\ &(4, 4, 4, 4, 4) & (4,
  5, 5, 8, 8) & (4, 6, 6, 12, 12) & (4, 7, 7, 16, 16) & (4, 8, 8, 20,
  20)\\
\end{supertabular}
\end{center}

\newpage


%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_4$ }\\ \hline}
%\tablehead{\multicolumn{6}{l}{{\it cont. from previous page }}\\
%  \hline}
%\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline}
%%\topcaption{The elements of $G_4$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 1, 2, 3, 4) & (0, 2, 4, 6, 8) & (0, 3, 6, 9,
  12) & (0, 4, 8, 12, 16)\\ &(0, 1, 1, 4, 4) & (0, 2, 3, 7, 8) & (0,
  3, 5, 10, 12) & (0, 4, 7, 13, 16) & (0, 5, 9, 16, 20)\\ &(0, 2, 2,
  8, 8) & (0, 3, 4, 11, 12) & (0, 4, 6, 14, 16) & (0, 5, 8, 17, 20) &
  (0, 6, 10, 20, 24)\\ &(0, 3, 3, 12, 12) & (0, 4, 5, 15, 16) & (0, 5,
  7, 18, 20) & (0, 6, 9, 21, 24) & (0, 7, 11, 24, 28)\\ &(0, 4, 4, 16,
  16) & (0, 5, 6, 19, 20) & (0, 6, 8, 22, 24) & (0, 7, 10, 25, 28) &
  (0, 8, 12, 28, 32)\\ &(1, 1, 1, 1, 1) & (1, 2, 3, 4, 5) & (1, 3, 5,
  7, 9) & (1, 4, 7, 10, 13) & (1, 5, 9, 13, 17)\\ &(1, 2, 2, 5, 5) &
  (1, 3, 4, 8, 9) & (1, 4, 6, 11, 13) & (1, 5, 8, 14, 17) & (1, 6, 10,
  17, 21)\\ &(1, 3, 3, 9, 9) & (1, 4, 5, 12, 13) & (1, 5, 7, 15, 17) &
  (1, 6, 9, 18, 21) & (1, 7, 11, 21, 25)\\ &(1, 4, 4, 13, 13) & (1, 5,
  6, 16, 17) & (1, 6, 8, 19, 21) & (1, 7, 10, 22, 25) & (1, 8, 12, 25,
  29)\\ &(1, 5, 5, 17, 17) & (1, 6, 7, 20, 21) & (1, 7, 9, 23, 25) &
  (1, 8, 11, 26, 29) & (1, 9, 13, 29, 33)\\ &(2, 2, 2, 2, 2) & (2, 3,
  4, 5, 6) & (2, 4, 6, 8, 10) & (2, 5, 8, 11, 14) & (2, 6, 10, 14,
  18)\\ &(2, 3, 3, 6, 6) & (2, 4, 5, 9, 10) & (2, 5, 7, 12, 14) & (2,
  6, 9, 15, 18) & (2, 7, 11, 18, 22)\\ &(2, 4, 4, 10, 10) & (2, 5, 6,
  13, 14) & (2, 6, 8, 16, 18) & (2, 7, 10, 19, 22) & (2, 8, 12, 22,
  26)\\ &(2, 5, 5, 14, 14) & (2, 6, 7, 17, 18) & (2, 7, 9, 20, 22) &
  (2, 8, 11, 23, 26) & (2, 9, 13, 26, 30)\\ &(2, 6, 6, 18, 18) & (2,
  7, 8, 21, 22) & (2, 8, 10, 24, 26) & (2, 9, 12, 27, 30) & (2, 10,
  14, 30, 34)\\ &(3, 3, 3, 3, 3) & (3, 4, 5, 6, 7) & (3, 5, 7, 9, 11)
  & (3, 6, 9, 12, 15) & (3, 7, 11, 15, 19)\\ &(3, 4, 4, 7, 7) & (3, 5,
  6, 10, 11) & (3, 6, 8, 13, 15) & (3, 7, 10, 16, 19) & (3, 8, 12, 19,
  23)\\ &(3, 5, 5, 11, 11) & (3, 6, 7, 14, 15) & (3, 7, 9, 17, 19) &
  (3, 8, 11, 20, 23) & (3, 9, 13, 23, 27)\\ &(3, 6, 6, 15, 15) & (3,
  7, 8, 18, 19) & (3, 8, 10, 21, 23) & (3, 9, 12, 24, 27) & (3, 10,
  14, 27, 31)\\ &(3, 7, 7, 19, 19) & (3, 8, 9, 22, 23) & (3, 9, 11,
  25, 27) & (3, 10, 13, 28, 31) & (3, 11, 15, 31, 35)\\ &(4, 4, 4, 4,
  4) & (4, 5, 6, 7, 8) & (4, 6, 8, 10, 12) & (4, 7, 10, 13, 16) & (4,
  8, 12, 16, 20)\\ &(4, 5, 5, 8, 8) & (4, 6, 7, 11, 12) & (4, 7, 9,
  14, 16) & (4, 8, 11, 17, 20) & (4, 9, 13, 20, 24)\\ &(4, 6, 6, 12,
  12) & (4, 7, 8, 15, 16) & (4, 8, 10, 18, 20) & (4, 9, 12, 21, 24) &
  (4, 10, 14, 24, 28)\\ &(4, 7, 7, 16, 16) & (4, 8, 9, 19, 20) & (4,
  9, 11, 22, 24) & (4, 10, 13, 25, 28) & (4, 11, 15, 28, 32)\\ &(4, 8,
  8, 20, 20) & (4, 9, 10, 23, 24) & (4, 10, 12, 26, 28) & (4, 11, 14,
  29, 32) & (4, 12, 16, 32, 36)\\
\end{supertabular}
\end{center}

\newpage


%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_5$ }\\ \hline}
\tablelasttail{\hline}
%%\topcaption{The elements of $G_5$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 1, 1, 0, 3) & (0, 2, 2, 0, 6) & (0, 3, 3, 0,
  9) & (0, 4, 4, 0, 12)\\ &(0, 1, 3, 1, 0) & (0, 2, 4, 1, 3) & (0, 3,
  5, 1, 6) & (0, 4, 6, 1, 9) & (0, 5, 7, 1, 12)\\ &(0, 2, 6, 2, 0) &
  (0, 3, 7, 2, 3) & (0, 4, 8, 2, 6) & (0, 5, 9, 2, 9) & (0, 6, 10, 2,
  12)\\ &(0, 3, 9, 3, 0) & (0, 4, 10, 3, 3) & (0, 5, 11, 3, 6) & (0,
  6, 12, 3, 9) & (0, 7, 13, 3, 12)\\ &(0, 4, 12, 4, 0) & (0, 5, 13, 4,
  3) & (0, 6, 14, 4, 6) & (0, 7, 15, 4, 9) & (0, 8, 16, 4, 12)\\ &(1,
  1, 1, 1, 1) & (1, 2, 2, 1, 4) & (1, 3, 3, 1, 7) & (1, 4, 4, 1, 10) &
  (1, 5, 5, 1, 13)\\ &(1, 2, 4, 2, 1) & (1, 3, 5, 2, 4) & (1, 4, 6, 2,
  7) & (1, 5, 7, 2, 10) & (1, 6, 8, 2, 13)\\ &(1, 3, 7, 3, 1) & (1, 4,
  8, 3, 4) & (1, 5, 9, 3, 7) & (1, 6, 10, 3, 10) & (1, 7, 11, 3, 13)\\
  &(1, 4, 10, 4, 1) & (1, 5, 11, 4, 4) & (1, 6, 12, 4, 7) & (1, 7, 13,
  4, 10) & (1, 8, 14, 4, 13)\\ &(1, 5, 13, 5, 1) & (1, 6, 14, 5, 4) &
  (1, 7, 15, 5, 7) & (1, 8, 16, 5, 10) & (1, 9, 17, 5, 13)\\ &(2, 2,
  2, 2, 2) & (2, 3, 3, 2, 5) & (2, 4, 4, 2, 8) & (2, 5, 5, 2, 11) &
  (2, 6, 6, 2, 14)\\ &(2, 3, 5, 3, 2) & (2, 4, 6, 3, 5) & (2, 5, 7, 3,
  8) & (2, 6, 8, 3, 11) & (2, 7, 9, 3, 14)\\ &(2, 4, 8, 4, 2) & (2, 5,
  9, 4, 5) & (2, 6, 10, 4, 8) & (2, 7, 11, 4, 11) & (2, 8, 12, 4,
  14)\\ &(2, 5, 11, 5, 2) & (2, 6, 12, 5, 5) & (2, 7, 13, 5, 8) & (2,
  8, 14, 5, 11) & (2, 9, 15, 5, 14)\\ &(2, 6, 14, 6, 2) & (2, 7, 15,
  6, 5) & (2, 8, 16, 6, 8) & (2, 9, 17, 6, 11) & (2, 10, 18, 6, 14)\\
  &(3, 3, 3, 3, 3) & (3, 4, 4, 3, 6) & (3, 5, 5, 3, 9) & (3, 6, 6, 3,
  12) & (3, 7, 7, 3, 15)\\ &(3, 4, 6, 4, 3) & (3, 5, 7, 4, 6) & (3, 6,
  8, 4, 9) & (3, 7, 9, 4, 12) & (3, 8, 10, 4, 15)\\ &(3, 5, 9, 5, 3) &
  (3, 6, 10, 5, 6) & (3, 7, 11, 5, 9) & (3, 8, 12, 5, 12) & (3, 9, 13,
  5, 15)\\ &(3, 6, 12, 6, 3) & (3, 7, 13, 6, 6) & (3, 8, 14, 6, 9) &
  (3, 9, 15, 6, 12) & (3, 10, 16, 6, 15)\\ &(3, 7, 15, 7, 3) & (3, 8,
  16, 7, 6) & (3, 9, 17, 7, 9) & (3, 10, 18, 7, 12) & (3, 11, 19, 7,
  15)\\ &(4, 4, 4, 4, 4) & (4, 5, 5, 4, 7) & (4, 6, 6, 4, 10) & (4, 7,
  7, 4, 13) & (4, 8, 8, 4, 16)\\ &(4, 5, 7, 5, 4) & (4, 6, 8, 5, 7) &
  (4, 7, 9, 5, 10) & (4, 8, 10, 5, 13) & (4, 9, 11, 5, 16)\\ &(4, 6,
  10, 6, 4) & (4, 7, 11, 6, 7) & (4, 8, 12, 6, 10) & (4, 9, 13, 6, 13)
  & (4, 10, 14, 6, 16)\\ &(4, 7, 13, 7, 4) & (4, 8, 14, 7, 7) & (4, 9,
  15, 7, 10) & (4, 10, 16, 7, 13) & (4, 11, 17, 7, 16)\\ &(4, 8, 16,
  8, 4) & (4, 9, 17, 8, 7) & (4, 10, 18, 8, 10) & (4, 11, 19, 8, 13) &
  (4, 12, 20, 8, 16)\\
\end{supertabular}
\end{center}

\newpage


%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_6$ }\\ \hline}
\tablelasttail{\hline}
%%\topcaption{The elements of $G_6$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 3, 0, 1, 1) & (0, 6, 0, 2, 2) & (0, 9, 0, 3,
  3) & (0, 12, 0, 4, 4)\\ &(0, 1, 4, 0, 0) & (0, 4, 4, 1, 1) & (0, 7,
  4, 2, 2) & (0, 10, 4, 3, 3) & (0, 13, 4, 4, 4)\\ &(0, 2, 8, 0, 0) &
  (0, 5, 8, 1, 1) & (0, 8, 8, 2, 2) & (0, 11, 8, 3, 3) & (0, 14, 8, 4,
  4)\\ &(0, 3, 12, 0, 0) & (0, 6, 12, 1, 1) & (0, 9, 12, 2, 2) & (0,
  12, 12, 3, 3) & (0, 15, 12, 4, 4)\\ &(0, 4, 16, 0, 0) & (0, 7, 16,
  1, 1) & (0, 10, 16, 2, 2) & (0, 13, 16, 3, 3) & (0, 16, 16, 4, 4)\\
  &(1, 1, 1, 1, 1) & (1, 4, 1, 2, 2) & (1, 7, 1, 3, 3) & (1, 10, 1, 4,
  4) & (1, 13, 1, 5, 5)\\ &(1, 2, 5, 1, 1) & (1, 5, 5, 2, 2) & (1, 8,
  5, 3, 3) & (1, 11, 5, 4, 4) & (1, 14, 5, 5, 5)\\ &(1, 3, 9, 1, 1) &
  (1, 6, 9, 2, 2) & (1, 9, 9, 3, 3) & (1, 12, 9, 4, 4) & (1, 15, 9, 5,
  5)\\ &(1, 4, 13, 1, 1) & (1, 7, 13, 2, 2) & (1, 10, 13, 3, 3) & (1,
  13, 13, 4, 4) & (1, 16, 13, 5, 5)\\ &(1, 5, 17, 1, 1) & (1, 8, 17,
  2, 2) & (1, 11, 17, 3, 3) & (1, 14, 17, 4, 4) & (1, 17, 17, 5, 5)\\
  &(2, 2, 2, 2, 2) & (2, 5, 2, 3, 3) & (2, 8, 2, 4, 4) & (2, 11, 2, 5,
  5) & (2, 14, 2, 6, 6)\\ &(2, 3, 6, 2, 2) & (2, 6, 6, 3, 3) & (2, 9,
  6, 4, 4) & (2, 12, 6, 5, 5) & (2, 15, 6, 6, 6)\\ &(2, 4, 10, 2, 2) &
  (2, 7, 10, 3, 3) & (2, 10, 10, 4, 4) & (2, 13, 10, 5, 5) & (2, 16,
  10, 6, 6)\\ &(2, 5, 14, 2, 2) & (2, 8, 14, 3, 3) & (2, 11, 14, 4, 4)
  & (2, 14, 14, 5, 5) & (2, 17, 14, 6, 6)\\ &(2, 6, 18, 2, 2) & (2, 9,
  18, 3, 3) & (2, 12, 18, 4, 4) & (2, 15, 18, 5, 5) & (2, 18, 18, 6,
  6)\\ &(3, 3, 3, 3, 3) & (3, 6, 3, 4, 4) & (3, 9, 3, 5, 5) & (3, 12,
  3, 6, 6) & (3, 15, 3, 7, 7)\\ &(3, 4, 7, 3, 3) & (3, 7, 7, 4, 4) &
  (3, 10, 7, 5, 5) & (3, 13, 7, 6, 6) & (3, 16, 7, 7, 7)\\ &(3, 5, 11,
  3, 3) & (3, 8, 11, 4, 4) & (3, 11, 11, 5, 5) & (3, 14, 11, 6, 6) &
  (3, 17, 11, 7, 7)\\ &(3, 6, 15, 3, 3) & (3, 9, 15, 4, 4) & (3, 12,
  15, 5, 5) & (3, 15, 15, 6, 6) & (3, 18, 15, 7, 7)\\ &(3, 7, 19, 3,
  3) & (3, 10, 19, 4, 4) & (3, 13, 19, 5, 5) & (3, 16, 19, 6, 6) & (3,
  19, 19, 7, 7)\\ &(4, 4, 4, 4, 4) & (4, 7, 4, 5, 5) & (4, 10, 4, 6,
  6) & (4, 13, 4, 7, 7) & (4, 16, 4, 8, 8)\\ &(4, 5, 8, 4, 4) & (4, 8,
  8, 5, 5) & (4, 11, 8, 6, 6) & (4, 14, 8, 7, 7) & (4, 17, 8, 8, 8)\\
  &(4, 6, 12, 4, 4) & (4, 9, 12, 5, 5) & (4, 12, 12, 6, 6) & (4, 15,
  12, 7, 7) & (4, 18, 12, 8, 8)\\ &(4, 7, 16, 4, 4) & (4, 10, 16, 5,
  5) & (4, 13, 16, 6, 6) & (4, 16, 16, 7, 7) & (4, 19, 16, 8, 8)\\
  &(4, 8, 20, 4, 4) & (4, 11, 20, 5, 5) & (4, 14, 20, 6, 6) & (4, 17,
  20, 7, 7) & (4, 20, 20, 8, 8)\\
\end{supertabular}
\end{center}

\newpage



%\topcaption{}
\tablefirsthead{\hline \multicolumn{6}{|c|}{$G_7$ }\\ \hline}
\tablehead{\multicolumn{6}{l}{{\it cont. from previous page }}\\
  \hline}
\tabletail{\hline \multicolumn{6}{r}{{\it cont. to next page }}\\ }
\tablelasttail{\hline\multicolumn{6}{r}{{ \it end of table }}\\}
%%\topcaption{The elements of $G_7$}
\begin{center}
\begin{supertabular}{|llllll|}
  &(0, 0, 0, 0, 0) & (0, 0, 0, 1, 4) & (0, 0, 0, 2, 8) & (0, 0, 0, 3,
  12) & (0, 0, 0, 4, 16)\\ &(0, 1, 1, 4, 4) & (0, 1, 1, 5, 8) & (0, 1,
  1, 6, 12) & (0, 1, 1, 7, 16) & (0, 1, 1, 8, 20)\\ &(0, 2, 2, 8, 8) &
  (0, 2, 2, 9, 12) & (0, 2, 2, 10, 16) & (0, 2, 2, 11, 20) & (0, 2, 2,
  12, 24)\\ &(0, 3, 3, 12, 12) & (0, 3, 3, 13, 16) & (0, 3, 3, 14, 20)
  & (0, 3, 3, 15, 24) & (0, 3, 3, 16, 28)\\ &(0, 4, 4, 16, 16) & (0,
  4, 4, 17, 20) & (0, 4, 4, 18, 24) & (0, 4, 4, 19, 28) & (0, 4, 4,
  20, 32)\\ &(0, 1, 2, 3, 4) & (0, 1, 2, 4, 8) & (0, 1, 2, 5, 12) &
  (0, 1, 2, 6, 16) & (0, 1, 2, 7, 20)\\ &(0, 2, 3, 7, 8) & (0, 2, 3,
  8, 12) & (0, 2, 3, 9, 16) & (0, 2, 3, 10, 20) & (0, 2, 3, 11, 24)\\
  &(0, 3, 4, 11, 12) & (0, 3, 4, 12, 16) & (0, 3, 4, 13, 20) & (0, 3,
  4, 14, 24) & (0, 3, 4, 15, 28)\\ &(0, 4, 5, 15, 16) & (0, 4, 5, 16,
  20) & (0, 4, 5, 17, 24) & (0, 4, 5, 18, 28) & (0, 4, 5, 19, 32)\\
  &(0, 5, 6, 19, 20) & (0, 5, 6, 20, 24) & (0, 5, 6, 21, 28) & (0, 5,
  6, 22, 32) & (0, 5, 6, 23, 36)\\ &(0, 2, 4, 6, 8) & (0, 2, 4, 7, 12)
  & (0, 2, 4, 8, 16) & (0, 2, 4, 9, 20) & (0, 2, 4, 10, 24)\\ &(0, 3,
  5, 10, 12) & (0, 3, 5, 11, 16) & (0, 3, 5, 12, 20) & (0, 3, 5, 13,
  24) & (0, 3, 5, 14, 28)\\ &(0, 4, 6, 14, 16) & (0, 4, 6, 15, 20) &
  (0, 4, 6, 16, 24) & (0, 4, 6, 17, 28) & (0, 4, 6, 18, 32)\\ &(0, 5,
  7, 18, 20) & (0, 5, 7, 19, 24) & (0, 5, 7, 20, 28) & (0, 5, 7, 21,
  32) & (0, 5, 7, 22, 36)\\ &(0, 6, 8, 22, 24) & (0, 6, 8, 23, 28) &
  (0, 6, 8, 24, 32) & (0, 6, 8, 25, 36) & (0, 6, 8, 26, 40)\\ &(0, 3,
  6, 9, 12) & (0, 3, 6, 10, 16) & (0, 3, 6, 11, 20) & (0, 3, 6, 12,
  24) & (0, 3, 6, 13, 28)\\ &(0, 4, 7, 13, 16) & (0, 4, 7, 14, 20) &
  (0, 4, 7, 15, 24) & (0, 4, 7, 16, 28) & (0, 4, 7, 17, 32)\\ &(0, 5,
  8, 17, 20) & (0, 5, 8, 18, 24) & (0, 5, 8, 19, 28) & (0, 5, 8, 20,
  32) & (0, 5, 8, 21, 36)\\ &(0, 6, 9, 21, 24) & (0, 6, 9, 22, 28) &
  (0, 6, 9, 23, 32) & (0, 6, 9, 24, 36) & (0, 6, 9, 25, 40)\\ &(0, 7,
  10, 25, 28) & (0, 7, 10, 26, 32) & (0, 7, 10, 27, 36) & (0, 7, 10,
  28, 40) & (0, 7, 10, 29, 44)\\ &(0, 4, 8, 12, 16) & (0, 4, 8, 13,
  20) & (0, 4, 8, 14, 24) & (0, 4, 8, 15, 28) & (0, 4, 8, 16, 32)\\
  &(0, 5, 9, 16, 20) & (0, 5, 9, 17, 24) & (0, 5, 9, 18, 28) & (0, 5,
  9, 19, 32) & (0, 5, 9, 20, 36)\\ &(0, 6, 10, 20, 24) & (0, 6, 10,
  21, 28) & (0, 6, 10, 22, 32) & (0, 6, 10, 23, 36) & (0, 6, 10, 24,
  40)\\ &(0, 7, 11, 24, 28) & (0, 7, 11, 25, 32) & (0, 7, 11, 26, 36)
  & (0, 7, 11, 27, 40) & (0, 7, 11, 28, 44)\\ &(0, 8, 12, 28, 32) &
  (0, 8, 12, 29, 36) & (0, 8, 12, 30, 40) & (0, 8, 12, 31, 44) & (0,
  8, 12, 32, 48)\\ &(1, 1, 1, 1, 1) & (1, 1, 1, 2, 5) & (1, 1, 1, 3,
  9) & (1, 1, 1, 4, 13) & (1, 1, 1, 5, 17)\\ &(1, 2, 2, 5, 5) & (1, 2,
  2, 6, 9) & (1, 2, 2, 7, 13) & (1, 2, 2, 8, 17) & (1, 2, 2, 9, 21)\\
  &(1, 3, 3, 9, 9) & (1, 3, 3, 10, 13) & (1, 3, 3, 11, 17) & (1, 3, 3,
  12, 21) & (1, 3, 3, 13, 25)\\ &(1, 4, 4, 13, 13) & (1, 4, 4, 14, 17)
  & (1, 4, 4, 15, 21) & (1, 4, 4, 16, 25) & (1, 4, 4, 17, 29)\\ &(1,
  5, 5, 17, 17) & (1, 5, 5, 18, 21) & (1, 5, 5, 19, 25) & (1, 5, 5,
  20, 29) & (1, 5, 5, 21, 33)\\ &(1, 2, 3, 4, 5) & (1, 2, 3, 5, 9) &
  (1, 2, 3, 6, 13) & (1, 2, 3, 7, 17) & (1, 2, 3, 8, 21)\\ &(1, 3, 4,
  8, 9) & (1, 3, 4, 9, 13) & (1, 3, 4, 10, 17) & (1, 3, 4, 11, 21) &
  (1, 3, 4, 12, 25)\\ &(1, 4, 5, 12, 13) & (1, 4, 5, 13, 17) & (1, 4,
  5, 14, 21) & (1, 4, 5, 15, 25) & (1, 4, 5, 16, 29)\\ &(1, 5, 6, 16,
  17) & (1, 5, 6, 17, 21) & (1, 5, 6, 18, 25) & (1, 5, 6, 19, 29) &
  (1, 5, 6, 20, 33)\\ &(1, 6, 7, 20, 21) & (1, 6, 7, 21, 25) & (1, 6,
  7, 22, 29) & (1, 6, 7, 23, 33) & (1, 6, 7, 24, 37)\\ &(1, 3, 5, 7,
  9) & (1, 3, 5, 8, 13) & (1, 3, 5, 9, 17) & (1, 3, 5, 10, 21) & (1,
  3, 5, 11, 25)\\ &(1, 4, 6, 11, 13) & (1, 4, 6, 12, 17) & (1, 4, 6,
  13, 21) & (1, 4, 6, 14, 25) & (1, 4, 6, 15, 29)\\ &(1, 5, 7, 15, 17)
  & (1, 5, 7, 16, 21) & (1, 5, 7, 17, 25) & (1, 5, 7, 18, 29) & (1, 5,
  7, 19, 33)\\ &(1, 6, 8, 19, 21) & (1, 6, 8, 20, 25) & (1, 6, 8, 21,
  29) & (1, 6, 8, 22, 33) & (1, 6, 8, 23, 37)\\ &(1, 7, 9, 23, 25) &
  (1, 7, 9, 24, 29) & (1, 7, 9, 25, 33) & (1, 7, 9, 26, 37) & (1, 7,
  9, 27, 41)\\ &(1, 4, 7, 10, 13) & (1, 4, 7, 11, 17) & (1, 4, 7, 12,
  21) & (1, 4, 7, 13, 25) & (1, 4, 7, 14, 29)\\ &(1, 5, 8, 14, 17) &
  (1, 5, 8, 15, 21) & (1, 5, 8, 16, 25) & (1, 5, 8, 17, 29) & (1, 5,
  8, 18, 33)\\ &(1, 6, 9, 18, 21) & (1, 6, 9, 19, 25) & (1, 6, 9, 20,
  29) & (1, 6, 9, 21, 33) & (1, 6, 9, 22, 37)\\ &(1, 7, 10, 22, 25) &
  (1, 7, 10, 23, 29) & (1, 7, 10, 24, 33) & (1, 7, 10, 25, 37) & (1,
  7, 10, 26, 41)\\ &(1, 8, 11, 26, 29) & (1, 8, 11, 27, 33) & (1, 8,
  11, 28, 37) & (1, 8, 11, 29, 41) & (1, 8, 11, 30, 45)\\ &(1, 5, 9,
  13, 17) & (1, 5, 9, 14, 21) & (1, 5, 9, 15, 25) & (1, 5, 9, 16, 29)
  & (1, 5, 9, 17, 33)\\ &(1, 6, 10, 17, 21) & (1, 6, 10, 18, 25) & (1,
  6, 10, 19, 29) & (1, 6, 10, 20, 33) & (1, 6, 10, 21, 37)\\ &(1, 7,
  11, 21, 25) & (1, 7, 11, 22, 29) & (1, 7, 11, 23, 33) & (1, 7, 11,
  24, 37) & (1, 7, 11, 25, 41)\\ &(1, 8, 12, 25, 29) & (1, 8, 12, 26,
  33) & (1, 8, 12, 27, 37) & (1, 8, 12, 28, 41) & (1, 8, 12, 29, 45)\\
  &(1, 9, 13, 29, 33) & (1, 9, 13, 30, 37) & (1, 9, 13, 31, 41) & (1,
  9, 13, 32, 45) & (1, 9, 13, 33, 49)\\ &(2, 2, 2, 2, 2) & (2, 2, 2,
  3, 6) & (2, 2, 2, 4, 10) & (2, 2, 2, 5, 14) & (2, 2, 2, 6, 18)\\
  &(2, 3, 3, 6, 6) & (2, 3, 3, 7, 10) & (2, 3, 3, 8, 14) & (2, 3, 3,
  9, 18) & (2, 3, 3, 10, 22)\\ &(2, 4, 4, 10, 10) & (2, 4, 4, 11, 14)
  & (2, 4, 4, 12, 18) & (2, 4, 4, 13, 22) & (2, 4, 4, 14, 26)\\ &(2,
  5, 5, 14, 14) & (2, 5, 5, 15, 18) & (2, 5, 5, 16, 22) & (2, 5, 5,
  17, 26) & (2, 5, 5, 18, 30)\\ &(2, 6, 6, 18, 18) & (2, 6, 6, 19, 22)
  & (2, 6, 6, 20, 26) & (2, 6, 6, 21, 30) & (2, 6, 6, 22, 34)\\ &(2,
  3, 4, 5, 6) & (2, 3, 4, 6, 10) & (2, 3, 4, 7, 14) & (2, 3, 4, 8, 18)
  & (2, 3, 4, 9, 22)\\ &(2, 4, 5, 9, 10) & (2, 4, 5, 10, 14) & (2, 4,
  5, 11, 18) & (2, 4, 5, 12, 22) & (2, 4, 5, 13, 26)\\ &(2, 5, 6, 13,
  14) & (2, 5, 6, 14, 18) & (2, 5, 6, 15, 22) & (2, 5, 6, 16, 26) &
  (2, 5, 6, 17, 30)\\ &(2, 6, 7, 17, 18) & (2, 6, 7, 18, 22) & (2, 6,
  7, 19, 26) & (2, 6, 7, 20, 30) & (2, 6, 7, 21, 34)\\ &(2, 7, 8, 21,
  22) & (2, 7, 8, 22, 26) & (2, 7, 8, 23, 30) & (2, 7, 8, 24, 34) &
  (2, 7, 8, 25, 38)\\ &(2, 4, 6, 8, 10) & (2, 4, 6, 9, 14) & (2, 4, 6,
  10, 18) & (2, 4, 6, 11, 22) & (2, 4, 6, 12, 26)\\ &(2, 5, 7, 12, 14)
  & (2, 5, 7, 13, 18) & (2, 5, 7, 14, 22) & (2, 5, 7, 15, 26) & (2, 5,
  7, 16, 30)\\ &(2, 6, 8, 16, 18) & (2, 6, 8, 17, 22) & (2, 6, 8, 18,
  26) & (2, 6, 8, 19, 30) & (2, 6, 8, 20, 34)\\ &(2, 7, 9, 20, 22) &
  (2, 7, 9, 21, 26) & (2, 7, 9, 22, 30) & (2, 7, 9, 23, 34) & (2, 7,
  9, 24, 38)\\ &(2, 8, 10, 24, 26) & (2, 8, 10, 25, 30) & (2, 8, 10,
  26, 34) & (2, 8, 10, 27, 38) & (2, 8, 10, 28, 42)\\ &(2, 5, 8, 11,
  14) & (2, 5, 8, 12, 18) & (2, 5, 8, 13, 22) & (2, 5, 8, 14, 26) &
  (2, 5, 8, 15, 30)\\ &(2, 6, 9, 15, 18) & (2, 6, 9, 16, 22) & (2, 6,
  9, 17, 26) & (2, 6, 9, 18, 30) & (2, 6, 9, 19, 34)\\ &(2, 7, 10, 19,
  22) & (2, 7, 10, 20, 26) & (2, 7, 10, 21, 30) & (2, 7, 10, 22, 34) &
  (2, 7, 10, 23, 38)\\ &(2, 8, 11, 23, 26) & (2, 8, 11, 24, 30) & (2,
  8, 11, 25, 34) & (2, 8, 11, 26, 38) & (2, 8, 11, 27, 42)\\ &(2, 9,
  12, 27, 30) & (2, 9, 12, 28, 34) & (2, 9, 12, 29, 38) & (2, 9, 12,
  30, 42) & (2, 9, 12, 31, 46)\\ &(2, 6, 10, 14, 18) & (2, 6, 10, 15,
  22) & (2, 6, 10, 16, 26) & (2, 6, 10, 17, 30) & (2, 6, 10, 18, 34)\\
  &(2, 7, 11, 18, 22) & (2, 7, 11, 19, 26) & (2, 7, 11, 20, 30) & (2,
  7, 11, 21, 34) & (2, 7, 11, 22, 38)\\ &(2, 8, 12, 22, 26) & (2, 8,
  12, 23, 30) & (2, 8, 12, 24, 34) & (2, 8, 12, 25, 38) & (2, 8, 12,
  26, 42)\\ &(2, 9, 13, 26, 30) & (2, 9, 13, 27, 34) & (2, 9, 13, 28,
  38) & (2, 9, 13, 29, 42) & (2, 9, 13, 30, 46)\\ &(2, 10, 14, 30, 34)
  & (2, 10, 14, 31, 38) & (2, 10, 14, 32, 42) & (2, 10, 14, 33, 46) &
  (2, 10, 14, 34, 50)\\ &(3, 3, 3, 3, 3) & (3, 3, 3, 4, 7) & (3, 3, 3,
  5, 11) & (3, 3, 3, 6, 15) & (3, 3, 3, 7, 19)\\ &(3, 4, 4, 7, 7) &
  (3, 4, 4, 8, 11) & (3, 4, 4, 9, 15) & (3, 4, 4, 10, 19) & (3, 4, 4,
  11, 23)\\ &(3, 5, 5, 11, 11) & (3, 5, 5, 12, 15) & (3, 5, 5, 13, 19)
  & (3, 5, 5, 14, 23) & (3, 5, 5, 15, 27)\\ &(3, 6, 6, 15, 15) & (3,
  6, 6, 16, 19) & (3, 6, 6, 17, 23) & (3, 6, 6, 18, 27) & (3, 6, 6,
  19, 31)\\ &(3, 7, 7, 19, 19) & (3, 7, 7, 20, 23) & (3, 7, 7, 21, 27)
  & (3, 7, 7, 22, 31) & (3, 7, 7, 23, 35)\\ &(3, 4, 5, 6, 7) & (3, 4,
  5, 7, 11) & (3, 4, 5, 8, 15) & (3, 4, 5, 9, 19) & (3, 4, 5, 10,
  23)\\ &(3, 5, 6, 10, 11) & (3, 5, 6, 11, 15) & (3, 5, 6, 12, 19) &
  (3, 5, 6, 13, 23) & (3, 5, 6, 14, 27)\\ &(3, 6, 7, 14, 15) & (3, 6,
  7, 15, 19) & (3, 6, 7, 16, 23) & (3, 6, 7, 17, 27) & (3, 6, 7, 18,
  31)\\ &(3, 7, 8, 18, 19) & (3, 7, 8, 19, 23) & (3, 7, 8, 20, 27) &
  (3, 7, 8, 21, 31) & (3, 7, 8, 22, 35)\\ &(3, 8, 9, 22, 23) & (3, 8,
  9, 23, 27) & (3, 8, 9, 24, 31) & (3, 8, 9, 25, 35) & (3, 8, 9, 26,
  39)\\ &(3, 5, 7, 9, 11) & (3, 5, 7, 10, 15) & (3, 5, 7, 11, 19) &
  (3, 5, 7, 12, 23) & (3, 5, 7, 13, 27)\\ &(3, 6, 8, 13, 15) & (3, 6,
  8, 14, 19) & (3, 6, 8, 15, 23) & (3, 6, 8, 16, 27) & (3, 6, 8, 17,
  31)\\ &(3, 7, 9, 17, 19) & (3, 7, 9, 18, 23) & (3, 7, 9, 19, 27) &
  (3, 7, 9, 20, 31) & (3, 7, 9, 21, 35)\\ &(3, 8, 10, 21, 23) & (3, 8,
  10, 22, 27) & (3, 8, 10, 23, 31) & (3, 8, 10, 24, 35) & (3, 8, 10,
  25, 39)\\ &(3, 9, 11, 25, 27) & (3, 9, 11, 26, 31) & (3, 9, 11, 27,
  35) & (3, 9, 11, 28, 39) & (3, 9, 11, 29, 43)\\ &(3, 6, 9, 12, 15) &
  (3, 6, 9, 13, 19) & (3, 6, 9, 14, 23) & (3, 6, 9, 15, 27) & (3, 6,
  9, 16, 31)\\ &(3, 7, 10, 16, 19) & (3, 7, 10, 17, 23) & (3, 7, 10,
  18, 27) & (3, 7, 10, 19, 31) & (3, 7, 10, 20, 35)\\ &(3, 8, 11, 20,
  23) & (3, 8, 11, 21, 27) & (3, 8, 11, 22, 31) & (3, 8, 11, 23, 35) &
  (3, 8, 11, 24, 39)\\ &(3, 9, 12, 24, 27) & (3, 9, 12, 25, 31) & (3,
  9, 12, 26, 35) & (3, 9, 12, 27, 39) & (3, 9, 12, 28, 43)\\ &(3, 10,
  13, 28, 31) & (3, 10, 13, 29, 35) & (3, 10, 13, 30, 39) & (3, 10,
  13, 31, 43) & (3, 10, 13, 32, 47)\\ &(3, 7, 11, 15, 19) & (3, 7, 11,
  16, 23) & (3, 7, 11, 17, 27) & (3, 7, 11, 18, 31) & (3, 7, 11, 19,
  35)\\ &(3, 8, 12, 19, 23) & (3, 8, 12, 20, 27) & (3, 8, 12, 21, 31)
  & (3, 8, 12, 22, 35) & (3, 8, 12, 23, 39)\\ &(3, 9, 13, 23, 27) &
  (3, 9, 13, 24, 31) & (3, 9, 13, 25, 35) & (3, 9, 13, 26, 39) & (3,
  9, 13, 27, 43)\\ &(3, 10, 14, 27, 31) & (3, 10, 14, 28, 35) & (3,
  10, 14, 29, 39) & (3, 10, 14, 30, 43) & (3, 10, 14, 31, 47)\\ &(3,
  11, 15, 31, 35) & (3, 11, 15, 32, 39) & (3, 11, 15, 33, 43) & (3,
  11, 15, 34, 47) & (3, 11, 15, 35, 51)\\ &(4, 4, 4, 4, 4) & (4, 4, 4,
  5, 8) & (4, 4, 4, 6, 12) & (4, 4, 4, 7, 16) & (4, 4, 4, 8, 20)\\
  &(4, 5, 5, 8, 8) & (4, 5, 5, 9, 12) & (4, 5, 5, 10, 16) & (4, 5, 5,
  11, 20) & (4, 5, 5, 12, 24)\\ &(4, 6, 6, 12, 12) & (4, 6, 6, 13, 16)
  & (4, 6, 6, 14, 20) & (4, 6, 6, 15, 24) & (4, 6, 6, 16, 28)\\ &(4,
  7, 7, 16, 16) & (4, 7, 7, 17, 20) & (4, 7, 7, 18, 24) & (4, 7, 7,
  19, 28) & (4, 7, 7, 20, 32)\\ &(4, 8, 8, 20, 20) & (4, 8, 8, 21, 24)
  & (4, 8, 8, 22, 28) & (4, 8, 8, 23, 32) & (4, 8, 8, 24, 36)\\ &(4,
  5, 6, 7, 8) & (4, 5, 6, 8, 12) & (4, 5, 6, 9, 16) & (4, 5, 6, 10,
  20) & (4, 5, 6, 11, 24)\\ &(4, 6, 7, 11, 12) & (4, 6, 7, 12, 16) &
  (4, 6, 7, 13, 20) & (4, 6, 7, 14, 24) & (4, 6, 7, 15, 28)\\ &(4, 7,
  8, 15, 16) & (4, 7, 8, 16, 20) & (4, 7, 8, 17, 24) & (4, 7, 8, 18,
  28) & (4, 7, 8, 19, 32)\\ &(4, 8, 9, 19, 20) & (4, 8, 9, 20, 24) &
  (4, 8, 9, 21, 28) & (4, 8, 9, 22, 32) & (4, 8, 9, 23, 36)\\ &(4, 9,
  10, 23, 24) & (4, 9, 10, 24, 28) & (4, 9, 10, 25, 32) & (4, 9, 10,
  26, 36) & (4, 9, 10, 27, 40)\\ &(4, 6, 8, 10, 12) & (4, 6, 8, 11,
  16) & (4, 6, 8, 12, 20) & (4, 6, 8, 13, 24) & (4, 6, 8, 14, 28)\\
  &(4, 7, 9, 14, 16) & (4, 7, 9, 15, 20) & (4, 7, 9, 16, 24) & (4, 7,
  9, 17, 28) & (4, 7, 9, 18, 32)\\ &(4, 8, 10, 18, 20) & (4, 8, 10,
  19, 24) & (4, 8, 10, 20, 28) & (4, 8, 10, 21, 32) & (4, 8, 10, 22,
  36)\\ &(4, 9, 11, 22, 24) & (4, 9, 11, 23, 28) & (4, 9, 11, 24, 32)
  & (4, 9, 11, 25, 36) & (4, 9, 11, 26, 40)\\ &(4, 10, 12, 26, 28) &
  (4, 10, 12, 27, 32) & (4, 10, 12, 28, 36) & (4, 10, 12, 29, 40) &
  (4, 10, 12, 30, 44)\\ &(4, 7, 10, 13, 16) & (4, 7, 10, 14, 20) & (4,
  7, 10, 15, 24) & (4, 7, 10, 16, 28) & (4, 7, 10, 17, 32)\\ &(4, 8,
  11, 17, 20) & (4, 8, 11, 18, 24) & (4, 8, 11, 19, 28) & (4, 8, 11,
  20, 32) & (4, 8, 11, 21, 36)\\ &(4, 9, 12, 21, 24) & (4, 9, 12, 22,
  28) & (4, 9, 12, 23, 32) & (4, 9, 12, 24, 36) & (4, 9, 12, 25, 40)\\
  &(4, 10, 13, 25, 28) & (4, 10, 13, 26, 32) & (4, 10, 13, 27, 36) &
  (4, 10, 13, 28, 40) & (4, 10, 13, 29, 44)\\ &(4, 11, 14, 29, 32) &
  (4, 11, 14, 30, 36) & (4, 11, 14, 31, 40) & (4, 11, 14, 32, 44) &
  (4, 11, 14, 33, 48)\\ &(4, 8, 12, 16, 20) & (4, 8, 12, 17, 24) & (4,
  8, 12, 18, 28) & (4, 8, 12, 19, 32) & (4, 8, 12, 20, 36)\\ &(4, 9,
  13, 20, 24) & (4, 9, 13, 21, 28) & (4, 9, 13, 22, 32) & (4, 9, 13,
  23, 36) & (4, 9, 13, 24, 40)\\ &(4, 10, 14, 24, 28) & (4, 10, 14,
  25, 32) & (4, 10, 14, 26, 36) & (4, 10, 14, 27, 40) & (4, 10, 14,
  28, 44)\\ &(4, 11, 15, 28, 32) & (4, 11, 15, 29, 36) & (4, 11, 15,
  30, 40) & (4, 11, 15, 31, 44) & (4, 11, 15, 32, 48)\\ &(4, 12, 16,
  32, 36) & (4, 12, 16, 33, 40) & (4, 12, 16, 34, 44) & (4, 12, 16,
  35, 48) & (4, 12, 16, 36, 52)\\
\end{supertabular}
\end{center}

\newpage

%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_0}$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_0]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
$ 1 + 101y + 101{y^2} + {y^3},-{y^3},-{y^2},-y,-1$\\
\end{supertabular}
\end{center}
\vspace{2cm}

%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_1}$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_1]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
 $1+25y+25{y^2}+{y^3},6y+6{y^2},6y+6{y^2},6y+6{y^2},6y+6{y^2},$\\
  $ -{y^3},0,-{y^2},-{y^2},0,$\\
  $ -{y^2},-{y^2},0,0,-{y^2},$\\
  $ -y,-y,0,0,-y,$\\
 $ -1,0,-y,-y,0 $ \\
\end{supertabular}
\end{center}
\vspace{2cm}

%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_2}$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_2]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
  $1+21y+21{y^2}+{y^3},0,0,0,0,$\\
  $ -{y^3},0,0,0,0,$\\
  $ -{y^2},0,0,0,0,$\\
  $ -y,0,0,0,0,$\\
  $ -1,0,0,0,0 $\\
\end{supertabular}
\end{center}
\vspace{2cm}


%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_3}$}
\tablefirsthead{\hline \multicolumn{1}{|c|}{$\chi_y^\alpha[W/\!/G_3]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|l|}
$ 1+17y+17{y^2}+{y^3},0,0,0,0,$\\
  $ -{y^3},-4{y^2},-y,-y,-4{y^2},$\\
  $ -{y^2},-{y^2},-4y,-4y,-{y^2},$\\
  $ -y,-y,-4{y^2},-4{y^2},-y,$\\
  $ -1,-4y,-{y^2},-{y^2},-4y $\\
\end{supertabular}
\end{center}
\newpage

%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_4}$}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$\chi_y^\alpha[W/\!/G_4]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
  &$ 1+5y+5{y^2}+{y^3},0,0,0,0,$
  &$ 0,0,2y+2{y^2},0,2y+2{y^2},$
  &$ 0,0,0,2y+2{y^2},2y+2{y^2},$\\
  &$ 0,2y+2{y^2},2y+2{y^2},0,0,$
  &$ 0,2y+2{y^2},0,2y+2{y^2},0,$
  &$ -{y^3},0,0,0,0,$\\
 &$  0,-y,-{y^2},0,0,$
  &$ -y,0,0,-{y^2},0,$
  &$ -y,0,-{y^2},0,0,$\\
 &$  0,0,0,-{y^2},-y,$
 &$  -{y^2},0,0,0,0,$
 &$  -{y^2},0,0,0,-{y^2},$\\
 &$  0,0,-{y^2},0,-{y^2},$
 &$  0,-{y^2},0,-{y^2},0,$
 &$  -{y^2},-{y^2},0,0,0,$\\
 &$  -y,0,0,0,0,$
 &$  -y,0,0,0,-y,$
 &$  0,0,-y,0,-y,$\\
  &$ 0,-y,0,-y,0,$
 &$  -y,-y,0,0,0,$
  &$ -1,0,0,0,0,$\\
 &$  0,-{y^2},-y,0,0,$
 &$  -{y^2},0,0,-y,0,$
 &$  -{y^2},0,-y,0,0,$\\
 &$  0,0,0,-y,-{y^2}$& & \\
\end{supertabular}
\end{center}
\vspace{.5cm}


%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_5}$}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$\chi_y^\alpha[W/\!/G_5]$ }\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
&$ 1+y+{y^2}+{y^3},0,0,0,0,$
 &$  0,0,0,0,0,$
 &$  0,0,0,0,0,$\\
 &$  0,0,0,0,0,$
 &$  0,0,0,0,0,$
 &$ -{y^3},-{y^2},-{y^2},0,0,$\\
 &$ -{y^2},0,-{y^2},0,-{y^2},$
 &$ -{y^2},-y,0,0,-{y^2},$
 &$  0,0,-{y^2},0,-y,$\\
  &$ 0,0,0,0,0,$
 &$  -{y^2},0,-y,0,-{y^2},$
  &$ 0,0,0,-{y^2},-y,$\\
 &$  -y,0,0,-{y^2},-y,$
 &$  0,0,0,0,0,$
  &$ -{y^2},0,-{y^2},-{y^2},0,$\\
 &$  -y,-y,0,-{y^2},0,$
  &$ -y,0,-y,-y,0,$
 &$  0,0,0,0,0,$\\
  &$ -{y^2},-{y^2},-y,0,0,$
 &$  0,-{y^2},-y,0,0,$
 &$  -1,0,0,-y,-y,$\\
 &$  0,0,0,0,0,$
 &$  0,-{y^2},0,-y,0,$
 &$  -y,-y,0,0,-{y^2},$\\
 &$  -y,-y,0,-y,0$&  &\\
\end{supertabular}
\end{center}
\vspace{.5cm}









%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_6}$}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$\chi_y^\alpha[W/\!/G_6]$ }\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
 &$ 1+5y+5{y^2}+{y^3},0,0,0,0,$
  &$  0,0,0,0,0,$
  &$  0,0,0,0,0,$\\
  &$  0,0,0,0,0,$
   &$ 0,0,0,0,0,$
   &$ -{y^3},-{y^2},-{y^2},0,-4{y^2},$\\
  &$  0,0,0,0,0,$
   &$ -{y^2},-{y^2},-y,-y,0,$
  &$  -{y^2},-{y^2},0,-y,-4{y^2},$\\
  &$  0,-{y^2},-{y^2},-y,-4{y^2},$
 &$  -{y^2},0,-y,-4{y^2},-{y^2},$
 &$  -{y^2},-{y^2},-y,-4{y^2},0,$\\
  &$  0,0,0,0,0,$
   &$ 0,-y,-y,-4y,-{y^2},$
   &$ -{y^2},-y,-y,0,-{y^2},$\\
  &$  -y,-y,-4y,-{y^2},0,$
  &$  -y,-y,0,-{y^2},-{y^2},$
  &$  0,-y,-4{y^2},-{y^2},-{y^2},$\\
  &$  0,0,0,0,0,$
  &$  -y,0,-4y,-{y^2},-y,$
  &$  -1,-4y,0,-y,-y,$\\
  &$  0,-4y,-{y^2},-y,-y,$
  &$  -y,-4y,-{y^2},0,-y,$
  &$  -y,0,-{y^2},-{y^2},-y,$\\
  &$  0,0,0,0,0 $ & &\\
\end{supertabular}
\end{center}








\newpage


%\topcaption{Twisted sector contributions to $\chi_y$ of ${\cal M}_{G_7}$}
\tablefirsthead{\hline \multicolumn{4}{|c|}{$\chi_y^\alpha[W/\!/G_7]$}\\
\hline}
%\tablehead{\hline}
%\tabletail{\hline\\}
\tablelasttail{\hline}
\begin{center}
\begin{supertabular}{|llll|}
  &$ 1 + y + {y^2} + {y^3},0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ -{y^3},0,-{y^2},-{y^2},0,$
  &$ 0,-{y^2},-{y^2},-{y^2},-{y^2},$\\
  &$ -y,0,-{y^2},-{y^2},0,$
  &$ -y,-y,0,0,-y,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,-{y^2},-{y^2},-{y^2},$
  &$ -y,-y,0,-{y^2},0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,-{y^2},-{y^2},-{y^2},$
  &$ 0,0,0,0,0,$\\
  &$ -{y^2},-{y^2},-{y^2},0,0,$
  &$ 0,0,0,0,0,$
  &$ -{y^2},0,-{y^2},-{y^2},-{y^2},$\\
  &$ 0,-y,0,-{y^2},-{y^2},$
  &$ 0,-{y^2},-{y^2},0,-y,$
  &$ 0,0,0,0,0,$\\
  &$ -{y^2},0,0,-{y^2},-{y^2},$
  &$ 0,0,0,0,0,$
  &$ -{y^2},-{y^2},-{y^2},-{y^2},0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$
  &$ 0,-{y^2},-{y^2},-{y^2},0,$\\
  &$ 0,-{y^2},-{y^2},-{y^2},0,$
  &$ -y,0,-{y^2},0,-y,$
  &$ -{y^2},-{y^2},0,0,-{y^2},$\\
  &$ -{y^2},0,-y,-y,0,$
  &$ 0,-y,-y,-y,-y,$
  &$ 0,0,0,0,0,$\\
  &$ -{y^2},-{y^2},0,0,-{y^2},$
  &$ 0,0,-y,-y,-y,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ -{y^2},-{y^2},-{y^2},0,-{y^2},$
  &$ -{y^2},-{y^2},0,-y,0,$\\
  &$ 0,-y,0,-{y^2},-{y^2},$
  &$ 0,0,0,0,0,$
  &$ -{y^2},-{y^2},-{y^2},0,0,$\\
  &$ 0,-{y^2},0,-y,-y,$
  &$ 0,0,0,0,0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,-{y^2},-{y^2},0,-y,$
  &$ 0,0,0,0,0,$
 &$ -{y^2},0,0,-{y^2},-{y^2},$\\
  &$ 0,-y,-y,0,-{y^2},$
  &$ 0,0,0,0,0,$
 &$  -{y^2},-{y^2},0,-{y^2},-{y^2},$\\
  &$ -{y^2},0,-y,0,-{y^2},$
  &$ 0,-y,-y,-y,0,$
  &$ 0,0,0,0,0,$\\
  &$ -y,-y,0,0,-y,$
  &$ -y,-y,0,0,-y,$
  &$ 0,0,0,0,0,$\\
  &$ 0,-{y^2},-{y^2},-{y^2},-{y^2},$
  &$ -y,0,-{y^2},-{y^2},0,$
  &$ 0,0,0,0,0,$\\
  &$ 0,0,0,0,0,$
  &$ 0,0,-{y^2},-{y^2},-{y^2},$
  &$ -y,-y,0,-{y^2},0,$\\
  &$ -y,-y,-y,0,-y,$
  &$ 0,0,0,0,0,$
  &$ 0,-y,0,-{y^2},-{y^2},$\\
 &$  -y,-y,-y,0,0,$
  &$ 0,0,0,0,0,$
 &$  0,-{y^2},0,-y,-y,$\\
  &$ 0,-y,-y,0,-{y^2},$
  &$ 0,0,0,0,0,$
  &$ 0,-{y^2},-{y^2},0,-y,$\\
  &$ -y,0,0,-y,-y,$
 &$  0,0,0,0,0,$
  &$ 0,-{y^2},-{y^2},-{y^2},0,$\\
  &$ -y,0,-{y^2},0,-y,$
  &$ -y,-y,0,-y,-y,$
 &$  0,0,0,0,0,$\\
 &$  0,0,0,0,0,$
 &$  -1,0,-y,-y,0,$
 &$  0,0,0,0,0,$\\
 &$  -{y^2},-{y^2},0,0,-{y^2},$
 &$  -{y^2},0,-y,-y,0,$
 &$  0,-y,-y,-y,-y,$\\
 &$  0,0,0,0,0,$
 &$  -{y^2},-{y^2},0,-y,0,$
 &$  0,0,-y,-y,-y,$\\
 &$  0,0,-y,-y,-y,$
 &$  0,0,0,0,0,$
 &$  0,-{y^2},0,-y,-y,$\\
 &$  -y,0,-y,-y,-y,$
 &$  0,0,0,0,0,$
 &$  -y,-y,-y,0,0,$\\
 &$  0,0,0,0,0,$
 &$  0,0,0,0,0,$
 &$  0,-y,-y,0,-{y^2},$\\
 &$  -y,-y,-y,-y,0,$
 &$  0,0,0,0,0,$
 &$  -y,0,0,-y,-y,$\\
 &$  0,-y,-y,-y,0,$
 &$  0,-y,-y,-y,0,$
 &$  0,0,0,0,0,$\\
 &$  0,0,0,0,0,$
 &$  -{y^2},0,-y,0,-{y^2} $
 &\\
\end{supertabular}
\end{center}



\newpage
\normalsize
\section{Discussions}
We have pointed out in the previous section that there exist instances
in which the elliptic genera for a pair of Landau-Ginzburg orbifolds
obey the relation (\ref{dualeg}) (or equivalently (\ref{dualchiy}))
and moreover the roles of the untwisted and twisted sectors are
exchanged. To consider this phenomenon a little bit further we shall
now concentrate on the case where two Landau-Ginzburg orbifolds are in
correspondence with sigma models as investigated in sects. 4.3 and
4.4.

Let us first extend the $\chi_y$-genus to describe the full
$U(1)\times U(1)$ charge spectrum for the $(c,c)$ ring.  For a
Calabi-Yau $\hat c$-fold ${\cal M}$ we define
\begin{equation}
  \chi[{\cal M}](y,\bar y)=\sum_{q_{\rm L},q_{\rm R}=0}^{\hat c}
  (-1)^{q_{\rm L}+q_{\rm R}}h_{q_{\rm L},\hat c-q_{\rm R}}y^{q_{\rm
    L}}\bar y^{q_{\rm R}}\,,
\label{extdgenus}
\end{equation}
where $h_{q_{\rm L},\hat c-q_{\rm R}}$ is the number of states with
charge $(q_{\rm L},q_{\rm R})$ and is also equal to the Hodge number,
$\mathop{\rm dim} H^{q_{\rm L},\hat c-q_{\rm R}}({\cal M})$.  Suppose that
a pair of Calabi-Yau $\hat c$-folds $({\cal M},\tilde{\cal M})$
consist of a mirror pair, then we have
\begin{equation}
  \tilde h_{p,q}=h_{p,\hat c-q}\,,
\end{equation}
where $\tilde h_{p,q}$ are the Hodge numbers for $\tilde{\cal M}$.
Hence the extended genus (\ref{extdgenus}) for ${\cal M}$ and
$\tilde{\cal M}$ are related through
\begin{equation}
  \chi[{\cal M}](y,\bar y)=(-1)^{\hat c}\bar y^{\hat c}\chi[\tilde{\cal
    M}](y,1/\bar y)\,.
\end{equation}
Setting $\bar y=1$ yields the relation for $\chi_y$-genera
\begin{equation}
  \chi_y[{\cal M}]=(-1)^{\hat c}\chi_y[\tilde{\cal M}]\,.
\end{equation}


When the $\hat c$-fold ${\cal M}$ has a correspondence with a
Landau-Ginzburg orbifold $W/\!/G$ one can write down $\chi[{\cal
  M}](y,\bar y)=\pm\chi[W/\!/G](y,\bar y)$ explicitly.
Following the reasoning in \cite{rVafaii}\cite{rIV}\ we find from
(\ref{chiygenus}) that
\begin{eqnarray}
 &&\chi[W/\!/G](y,\bar y)=\frac{(-1)^N}{\vert
    G\vert}\sum_{{\boldsymbol \alpha},{\boldsymbol \beta}\in G}
  \prod_{\stackrel{\scriptstyle i}{\omega_i\alpha_i\not\in{\bf Z}}}
  (y\bar y)^{\frac{1}{2}(1-2\omega_i)}(y/\bar
  y)^{-(\!(\omega_i\alpha_i)\!)}\nonumber\\
  &&\hspace{1.5cm}\times \prod_{\stackrel{\scriptstyle
      i}{\omega_i\alpha_i\in{\bf Z}}} \bfe{\omega_i\beta_i+\frac{1}{2}}
\frac{1-\bfe{(1-\omega_i)\beta_i}(y\bar y)^{1-\omega_i}}
{1-\bfe{\omega_i\beta_i}(y\bar y)^{\omega_i}}\,.
  \label{extdgenuslg}
\end{eqnarray}

We now consider a $3$-fold ${\cal M}=\hat {\cal M}_G$ where ${\cal
  M}_G$ is given by (\ref{fermat}) with $d=5$. Inspecting the tables
for $G_k\ (k=0,1,\ldots,7)$ we see that
$\sum_{\omega_i\alpha_i\not\in{\bf Z}}(\!(\omega_i\alpha_i)\!)=0$
occurs only in the untwisted sectors\footnote{This is in no way a
  generic situation.}. Thus it is clear from (\ref{extdgenuslg}) that
the states with $(q_{\rm L},q_{\rm R})=(1,1)$ corresponding to
$H^{1,2}({\cal M})$, come from the untwisted sectors while the states
with $(q_{\rm L},q_{\rm R})=(1,2)$ corresponding to $H^{1,1}({\cal
  M})$, from the twisted sectors.  Similarly, for its mirror partner
$\tilde {\cal M}$ described as $W/\!/G^*$, the elements of
$H^{1,2}(\tilde {\cal M})$ (or $ H^{1,1}(\tilde {\cal M})$) arise from
the untwisted (or twisted) sectors.  To see more explicitly let us
evaluate (\ref{extdgenuslg}) for $G=G_2$ and $G^*=G_5$.  We obtain
\begin{equation}
  \chi[W/\!/G](y,\bar y)= \chi_u[W/\!/G](y,\bar y)+
  \chi_t[W/\!/G](y,\bar y)\,,
\end{equation}
where $\chi_u$ (or $\chi_t$) stands for the contribution from the
untwisted (or twisted) sectors:
\begin{eqnarray}
 \chi_u[W/\!/G](y,\bar y)&=&1+21y\bar y+21y^2\bar y^2+y^3\bar y^3\,,\\
 \chi_t[W/\!/G](y,\bar y)&=&-y^3-y^2\bar y-y\bar y^2-\bar y^3\,.
\end{eqnarray}
For $W/\!/G^*$ we get
\begin{eqnarray}
 \chi_u[W/\!/G^*](y,\bar y)&=&1+y\bar y+y^2\bar y^2+y^3\bar y^3\,,\\
 \chi_t[W/\!/G^*](y,\bar y)&=&-y^3-21y^2\bar y-21y\bar y^2-\bar y^3\,.
\end{eqnarray}
Hence
\begin{eqnarray}
  \chi_u[W/\!/G](y,\bar y)&=&-\bar y^3\chi_t[W/\!/G^*](y,1/\bar y)\,,\\
  \chi_t[W/\!/G](y,\bar
  y)&=&-\bar y^3\chi_u[W/\!/G^*](y,1/\bar y)\,.
\end{eqnarray}
Upon setting $\bar y=1$ this reduces to the relation for
$\chi_y$-genera we found in sect 4.4. We have checked using
(\ref{extdgenuslg}) that similar results hold for all the examples
given in sect.4.4.  Therefore what we have observed seems to be
natural whenever a mirror pair has a corresponding pair of
Landau-Ginzburg orbifolds.



\newpage
\normalsize
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%\input yitp94table.tex

\footnotesize

\landscape

\begin{table}[tbp]
  \begin{center}
    \leavevmode
\begin{tabular}{lll}\hline
  $(h,d_1,d_2,d_3)$&$W(z_1,z_2,z_3)$&$\chi_y[W] \qquad(t=y^{1/h})$\\
  \hline &&\\ $(12,4,4,3)$&$z_1^3+z_2^3+z_3^4$&$1 + {t^3} + 2\,{t^4} +
  {t^6} + 2\,{t^7} + {t^8} + 2\,{t^{10}} + {t^{11}} + {t^{14}}$\\
  $(13,4,3,5)$&$z_1^2z_3+z_2z_3^2+z_1z_2^3$&$1 + {t^3} + {t^4} + {t^5}
  + {t^6} + {t^87} + {t^8} + {t^9} + {t^{10}} + {t^{11}} + {t^{12}} +
  {t^{15}}$\\ $(15,6,5,3)$&$z_1^2z_3+z_2^3+z_2^5$&$1 + {t^3} + {t^5} +
  2\,{t^6} + {t^8} + {t^9} + 2\,{t^{11}} + {t^{12}} + {t^{14}} +
  {t^{17}}$\\ $(16,5,4,6)$&$z_1^2z_3+z_2z_3^2+z_2^4$&$1 + {t^4} +
  {t^5} + {t^6} + {t^8} + {t^9} + {t^{10}} + {t^{12}} + {t^{13}} +
  {t^{14}} + {t^{18}}$\\ $(16,4,3,8)$&$z_1^4+z_1z_2^4+z_3^2$&$ 1 +
  {t^3} + {t^4} + {t^6} + {t^7} + {t^8} + {t^9} + {t^{10}} + {t^{11}}
  + {t^{12}} + {t^{14}} + {t^{15}} + {t^{18}}$\\
  $(18,7,6,4)$&$z_1^2z_3+z_2^3+z_2z_3^3$&$ 1 + {t^4} + {t^6} + {t^7} +
  {t^8} + {t^{10}} + {t^{12}} + {t^{13}} + {t^{14}} + {t^{16}} +
  {t^{20}}$\\ $(18,5,3,9)$&$z_1^3z_2+z_2^6+z_3^2$&$ 1 + {t^3} + {t^5}
  + {t^6} + {t^8} + {t^9} + {t^{10}} + {t^{11}} + {t^{12}} + {t^{14}}
  + {t^{15}} + {t^{17}} + {t^{20}}$\\
  $(20,5,4,10)$&$z_1^4+z_2^5+z_3^2$&$ 1 + {t^4} + {t^5} + {t^8} +
  {t^9} + {t^{10}} + {t^{12}} + {t^{13}} + {t^{14}} + {t^{17}} +
  {t^{18}} + {t^{22}}$\\ $(22,6,4,11)$&$z_1^3z_2+z_1z_2^4+z_3^2$&$ 1 +
  {t^4} + {t^6} + {t^8} + {t^{10}} + 2\,{t^{12}} + {t^{14}} + {t^{16}}
  + {t^{18}} + {t^{20}} + {t^{24}}$\\
  $(24,9,8,6)$&$z_1^2z_3+z_2^3+z_3^4$&$ 1 + {t^6} + {t^8} + {t^9} +
  {t^{12}} + {t^{14}} + {t^{17}} + {t^{18}} + {t^{20}} + {t^{26}}$\\
  $(24,8,3,12)$&$z_1^3+z_2^8+z_3^2$&$1 + {t^3} + {t^6} + {t^8} + {t^9}
  + {t^{11}} + {t^{12}} + {t^{14}} + {t^{15}} + {t^{17}} + {t^{18}} +
  {t^{20}} + {t^{23}} + {t^{26}}$\\
  $(30,8,6,15)$&$z_1^3z_2+z_2^5+z_3^2$&$1 + {t^6} + {t^8} + {t^{12}} +
  {t^{14}} + {t^{16}} + {t^{18}} + {t^{20}} + {t^{24}} + {t^{26}} +
  {t^{32}}$\\ $(30,10,4,15)$&$z_1^3+z_1z_2^5+z_3^2$&$ 1 + {t^4} +
  {t^8} + {t^{10}} + {t^{12}} + {t^{14}} + {t^{16}} + {t^{18}} +
  {t^{20}} + {t^{22}} + {t^{24}} + {t^{28}} + {t^{32}}$\\
  $(42,14,6,21)$&$z_1^3+z_2^7+z_3^2$&$ 1 + {t^6} + {t^{12}} + {t^{14}}
  + {t^{18}} + {t^{20}} + {t^{24}} + {t^{26}} + {t^{30}} + {t^{32}} +
  {t^{38}} + {t^{44}}$ \\ &&\\ \hline
    \end{tabular}
  \end{center}
  \caption{Exceptional singularities:  $\chi_y$-genera.}
  \label{tab:tabA}
\end{table}

\begin{table}[tbp]
  \begin{center}
    \leavevmode
\begin{tabular}{rl}\hline
$(h,d_1,d_2,d_3)$&$\{-\chi_y^0[W/\!/G_0],-\chi_y^1[W/\!/G_0],\ldots,
-\chi_y^{h-1}[W/\!/G_0]\}$\\
 \hline
&\\
$(12,4,4,3)$ & $\{ 0,{t^{14}},{t^3},2\,{t^4},0,{t^6},2\,{t^7},{t^8},0,2\,%
{t^{10}},{t^{11}},1\}$\\
$(13,4,3,5)$ & $\{ 0,{t^{15}},{t^3},{t^4},{t^5},{t^6},{t^7},{t^8},{t^9},%
{t^{10}},{t^{11}},{t^{12}},1\}$\\
$(15,6,5,3)$ & $\{ 0,{t^{17}},{t^3},0,{t^5},2\,{t^6},0,{t^8},{t^9},0,2\,%
{t^{11}},{t^{12}},0,{t^{14}},1\}$\\
$(16,5,4,6)$ & $\{ 0,{t^{18}},{t^3},{t^4},0,{t^6},{t^7},{t^8},{t^9},%
{t^{10}},{t^{11}},{t^{12}},0,{t^{14}},{t^{15}},1\}$\\
$(16,4,3,8)$ & $\{ 0,{t^{18}},0,{t^4},{t^5},{t^6},0,{t^8},{t^9},%
{t^{10}},0,{t^{12}},{t^{13}},{t^{14}},0,1\}$\\
$(18,7,6,4)$ & $\{ 0,{t^{20}},{t^3},0,{t^5},{t^6},0,{t^8},{t^9},{t^{10}},%
{t^{11}},{t^{12}},0,{t^{14}},{t^{15}},0,{t^{17}},1\}$\\
$(18,5,3,9)$ & $\{ 0,{t^{20}},0,{t^4},0,{t^6},{t^7},{t^8},0,{t^{10}},0,%
{t^{12}},{t^{13}},{t^{14}},0,{t^{16}},0,1\}$\\
$(20,5,4,10)$ & $\{ 0,{t^{22}},0,{t^4},{t^5},0,0,{t^8},{t^9},{t^{10}},0,%
{t^{12}},{t^{13}},{t^{14}},0,0,{t^{17}},{t^{18}},0,1\}$\\
$(22,6,4,11)$ & $\{ 0,{t^{24}},0,{t^4},0,{t^6},0,{t^8},0,{t^{10}},0,2\,%
{t^{12}},0,{t^{14}},0,{t^{16}},0,{t^{18}},0,{t^{20}},0,1\}$\\
$(24,9,8,6)$ & $\{ 0,{t^{26}},{t^3},0,0,{t^6},0,{t^8},{t^9},0,{t^{11}},%
{t^{12}},0,{t^{14}},{t^{15}},0,{t^{17}},{t^{18}},0,{t^{20}},0,0,%
{t^{23}},1\}$\\
$(24,8,3,12)$ & $\{ 0,{t^{26}},0,0,0,{t^6},0,{t^8},{t^9},0,0,{t^{12}},0,%
{t^{14}},0,0,{t^{17}},{t^{18}},0,{t^{20}},0,0,0,1\}$\\
$(30,8,6,15)$ & $\{ 0,{t^{32}},0,{t^4},0,0,0,{t^8},0,{t^{10}},0,{t^{12}},%
0,{t^{14}},0,{t^{16}},0,{t^{18}},0,{t^{20}},0,{t^{22}},0,{t^{24}},0,0,0,%
{t^{28}},0,1\}$\\
$(30,10,4,15)$ & $\{ 0,{t^{32}},0,0,0,{t^6},0,{t^8},0,0,0,{t^{12}},0,%
{t^{14}},0,{t^{16}},0,{t^{18}},0,{t^{20}},0,0,0,{t^{24}},0,%
{t^{26}},0,0,0,1\}$\\
$(42,14,6,21)$ & $\{ 0,{t^{44}},0,0,0,{t^6},0,0,0,0,0,{t^{12}},0,%
{t^{14}},0,0,0,{t^{18}},0,{t^{20}},0,0,0,{t^{24}},0,{t^{26}},0,0,0,%
{t^{30}},0,{t^{32}},0,0,0,0,0,{t^{38}},0,0,0,1\}$\\
&\\
\hline
\end{tabular}
  \end{center}
  \caption{Exceptional singularities:the untwisted and
    twisted sector contributions to
    the Landau-Ginzburg orbifold $\chi_y$-genera.}
  \label{tab:tabB}
\end{table}


\begin{table}[tbp]
  \begin{center}
    \leavevmode
\begin{tabular}{rl}\hline
$(h,d_1,d_2,d_3)$ &
$\{\chi_y^0[W/\!/G_0],\chi_y^1[W/\!/G_0],\ldots,\chi_y^{h-1}
[W/\!/G_0]\}$\\
 \hline
&\\
$(12,4,4,3)$&$
\{ 1 + 10y + {y^2},{y^2},y,2y,0,y,2y,y,0,2y,y,1\}$\\
$(13,4,3,5)$&$
\{ 1 + 10y + {y^2},{y^2},y,y,y,y,y,y,y,y,y,y,1\}$\\
$(15,6,5,3)$&$
\{ 1 + 10y + {y^2},{y^2},y,0,y,2y,0,y,y,0,2y,y,0,y,1\}$\\
$(16,5,4,6)$&$
\{ 1 + 9y + {y^2},{y^2},y,y,0,y,y,y,y,y,y,y,0,y,y,1\}$\\
$(16,4,3,8)$&$
\{ 1 + 11y + {y^2},{y^2},0,y,y,y,0,y,y,y,0,y,y,y,0,1\}$\\
$(18,7,6,4)$&$
\{ 1 + 9y + {y^2},{y^2},y,0,y,y,0,y,y,y,y,y,0,y,y,0,y,1\}$\\
$(18,5,3,9)$&$
\{ 1 + 11y + {y^2},{y^2},0,y,0,y,y,y,0,y,0,y,y,y,0,y,0,1\}$\\
$(20,5,4,10)$&$
\{ 1 + 10y + {y^2},{y^2},0,y,y,0,0,y,y,y,0,y,y,y,0,0,y,y,0,1\}$\\
$(22,6,4,11)$&$
\{ 1 + 10y + {y^2},{y^2},0,y,0,y,0,y,0,y,0,2y,0,y,0,y,0,y,0,y,0,1\}$\\
$(24,9,8,6)$&$
\{ 1 + 8y + {y^2},{y^2},y,0,0,y,0,y,y,0,y,y,0,y,y,0,y,y,0,y,0,0,y,1\}$\\
$(24,8,3,12)$&$
\{ 1 + 12y + {y^2},{y^2},0,0,0,y,0,y,y,0,0,y,0,y,0,0,y,y,0,y,0,0,0,1\}$\\
$(30,8,6,15)$&$
\{ 1 + 9y + {y^2},{y^2},0,y,0,0,0,y,0,y,0,y,0,y,0,y,0,y,0,y,0,
y,0,y,0,0,0,y,0,1\}$\\
$(30,10,4,15)$&$
\{ 1 + 11y + {y^2},{y^2},0,0,0,y,0,y,0,0,0,y,0,y,0,y,0,y,0,y,0,0,0,y,0,
y,0,0,0,1\}$\\
$(42,14,6,21)$&$
\{ 1 + 10y + {y^2},{y^2},0,0,0,y,0,0,0,0,0,y,0,y,0,0,0,y,0,y,0,0,0,y,0,
y,0,0,0,y,0,y,0,0,0,0,0,y,0,0,0,1\}$ \\
&\\
\hline
\end{tabular}
  \end{center}
  \caption{The $K3$ associated with exceptional singularities:
the  untwisted and twisted sector contributions to the Landau-Ginzburg
 orbifold $\chi_y$-genera.}
  \label{tab:tableC}
\end{table}

\end{document}
















