text:
\gad\Sno\ifd@bn@\Fno\z@\Lno\z@\fi\@fn@xt[\s@ct@e\s@cte{\gad\Sno\ifd@bn@\Fno\z@\Lno\z@\fi\@fn@xt[\s@ct@e\s@cte}
\let\Sect\gad\Sno\ifd@bn@\Fno\z@\Lno\z@\fi\@fn@xt[\s@ct@@\s@ct@ \let\sect\gad\Sno\ifd@bn@\Fno\z@\Lno\z@\fi\@fn@xt[\s@ct@e\s@cte
\def\@fn@xt*\subs@ct@\subs@ct{\@fn@xt*\subs@ct@\subs@ct}
\def\subs@ct#1{
 \s@bs@ct\empty{#1}}\wr@@c{\string\subcd{#1}{\the\pageno}}}
\def\subs@ct@*#1{\vsk->\vsk>
 \s@bs@ct\empty{#1}}\wr@@c{\string\subcd{#1}{\the\pageno}}}
\let\subsect\@fn@xt*\subs@ct@\subs@ct \def\Snodef#1{\Sno #1}

\def\l@b@l#1#2{\def\n@@{\csname #2no\endcsname}%
 \if*#1\gad\n@@ \@ft@\xdef\csname @#1@#2@\endcsname{\l@f@rm}\else\def\t@st{#1}%
 \ifx\t@st\empty\gad\n@@ \@ft@\xdef\csname @#1@#2@\endcsname{\l@f@rm}%
 \else\@ft@\ifx\csname @#1@#2@mark\endcsname\relax\gad\n@@
 \@ft@\xdef\csname @#1@#2@\endcsname{\l@f@rm}%
 \@ft@\gdef\csname @#1@#2@mark\endcsname{}%
 \wr@@x{\string\labeldef{#2}\space{\?#1@#2?}\space\ifnum\n@@<10 \space\fi{#1}}%
 \fi\fi\fi}
\def\labeldef#1#2#3{\@ft@\d@f} \def\d@f{\@ft@\def\?#3@#1?{#2}}
\def\Labeldef#1#2#3{\@ft@\d@f} \def\d@f{\@ft@\def\?#3@#1?{#2}\@ft@\gdef\csname @#3@#1@mark\endcsname{}}
\def\Tagdef#1#2{\Labeldef{F}{#1}{#2}}
\def\Lprdef#1#2{\Labeldef{L}{#1}{#2}}
\def\l@bel#1#2{\l@b@l{#1}{#2}\?#1@#2?}

\newcount\c@cite
\def\?#1?{\csname @#1@\endcsname}
\def\[{\@fn@xt:\c@t@sect\c@t@}
\def\c@t@#1]{{\c@cite\z@\@fwd@@{\?#1@L?}{\adv\c@cite1}{}%
 \@fwd@@{\?#1@F?}{\adv\c@cite1}{}\@fwd@@{\?#1?}{\adv\c@cite1}{}%
 \relax\ifnum\c@cite=\z@{\bf ???}\wrs@x{No label [#1]}\else
 \ifnum\c@cite=1\let\@@PS\relax\let\@@@\relax\else\let\@@PS\underbar
 \def\@@@{{\rm<}}\fi\@@PS{\?#1?\@@@\?#1@L?\@@@\?#1@F?}\fi}}
\def\(#1){{\rm(\c@t@#1])}}
\def\c@t@s@ct#1{\@fwd@@{\?#1@S?}{\?#1@S?\relax}%
 {{\bf ???}\wrs@x{No section label {#1}}}}
\def\c@t@sect:#1]{\c@t@s@ct{#1}} \let\SNo\c@t@s@ct

\newdimen\l@ftcd \newdimen\r@ghtcd \let\nlc\relax

\def\d@tt@d{\leaders\hbox to 1em{\kern.1em.\hfil}\hfill}
\def\entcd#1#2#3{\item{\l@bcdf@nt#1}{\entcdf@nt#2}\alb\kern.9em\hbox{}%
 \kern-.9em\d@tt@d\kern-.36em{\p@g@cdf@nt#3}\kern-\r@ghtcd\hbox{}\par}
\def\Entcd#1#2#3{\def\l@@b{\@fwd@@{#1}{\c@l@b{#1}}{}}\vsk.2>%
 \entcd{\l@@b}{#2}{#3}}
\def\subcd#1#2{{\adv\leftskip.333em\entcd{}{\s@bcdf@nt#1}{#2}}}
\def\Refcd#1#2{\def\t@@st{#1}\ifx\t@@st\empty\ifx\r@fl@ne\empty\relax\else
 \Bbb R@fcd{\r@fl@ne}{#2}\fi\else\Bbb R@fcd{#1}{#2}\fi}
\def\Bbb R@fcd#1#2{\sk@@p{.6\bls}\entcd{}{\hglue-\l@ftcd\Bbb R@fcdf@nt #1}{#2}}
\def\def\r@fl@ne} \def\Refempty{\let\r@fl@ne\empty{\def\r@fl@ne} \def\Refempty{\let\r@fl@ne\empty}
\def\Appencd{\par\adv\leftskip-\l@ftcd\adv\rightskip-\r@ghtcd\@ppl@ne
 \adv\leftskip\l@ftcd\adv\rightskip\r@ghtcd\let\Entcd\Appcd}
\def\def\@ppl@ne} \def\Appempty{\let\@ppl@ne\empty{\def\@ppl@ne} \def\Appempty{\let\@ppl@ne\empty}
\def\Appline#1{\def\@ppl@ne{\s@bs@ct{}{#1}}}
\def\leftcd#1{\adv\leftskip-\l@ftcd\s@twd@\l@ftcd{\c@l@b{#1}\enspace}
 \adv\leftskip\l@ftcd}
\def\rightcd#1{\adv\rightskip-\r@ghtcd\s@twd@\r@ghtcd{#1\enspace}
 \adv\rightskip\r@ghtcd}
\def\Bbb C@nt{Contents} \def\Ap@s{Appendices} \def\Bbb R@fcs{References}
\def\@fn@xt*\cont@@\cont@{\@fn@xt*\cont@@\cont@}
\def\cont@{\@fn@xt[\cnt@{\cnt@[\Bbb C@nt]}}
\def\cont@@*{\@fn@xt[\cnt@@{\cnt@@[\Bbb C@nt]}}
\def\cnt@[#1]{\c@nt@{M}{#1}{44}{\s@bs@ct{}{\@ppl@f@nt\Ap@s}}}
\def\cnt@@[#1]{\c@nt@{M}{#1}{44}{}}
\def\par\penalty-500\vsk>\vskip\z@\endgroup{\par\penalty-500\vsk>\vskip\z@\endgroup}
\def\@np@t{\jobname.cd}{\@np@t{\jobname.cd}}
\def\@fn@xt*\Cde@@\Cde@{\@fn@xt*\@fn@xt*\Cde@@\Cde@@@\@fn@xt*\Cde@@\Cde@@}
\def\@fn@xt*\Cde@@\Cde@@{\@fn@xt[\Cd@{\Cd@[\Bbb C@nt]}}
\def\@fn@xt*\Cde@@\Cde@@@*{\@fn@xt[\Cd@@{\Cd@@[\Bbb C@nt]}}
\def\Cd@[#1]{\cnt@[#1]\@np@t{\jobname.cd}\par\penalty-500\vsk>\vskip\z@\endgroup}
\def\Cd@@[#1]{\cnt@@[#1]\@np@t{\jobname.cd}\par\penalty-500\vsk>\vskip\z@\endgroup}
\def\def\c@l@b{\def\c@l@b}

\long\def\c@nt@#1#2#3#4{\s@twd@\l@ftcd{\c@l@b{#1}\enspace}
 \s@twd@\r@ghtcd{#3\enspace}\adv\r@ghtcd1.333em
 \def\@ppl@ne{#4}\def\r@fl@ne{\Bbb R@fcs}\s@ct{}{#2}\B@gr@\parindent\z@\let\nlc\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh
 \let\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\relax\parskip.2\bls\adv\leftskip\l@ftcd\adv\rightskip\r@ghtcd}

\def\immediate\openout\@@cd\jobname.cd \def\wr@@c{\write\@@cd{\immediate\openout\@@cd\jobname.cd \def\wr@@c{\write\@@cd}
 \def\cl@secd{\immediate\write\@@cd{\string\endinput}\immediate\closeout\@@cd}
 \def\cl@secd\global\let\cl@secd\relax}{\cl@secd\global\let\cl@secd\relax}}
\let\cl@secd\relax \def\wr@@c#1{} \let\cl@secd\global\let\cl@secd\relax}\relax

\def\@ft@\d@f} \def\d@f{\@ft@\def{\@ft@\d@f} \def\d@f{\@ft@\def}
\def\@ft@\ed@f} \def\ed@f{\@ft@\edef{\@ft@\cl@s@\ifed@s\@p@n\z@\else\ed@true\@p@n\ed@b@x\fi@f} \def\cl@s@\ifed@s\@p@n\z@\else\ed@true\@p@n\ed@b@x\fi@f{\@ft@\edef}
\def\defi#1#2{\def#1{#2}\wr@@x{\string\def\string#1{#2}}}

\def\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}{\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}}
\def\back#1 {\strut\kern-.33em #1\enspace\ignore}
\def\Text#1{\crcr\noalign{\alb\vsk>\normalbaselines\vsk->\vbox{\nt #1\strut}%
 \nobreak\nointerlineskip\vbox{\strut}\nobreak\vsk->\nobreak}}
\def\inText#1{\crcr\noalign{\penalty\postdisplaypenalty\vskip\belowdisplayskip
 \vbox{\normalbaselines\noindent#1}\penalty\predisplaypenalty
 \vskip\abovedisplayskip}}

\def\hcor#1{\advance\hoffset by #1}
\def\vcor#1{\advance\voffset by #1}
\let\bls\baselineskip \let\ignore\ignorespaces
\ifx\ic@\undefined \let\ic@\/\fi
\def\vsk#1>{\vskip#1\bls} \let\adv\advance
\def\vv#1>{\vadjust{\vsk#1>}\ignore}
\def\vvn#1>{\vadjust{\nobreak\vsk#1>\nobreak}\ignore}
\def\vvv#1>{\vskip\z@\vsk#1>\nt\ignore}
\def\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore{\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore}
\def\vsk.5>} \def\setparindent{\edef\Parindent{\the\parindent}{\vsk.5>} \def\setparindent{\edef\Parindent{\the\parindent}}
\def\Type{\vsk.5>\bgroup\parindent\z@\tt\rightskip\z@ plus1em minus1em%
 \spaceskip.3333em \xspaceskip.5em\relax}
\def\vsk.5>\egroup\nt} \def\Comment{\underbar{Comment}{\vsk.5>\egroup\nt} \def\Comment{\underbar{Comment}}

\let\Hat\widehat \let\Tilde\widetilde \let\dollar\$ \let\ampersand\&
\let\sss\scriptscriptstyle \let\hide\hidewidth \let\neq\undefined
\let\vp\vphantom \let\hp\hphantom \let\nt\noindent
\let\cline\centerline \let\lline\leftline \let\rline\rightline
\def\nn#1>{\noalign{\vskip#1\p@@}} \def\NN#1>{\openup#1\p@@}
\def\cnn#1>{\noalign{\vsk#1>}}
\def\bigcup\limits} \def\Cap{\bigcap\limits{\bigcup\limits} \def\Cap{\bigcap\limits}
\let\Lim\Lim\limits} \let\Sum\sum \def\sum{\Sum\limits \def\Lim\limits} \let\Sum\sum \def\sum{\Sum\limits{\Lim\limits} \let\Sum\sum \def\sum{\Sum\limits}
\def\bigoplus\limits} \def\Wedge{\bigwedge\nolimits{\bigoplus\limits} \def\Wedge{\bigwedge\nolimits}
\let\Prod\Prod\limits} \let\Int\int \def\int{\Int\limits \def\Prod\limits} \let\Int\int \def\int{\Int\limits{\Prod\limits} \let\Int\int \def\int{\Int\limits}
\def\bigotimes\limits{\bigotimes\limits}
\def\mathop{\tsize\Sum}\limits} \def\ssum{\mathop{\ssize\Sum}\limits{\mathop{\tsize\Sum}\limits} \def\ssum{\mathop{\ssize\Sum}\limits}
\def\mathop{\tsize\Prod}\limits{\mathop{\tsize\Prod}\limits}
\def\&{.\kern.1em} \def\>{{\!\;}} \def\]{{\!\!\;}} \def\){\>\]} \def\}{\]\]}
\def\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh{\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh}
\def\m@n@s~{\raise.15ex\mbox{-}} \def\@md@sh{\raise.13ex\hbox{--}}
\let\procent\% \def\%#1{\ifmmode\mathop{#1}\limits\else\procent#1\fi}
\let\@ml@t\" \def\"#1{\ifmmode ^{(#1)}\else\@ml@t#1\fi}
\let\@c@t@\' \def\'#1{\ifmmode _{(#1)}\else\@c@t@#1\fi}
\let\colon\: \def\:{^{\vp|}}

\let\texspace\ \def\ {\ifmmode\alb\fi\texspace}

\let\n@wp@ge\endgraf\n@wp@ge \def\endgraf\n@wp@ge{\endgraf\n@wp@ge}
\let\=\m@th \def\mbox#1{\hbox{\m@th$#1$}}
\def\mtext#1{\text{\m@th$#1$}} \def\^#1{\text{\m@th#1}}
\def\Line#1{\kern-.5\hsize\line{\m@th$\dsize#1$}\kern-.5\hsize}
\def\Lline#1{\kern-.5\hsize\lline{\m@th$\dsize#1$}\kern-.5\hsize}
\def\Cline#1{\kern-.5\hsize\cline{\m@th$\dsize#1$}\kern-.5\hsize}
\def\Rline#1{\kern-.5\hsize\rline{\m@th$\dsize#1$}\kern-.5\hsize}

\def\Ll@p#1{\llap{\m@th$#1$}} \def\Rl@p#1{\rlap{\m@th$#1$}}
\def\clap#1{\llap{#1\hss}} \def\C^{\>\ell}@p#1{\llap{\m@th$#1$\hss}}
\def\Llap#1{\mathchoice{\Ll@p{\dsize#1}}{\Ll@p{\tsize#1}}{\Ll@p{\ssize#1}}%
 {\Ll@p{\sss#1}}}
\def\Clap#1{\mathchoice{\C^{\>\ell}@p{\dsize#1}}{\C^{\>\ell}@p{\tsize#1}}{\C^{\>\ell}@p{\ssize#1}}%
 {\C^{\>\ell}@p{\sss#1}}}
\def\Rlap#1{\mathchoice{\Rl@p{\dsize#1}}{\Rl@p{\tsize#1}}{\Rl@p{\ssize#1}}%
 {\Rl@p{\sss#1}}}
\def\lrap#1#2{\llap{#1}\rlap{#2}} \def\LRap#1#2{\Llap{#1}\Rlap{#2}}
\def\LRtph#1#2{\setbox\z@\hbox{#1}\dimen\z@\wd\z@\hbox{\hbox to\dimen\z@{#2}}}
\def\LRph#1#2{\LRtph{\m@th$#1$}{\m@th$#2$}}
\def\Ctph#1#2{\LRtph{#1}{\hss#2\hss}}
\def\Ltph#1#2{\LRtph{#1}{\hss#2}} \def\Rtph#1#2{\LRtph{#1}{#2\hss}}
\def\CCph#1#2{\LRph{#1}{\hss#2\hss}}
\def\LLph#1#2{\LRph{#1}{\hss#2}} \def\RRph#1#2{\LRph{#1}{#2\hss}}
\def\Cph#1#2{\mathchoice{\CCph{\dsize#1}{\dsize#2}}{\CCph{\tsize#1}{\tsize#2}}
 {\CCph{\ssize#1}{\ssize#2}}{\CCph{\sss#1}{\sss#2}}}
\def\Lph#1#2{\mathchoice{\LLph{\dsize#1}{\dsize#2}}{\LLph{\tsize#1}{\tsize#2}}
 {\LLph{\ssize#1}{\ssize#2}}{\LLph{\sss#1}{\sss#2}}}
\def\Rph#1#2{\mathchoice{\RRph{\dsize#1}{\dsize#2}}{\RRph{\tsize#1}{\tsize#2}}
 {\RRph{\ssize#1}{\ssize#2}}{\RRph{\sss#1}{\sss#2}}}
\def\Lto#1{\setbox\z@\mbox{\tsize{#1}}%
 \mathrel{\mathop{\hbox to\wd\z@{\rightarrowfill}}\limits#1}}
\def\Lgets#1{\setbox\z@\mbox{\tsize{#1}}%
 \mathrel{\mathop{\hbox to\wd\z@{\leftarrowfill}}\limits#1}}
\def\vpb#1{{\vp{\big@\@ne(}}^{\]#1}} \def\vpp#1{{\vp{\big@\@ne]}}_{#1}}
\def\mathopen{[\![}} \def\rbc{\mathclose{]\!]}{\mathopen{[\![}} \def\rbc{\mathclose{]\!]}}
\def\mathopen{\big[\!\}\big[}} \def\Rbc{\mathclose{\big]\!\}\big]}{\mathopen{\big@\@ne[\!\}\big@\@ne[}} \def\Rbc{\mathclose{\big@\@ne]\!\}\big@\@ne]}}
\def\mathopen{\Big[\!\}\Big[}} \def\RBc{\mathclose{\Big]\!\}\Big]}{\mathopen{\big@{1.5}[\!\}\big@{1.5}[}} \def\RBc{\mathclose{\big@{1.5}]\!\}\big@{1.5}]}}
\def\mathopen{(\!(}} \def\rpt{\mathclose{)\!)}{\mathopen{(\!(}} \def\rpt{\mathclose{)\!)}}
\def\mathopen{\big(\!\}\big(}} \def\Rpt{\mathclose{\big)\!\}\big)}{\mathopen{\big@\@ne(\!\}\big@\@ne(}} \def\Rpt{\mathclose{\big@\@ne)\!\}\big@\@ne)}}
\def\mathopen{\Big(\!\}\Big(}} \def\RPt{\mathclose{\Big)\!\}\Big)}{\mathopen{\big@{1.5}(\!\}\big@{1.5}(}} \def\RPt{\mathclose{\big@{1.5})\!\}\big@{1.5})}}

\let\alb\allowbreak \def\hfil\alb\hfilneg{\hfil\alb\hfilneg}
\def\ald{\noalign{\alb}} \let\alds\allowdisplaybreaks

\let\o\circ \let\x\times \let\ox\otimes \let\plus\dotplus
\let\sub\subset \let\sup\supset \let\tabs\+
\let\le\leqslant \let\ge\geqslant
\let\der\partial \let\8\infty \let\*\star
\let\bra\langle \let\ket\rangle
\let\lto\longrightarrow \let\To\Rightarrow
\let\map\mapsto \let\lmap\longmapsto \let\hto\hookrightarrow
\let\down\downarrow \let\rto\rightrightarrows
\let\Vert\parallel \def\ |\ } \def\nin{\not\in{\ |\ } \def\nin{\not\in}
\let\Empty\varnothing \let\mstrut\mathstrut
\let\Rgoth\mathop{\roman{Re}\>}} \def\Im{\mathop{\roman{Im}\>} \let\Igoth\Im
\let\lb\lbrace \let\rb\rbrace
\let\lf\lfloor \let\rf\rfloor
\let\trileft\triangleleft \let\tright\triangleright

\def\lsym#1{#1\alb\ldots\relax#1\alb}
\def\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox{\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox}
\def\,,\alb\ {\ldots\ ,}\alb\ {\,,\alb\ {\ldots\ ,}\alb\ }
\def\mathop{\roman{Re}\>}} \def\Im{\mathop{\roman{Im}\>}{\mathop{\roman{Re}\>}} \def\Im{\mathop{\roman{Im}\>}}
\def\mathop{\roman{End}\>}} \def\Hom{\mathop{\roman{Hom}\>}{\mathop{\roman{End}\>}} \def\Hom{\mathop{\roman{Hom}\>}}
\def\mathop{\roman{im}\>}} \def\Sym{\mathop{\roman{Sym}\)}{\mathop{\roman{im}\>}} \def\Sym{\mathop{\roman{Sym}\)}}
\def\mathop{\roman{ker}\>}} \def\Ker{\mathop{\roman{Ker}\>}{\mathop{\roman{ker}\>}} \def\Ker{\mathop{\roman{Ker}\>}}
\def\mathop{\roman{Sing}\>}} \def\Span{\mathop{\roman{Span}\>}{\mathop{\roman{Sing}\>}} \def\Span{\mathop{\roman{Span}\>}}
\def\mathop{\roman{tr}}\nolimits} \def\Tr{\mathop{\roman{Tr}}\nolimits{\mathop{\roman{tr}}\nolimits} \def\Tr{\mathop{\roman{Tr}}\nolimits}
\def\mathop{\roman{Res}\>}\limits{\mathop{\roman{Res}\>}\limits}
\def\mathop{\roman{res}\>}\limits{\mathop{\roman{res}\>}\limits}
\def\mathop{\roman{sign}\)}\limits{\mathop{\roman{sign}\)}\limits}
\def\mathop{\roman{sgn}\)}\limits{\mathop{\roman{sgn}\)}\limits}
\def\det\nolimits_q\!} \def\Ent{\mathop{\italic{Entier}\>}{\det\nolimits_q\!} \def\Ent{\mathop{\italic{Entier}\>}}
\def\roman{id}} \def\const{\roman{const}} \def\for{\text{for }\,{\roman{id}} \def\const{\roman{const}} \def\for{\text{for }\,}
\def^{-1}} \def\_#1{_{\Rlap{#1}}{^{-1}} \def\_#1{_{\Rlap{#1}}}
\def\vst#1{{\lower1.9\p@@\mbox{\bigr|_{\raise.5\p@@\mbox{\ssize#1}}}}}
\def\vrp#1:#2>{{\vrule height#1 depth#2 width\z@}}
\def\vru#1>{\vrp#1:\z@>} \def\vrd#1>{\vrp\z@:#1>}
\def\qquad\quad} \def\respace{\hglue-.25em\ignore{\qquad\quad} \def\respace{\hglue-.25em\ignore}
\def\sscr#1{\raise.3ex\mbox{\sss#1}} \def\@@PS{\bold{OOPS!!!}}
\def\bigl(1+o(1)\bigr)} \def\Ono{\bigl(1+O(u^\tau)\bigr){\bigl(1+o(1)\bigr)} \def\Ono{\bigl(1+O(u^\tau)\bigr)}

\def\intcl{\mathop
 {\Rlap{\raise.3ex\mbox{\kern.12em\curvearrowleft}}\int}\limits}
\def\intcr{\mathop
 {\Rlap{\raise.3ex\mbox{\kern.24em\curvearrowright}}\int}\limits}

\def\arright#1{{\vp{#1}\smash{\%{#1}^{\Clap\to}}}}
\def\arleft#1{{\vp{#1}\smash{\%{#1}^{\Clap\gets}}}}

\def\raise.25ex\mbox{\ssize\pm}\>{\raise.25ex\mbox{\ssize\pm}\>}
\def\raise.25ex\mbox{\ssize\mp}\>{\raise.25ex\mbox{\ssize\mp}\>}
\def{\sscr+}} \def\mss{{\sscr-}{{\sscr+}} \def\mss{{\sscr-}}
\def{\sscr\pm}} \def\mpss{{\sscr\mp}{{\sscr\pm}} \def\mpss{{\sscr\mp}}

\let\al\alpha
\let\bt\beta
\let\gm\gamma \let\Gm\Gamma \let\Gms\varGamma
\let\dl\delta \let\Dl\Delta \let\Dlt\varDelta
\let\epe\epsilon \let\eps\varepsilon \let\epsilon\eps
\let\Xj\varXi
\let\zt\zeta
\let\tht\theta \let\Tht\Theta
\let\thi\vartheta \let\Tho\varTheta
\let\io\iota
\let\ka\kappa
\let\la\lambda \let\La\Lambda
\let\rhe\varrho
\let\si\sigma \let\Si\Sigma
\let\sig\varsigma \let\Sig\varSigma
\let\pho\phi \let\phi\varphi \let\Pho\varPhi
\let\Pss\varPsi
\let\ups\upsilon
\let\om\omega \let\Om\Omega \let\Ome\varOmega

\def\Bbb C{\Bbb C}
\def\Bbb R{\Bbb R}
\def\Bbb Z{\Bbb Z}

\def\Bbb A{\Bbb A}
\def\Bbb B{\Bbb B}
\def\Bbb D{\Bbb D}
\def\Bbb F{\Bbb F}
\def\Bbb I{\Bbb I}
\def\Bbb S{\Bbb S}
\def\Bbb T{\Bbb T}
\def\Bbb U{\Bbb U}
\def\Bbb X{\Bbb X}
\def\Bbb Y{\Bbb Y}

\def\Z_{\ge 0}} \def\Zpp{\Z_{>0}{\Bbb Z_{\ge 0}} \def\Zpp{\Bbb Z_{>0}}
\def\Z_{\le 0}} \def\Znn{\Z_{<0}{\Bbb Z_{\le 0}} \def\Znn{\Bbb Z_{<0}}

\def\difl/{differential} \def\dif/{difference}
\def\cf.{cf.\ \ignore} \def\Cf.{Cf.\ \ignore}
\def\egv/{eigenvector} \def\eva/{eigenvalue} \def\eq/{equation}
\def\lhs/{the left hand side} \def\rhs/{the right hand side}
\def\Lhs/{The left hand side} \def\Rhs/{The right hand side}
\def\gby/{generated by} \def\wrt/{with respect to} \def\st/{such that}
\def\resp/{respectively} \def\off/{offdiagonal} \def\wt/{weight}
\def\pol/{polynomial} \def\rat/{rational} \def\tri/{trigonometric}
\def\fn/{function} \def\var/{variable} \def\raf/{\rat/ \fn/}
\def\inv/{invariant} \def\hol/{holomorphic} \def\hof/{\hol/ \fn/}
\def\mer/{meromorphic} \def\mef/{\mer/ \fn/} \def\mult/{multiplicity}
\def\sym/{symmetric} \def\perm/{permutation} \def\fd/{finite-dimensional}
\def\rep/{representation} \def\irr/{irreducible} \def\irrep/{\irr/ \rep/}
\def\hom/{homomorphism} \def\aut/{automorphism} \def\iso/{isomorphism}
\def\lex/{lexicographical} \def\as/{asymptotic} \def\asex/{\as/ expansion}
\def\ndeg/{nondegenerate} \def\neib/{neighbourhood} \def\deq/{\dif/ \eq/}
\def\hw/{highest \wt/} \def\gv/{generating vector} \def\eqv/{equivalent}
\def\msd/{method of steepest descend} \def\pd/{pairwise distinct}
\def\wlg/{without loss of generality} \def\Wlg/{Without loss of generality}
\def\onedim/{one-dimensional} \def\qcl/{quasiclassical}
\def\hgeom/{hyper\-geometric} \def\hint/{\hgeom/ integral}
\def\hwm/{\hw/ module} \def\emod/{evaluation module} \def\Vmod/{Verma module}
\def\symg/{\sym/ group} \def\sol/{solution} \def\eval/{evaluation}
\def\anf/{analytic \fn/} \def\anco/{analytic continuation}
\def\qg/{quantum group} \def\qaff/{quantum affine algebra}

\def\Rm/{\^{$R$-}matrix} \def\Rms/{\^{$R$-}matrices} \def\YB/{Yang-Baxter \eq/}
\def\Ba/{Bethe ansatz} \def\Bv/{Bethe vector} \def\Bae/{\Ba/ \eq/}
\def\KZv/{Knizh\-nik-Zamo\-lod\-chi\-kov} \def\KZvB/{\KZv/-Bernard}
\def\KZ/{{\sl KZ\/}} \def\qKZ/{{\sl qKZ\/}}
\def\KZB/{{\sl KZB\/}} \def\qKZB/{{\sl qKZB\/}}
\def\qKZo/{\qKZ/ operator} \def\qKZc/{\qKZ/ connection}
\def\KZe/{\KZ/ \eq/} \def\qKZe/{\qKZ/ \eq/} \def\qKZBe/{\qKZB/ \eq/}

\def\h@ph{\discretionary{}{}{-}} \def\$#1$-{\,\^{$#1$}\h@ph}

\def\TFT/{Research Insitute for Theoretical Physics}
\def\HY/{University of Helsinki} \def\AoF/{the Academy of Finland}
\def\CNRS/{Supported in part by MAE\~MICECO\~CNRS Fellowship}
\def\LPT/{Laboratoire de Physique Th\'eorique ENSLAPP}
\def\ENSLyon/{\'Ecole Normale Sup\'erieure de Lyon}
\def\LPTaddr/{46, All\'ee d'Italie, 69364 Lyon Cedex 07, France}
\def\enslapp/{URA 14\~36 du CNRS, associ\'ee \`a l'E.N.S.\ de Lyon,
au LAPP d'Annecy et \`a l'Universit\`e de Savoie}
\def\ensemail/{vtarasov\@ enslapp.ens-lyon.fr}
\def\DMS/{Department of Mathematics, Faculty of Science}
\def\DMO/{\DMS/, Osaka University}
\def\DMOaddr/{Toyonaka, Osaka 560, Japan}
\def\dmoemail/{vt\@ math.sci.osaka-u.ac.jp}
\def\SPb/{St\&Peters\-burg}
\def\home/{\SPb/ Branch of Steklov Mathematical Institute}
\def\homeaddr/{Fontanka 27, \SPb/ \,191011, Russia}
\def\homemail/{vt\@ pdmi.ras.ru}
\def\absence/{On leave of absence from \home/}
\def\UNC/{Department of Mathematics, University of North Carolina}
\def\ChH/{Chapel Hill}
\def\UNCaddr/{\ChH/, NC 27599, USA} \def\avemail/{av\@ math.unc.edu}
\def\grant/{NSF grant DMS\~9501290}
\def\Grant/{Supported in part by \grant/}

\def\Aomoto/{K\&Aomoto}
\def\Dri/{V\]\&G\&Drin\-feld}
\def\Fadd/{L\&D\&Fad\-deev}
\def\Feld/{G\&Felder}
\def\Fre/{I\&B\&Fren\-kel}
\def\Gustaf/{R\&A\&Gustafson}
\def\Kazh/{D\&Kazhdan} \def\Kir/{A\&N\&Kiril\-lov}
\def\Kor/{V\]\&E\&Kore\-pin}
\def\Lusz/{G\&Lusztig}
\def\MN/{M\&Naza\-rov}
\def\Resh/{N\&Reshe\-ti\-khin} \def\Reshy/{N\&\]Yu\&Reshe\-ti\-khin}
\def\SchV/{V\]\&\]V\]\&Schecht\-man} \def\Sch/{V\]\&Schecht\-man}
\def\Skl/{E\&K\&Sklya\-nin}
\def\Smirn/{F\]\&Smirnov} \def\Smirnov/{F\]\&A\&Smirnov}
\def\Takh/{L\&A\&Takh\-tajan}
\def\VT/{V\]\&Ta\-ra\-sov} \def\VoT/{V\]\&O\&Ta\-ra\-sov}
\def\Varch/{A\&\]Var\-chenko} \def\Varn/{A\&N\&\]Var\-chenko}

\def\AMS/{Amer.\ Math.\ Society}
\def\CMP/{Comm.\ Math.\ Phys.{}}
\def\DMJ/{Duke.\ Math.\ J.{}}
\def\Inv/{Invent.\ Math.{}}
\def\IMRN/{Int.\ Math.\ Res.\ Notices}
\def\JPA/{J.\ Phys.\ A{}}
\def\JSM/{J.\ Soviet\ Math.{}}
\def\LMP/{Lett.\ Math.\ Phys.{}}
\def\LMJ/{Leningrad Math.\ J.{}}
\def\LpMJ/{\SPb/ Math.\ J.{}}
\def\SIAM/{SIAM J.\ Math.\ Anal.{}}
\def\SMNS/{Selecta Math., New Series}
\def\TMP/{Theor.\ Math.\ Phys.{}}
\def\ZNS/{Zap.\ nauch.\ semin. LOMI}

\def\ASMP/{Advanced Series in Math.\ Phys.{}}

\def\AMSa/{AMS \publaddr Providence}
\def\Birk/{Birkh\"auser}
\def\CUP/{Cambridge University Press} \def\CUPa/{\CUP/ \publaddr Cambridge}
\def\Spri/{Springer-Verlag} \def\Spria/{\Spri/ \publaddr Berlin}
\def\WS/{World Scientific} \def\WSa/{\WS/ \publaddr Singapore}

\newbox\lefthbox \newbox\righthbox

\let\sectsep. \let\labelsep. \let\contsep. \let\labelspace\relax
\let\sectpre\relax \let\contpre\relax
\def\sf@rm{\the\Sno} \def\sf@rm@{\sectpre\sf@rm\sectsep}
\def\c@l@b#1{\contpre#1\contsep}
\def\l@f@rm{\ifd@bn@\sf@rm\labelsep\fi\labelspace\the\n@@}

\def\def\sf@rm{\def\sf@rm}

\let\DoubleNum\d@bn@true \let\SingleNum\d@bn@false

\def\NoNewNum{\let\def\wr@@x{\immediate\write\@@x\relax\def\l@b@l##1##2{\if*##1%
 \@ft@\xdef\csname @##1@##2@\endcsname{\mbox{*{*}*}}\fi}}
\def\def\todaydef##1{\def\today{##1}{\def\todaydef##1{\def\tod@y\wr@@x{\string\todaydef{\tod@y}}{##1}}
 \def\nowtimedef##1{\def##1}}{##1}}}
\def\let\Input\input\let\writeldf\relax{\let\Input\input\let\def\wr@@x{\immediate\write\@@x\relax}
\def\NoNewTime\NoInput{\def\todaydef##1{\def\today{##1}\let\Input\input\let\writeldf\relax}

\def\DateTimeLabel{\vp1\vsk->{\ifMag\eightpoint\fi\vsk-3>
 \rline{\sl\tod@y\wr@@x{\string\todaydef{\tod@y}}}\rline{\sl##1}}}\vsk>}}

\def\sectfont#1{\def\s@cf@nt{#1}} \sectfont\bf
\def\subsectfont#1{\def\s@bf@nt{#1}} \subsectfont\it
\def\Entcdfont#1{\def\entcdf@nt{#1}} \Entcdfont\relax
\def\labelcdfont#1{\def\l@bcdf@nt{#1}} \labelcdfont\relax
\def\pagecdfont#1{\def\p@g@cdf@nt{#1}} \pagecdfont\relax
\def\subcdfont#1{\def\s@bcdf@nt{#1}} \subcdfont\it
\def\applefont#1{\def\@ppl@f@nt{#1}} \applefont\bf
\def\Refcdfont#1{\def\Bbb R@fcdf@nt{#1}} \Refcdfont\bf

\tenpoint

\csname beta.def\endcsname

\NoNewTime\NoInput

\expandafter\ifx\csname ident.def\endcsname\relax \else\endinput\fi
\expandafter\edef\csname ident.def\endcsname{%
 \catcode`\noexpand\@=\the\catcode`\@\space}

\catcode`\@=11

\font@\Beufm=eufm10 scaled 1440

\newif\ifMPIM

\def\Th#1{\pr@cl{Theorem \l@L{#1}}\ignore}
\def\Lm#1{\pr@cl{Lemma \l@L{#1}}\ignore}
\def\Cr#1{\pr@cl{Corollary \l@L{#1}}\ignore}
\def\Df#1{\pr@cl{Definition \l@L{#1}}\ignore}
\def\Cj#1{\pr@cl{Conjecture \l@L{#1}}\ignore}
\def\Prop#1{\pr@cl{Proposition \l@L{#1}}\ignore}

\def\Bbb E{\Bbb E}
\def\bold S{\bold S}
\def\Cal E{\Cal E}
\def\Cal P{\Cal P}
\def\frak l{\frak l}

\def\tilde v{\tilde v}

\def\C^{\>\ell}{\Bbb C^{\>\ell}}
\defq^{-H}} \def\q{q\1{q^{-H}} \def\q{q^{-1}} \def\_#1{_{\Rlap{#1}}}
\defA^{\sscr{e\]l\}l}}{A^{\sscr{e\]l\}l}}}

\def\Sb_\ell{\bold S_\ell}
\def\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}{\Cal P_{\ell,n}} \def\Plij{\Cal P_\ell^{i,j}}
\def\sum_{\,\si\in\Sl\!}{\sum_{\,\si\in\Sb_\ell\!}}
\def\sum_{\,\la\in\Pln\!}{\sum_{\,\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}\!}}
\def\sum_{\,\mu\in\Pln\!}{\sum_{\,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}\!}}

\def\l@inf{\lower.21ex\mbox{\ssize\8}} \def_{\kern-.02em\l@inf}{_{\kern-.02em\l@inf}}
\deft^{\sss\star}} \def\dtt{(dt/t)^\ell{t^{\sss\star}} \def\dtt{(dt/t)^\ell}
\def_{\]S}\:} \def\Of{_\Om\:{_{\]S}\:} \def\Of{_\Om\:}

\def\tsum_{m=1}^n} \def\tpron{\tprod_{m=1}^n{\mathop{\tsize\Sum}\limits} \def\ssum{\mathop{\ssize\Sum}\limits_{m=1}^n} \def\tpron{\mathop{\tsize\Prod}\limits_{m=1}^n}
\def\sum_{m=1}^n} \def\pron{\prod_{m=1}^n} \def\prmn{\prod_{m=1}^{n-1}{\sum_{m=1}^n} \def\pron{\Prod\limits} \let\Int\int \def\int{\Int\limits_{m=1}^n} \def\prmn{\Prod\limits} \let\Int\int \def\int{\Int\limits_{m=1}^{n-1}}
\def\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell{\Prod\limits} \let\Int\int \def\int{\Int\limits_{a=1}^\ell} \def\tpral{\mathop{\tsize\Prod}\limits_{a=1}^\ell}
\def\prod_{1\le a<b\le\ell}{\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le a<b\le\ell}}
\def\prod_{1\le j<m}} \def\tprjm{\!\tprod_{1\le j<m}\!{\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<m}} \def\tprjm{\!\mathop{\tsize\Prod}\limits_{1\le j<m}\!}
\def\prod_{s=0}^{\ell-1}} \def\prsl{\prod_{s=1-\ell}^{\ell-1}{\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\ell-1}} \def\prsl{\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1-\ell}^{\ell-1}}

\defm=1\lc n} \def\aell{a=1\lc\ell{m=1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox n} \def\aell{a=1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\ell}
\defx_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n{x_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_n} \def\yn{y_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox y_n} \def\zn{z_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox z_n}
\deft_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}{t_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox t_\ell} \def\twll{t_2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox t_\ell} \def\telm{t_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox t_{\ell-1}}
\defx\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}{x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}}
\def\mathop{\xt\]\italic{Res}\)}\limits{\mathop{x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\]\italic{Res}\)}\limits}
\def\mathop{\yt\]\italic{Res}\)}\limits{\mathop{\yt\]\italic{Res}\)}\limits}

\defv_1\lox v_n} \def\voxx{v_n\lox v_1{v_1\lox v_n} \def\voxx{v_n\lox v_1}
\defV_1(z_1)\lox V_n(z_n)} \def\Vozz{V_n(z_n)\lox V_1(z_1){V_1(z_1)\lox V_n(z_n)} \def\Vozz{V_n(z_n)\lox V_1(z_1)}
\defF\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n{F\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n}

\def\frak{sl}_2} \def\gl{\frak{gl}_2{\frak{sl}_2} \def\gl{\frak{gl}_2}
\defU_q(\gsl)} \def\Uqh{U_q(\Tilde{\gl}){U_q(\frak{sl}_2} \def\gl{\frak{gl}_2)} \def\Uqh{U_q(\Tilde{\gl})}
\defE_{\rho,\gm}(\gsl){E_{\rho,\gm}(\frak{sl}_2} \def\gl{\frak{gl}_2)}

\def\indet/{indeterminate}
\def\qlo/{quantum loop algebra} \def\eqg/{elliptic \qg/}
\def\Umod/{\$U_q(\gsl)} \def\Uqh{U_q(\Tilde{\gl})$-module} \def\Uhmod/{\$\Uqh$-module}
\def\Emod/{\$E_{\rho,\gm}(\gsl)$-module} \def\p-{\^{$\,p\)$}-}

\let\goodbm\relax \let\mgood\relax \let\mmgood\relax
\let\goodast\relax \let\agood\relax \let\aagood\relax
\let\nline\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh \def\vvm#1>{\relax} \let\qqm\qquad\quad} \def\respace{\hglue-.25em\ignore

\ifMag\let\goodbm\goodbreak \let\mgood\vgood \let\mmgood\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore
 \let\goodbreak\relax \let\vgood\relax \let\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore\relax \let\mline\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh
 \let\nline\relax \let\vvm\vv \let\qqm\qquad\fi

\csname ident.def\endcsname

\labeldef{S} {1} {Pi}
\labeldef{F} {1\labelsep \labelspace 1}  {P}
\labeldef{F} {1\labelsep \labelspace 2}  {P'}
\labeldef{L} {1\labelsep \labelspace 1}  {id}
\labeldef{F} {1\labelsep \labelspace 3}  {id1}

\labeldef{S} {2} {Pf}
\labeldef{L} {2\labelsep \labelspace 1}  {PP}
\labeldef{F} {2\labelsep \labelspace 1}  {N}
\labeldef{L} {2\labelsep \labelspace 2}  {Pxy}
\labeldef{F} {2\labelsep \labelspace 2}  {xRes}
\labeldef{L} {2\labelsep \labelspace 3}  {ResI}
\labeldef{F} {2\labelsep \labelspace 3}  {<>}
\labeldef{F} {2\labelsep \labelspace 4}  {PQ}
\labeldef{F} {2\labelsep \labelspace 5}  {MN}
\labeldef{F} {2\labelsep \labelspace 6}  {id2}
\labeldef{F} {2\labelsep \labelspace 7}  {detQ}
\labeldef{F} {2\labelsep \labelspace 8}  {detA}

\labeldef{S} {3} {Uq}
\labeldef{F} {3\labelsep \labelspace 1}  {Ughr}
\labeldef{L} {3\labelsep \labelspace 1}  {BC}
\labeldef{L} {3\labelsep \labelspace 2}  {BCC}
\labeldef{F} {3\labelsep \labelspace 2}  {BC1}
\labeldef{F} {3\labelsep \labelspace 3}  {BC2}
\labeldef{L} {3\labelsep \labelspace 3}  {KBI}

\labeldef{S} {4} {Ei}
\labeldef{F} {4\labelsep \labelspace 1}  {T}
\labeldef{F} {4\labelsep \labelspace 2}  {T'}
\labeldef{L} {4\labelsep \labelspace 1}  {idp}
\labeldef{F} {4\labelsep \labelspace 3}  {idp1}
\labeldef{F} {4\labelsep \labelspace 4}  {idp2}
\labeldef{L} {4\labelsep \labelspace 2}  {Res}
\labeldef{L} {4\labelsep \labelspace 3}  {XX}
\labeldef{F} {4\labelsep \labelspace 5}  {D}
\labeldef{F} {4\labelsep \labelspace 6}  {thi}
\labeldef{F} {4\labelsep \labelspace 7}  {Tho}
\labeldef{F} {4\labelsep \labelspace 8}  {detT}
\labeldef{F} {4\labelsep \labelspace 9}  {detAe}
\labeldef{L} {4\labelsep \labelspace 4}  {XT}

\Versiontrue

\ifMPIM
\hfuzz 15pt
\center
\vglue0pt
{\Twelvepoint \bls1.44\bls\Brm
Combinatorial Identities Related to Representations of
$\hbox{\Bmmi U}_{\tsize q}\hbox{(\mbox{\Tilde{\hbox{\Beufm gl}_{\tsize2}}})}$
\vsk1.07>
\smc \VT/}
\vsk1.7>
{\sl October \,16, 1998}
\vsk>
\endgraf\endgroup
\ftext{\=\tenpoint\sl E-mail\/{\rm:} \homemail/}
\else
\headline={\hfil\ifnum\pageno=1\hbox{\tenpoint MPI \,98\~~119}\fi}
\center
{\twelvepoint\bf \bls1.2\bls
Combinatorial Identities Related to Representations of $\Uqh$
\par}
\vsk1.5>
\VT/
\vsk1.5>
{\it \home/
\vsk.1>
\homeaddr/}
\vsk1.5>
{\sl October \,16, 1998}
\endgraf\endgroup
\ftext{\=\bls12pt{\tenpoint\sl E-mail\/{\rm:}\homemail/}}
\fi
\ifMag\vsk.4>\else\vsk>\fi
\vsk0>

\sect{Introduction}
Recently N\&Jing discovered in \Cite{J} the following combinatorial identity:
\ifMag\else\vv.3>\fi
$$
\ifMag
\Rline{\sum_{k=0}^\ell
\ \Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{k-1}\,{\eta^{\)\ell}-\eta^s\over1-\eta^{s+1}\}}\;
\sum_{\,\si\in\Sl\!}\]\Bigl(\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le a\le k}(t_{\si_a}\!-1)\,
\Prod\limits} \let\Int\int \def\int{\Int\limits_{k<b\le\ell}(t_{\si_b}\!-\eta^{\)\ell-1})\!\!
\prod_{1\le a<b\le\ell}{t_{\si_a}\!-\eta\)t_{\si_b}\over t_{\si_a}\!-t_{\si_b}}\Bigr)\)=\,0\,.\!}
\else
\sum_{k=0}^\ell
\ \Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{k-1}\,{\eta^{\)\ell}-\eta^s\over1-\eta^{s+1}\}}\;
\sum_{\,\si\in\Sl\!}\]\Bigl(\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le a\le k}(t_{\si_a}\!-1)\,
\Prod\limits} \let\Int\int \def\int{\Int\limits_{k<b\le\ell}(t_{\si_b}\!-\eta^{\)\ell-1})\!\!
\prod_{1\le a<b\le\ell}{t_{\si_a}\!-\eta\)t_{\si_b}\over t_{\si_a}\!-t_{\si_b}}\Bigr)\)=\,0\,.
\fi
\global\Tagdef{$\}\Bbb I\}$}{Ji}
\Tag{Ji}
$$
\ifMag\else\vvv.3>\fi
In his paper the identity comes from validity of the Serre relations in
some vertex \rep/s of quantum Kac-Moody algebras.
\par
In this note we are going to generalize this identity, see Theorems \[id],
\[idp]. The obtained identities are \eqv/ to existence of a singular vector
in certain tensor products of evaluation \rep/s of the quantum loop algebra
$\Uqh$, see Proposition \[BC], or more generally, of the \eqg/ $E_{\rho,\gm}(\gsl)$.
\par
There are two directions for generalizing the identity \(Ji). First one can
replace linear \fn/s of $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$ by \pol/s of larger degree. This is done
in Section \[:Pi], see \(id1).
\par
Furthermore, it is possible to take elliptic theta-\fn/s instead of \pol/s,
see Section \[:Ei]. The resulting identities depend on two extra parameters:
the elliptic modulus $p$ and the dynamical parameter $\al$. In the limit
$p\to 0$ the elliptic \fn/s degenerate into \pol/s and we get a family of
\pol/ identities depending on $\al$ and turning into the identities \(id1)
if either $\al\to 0$ or $\al\to\8$.
\par
The author is grateful to the Max-Planck-Institut f\"ur Mathematik in Bonn,
where the basic part of the paper had been written, for hospitality.

\Sno 0
\Sect[Pi]{Polynomial identities}
Given nonnegative integers $\ell$ and $n$ let $\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ be a set of partitions
\vv.1>
${\la=(\la_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\la_\ell)}$ \st/ ${n\ge{}}\alb{\la_1\lsym\ge\la_\ell\ge1}$.
For a partition $\la$ let $\om_{k,\la}=\#\)\lb\)j\ |\ } \def\nin{\not\in\la_j=k\)\rb$.
\vsk.2>
Introduce \indet/s $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$, $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\yn$, $\eta$. In the paper we use
the following compact notations:
\vv.1>
$$
t=(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,,\qquad\quad} \def\respace{\hglue-.25em\ignore x=(x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n)\,, \qquad\quad} \def\respace{\hglue-.25em\ignore y=(\yn)\,.
$$
\vsk.2>
For any $m=1\lc n} \def\aell{a=1\lc\ell$ set
$$
\gather
\\
\ifMag\cnn-1.3>\else\cnn-1.4>\fi
X_m(u;x;y)\,=\,u\prod_{1\le j<m}} \def\tprjm{\!\tprod_{1\le j<m}\!\!(u-y_j)\}\Prod\limits} \let\Int\int \def\int{\Int\limits_{m<k\le n}\!(u-x_k)\,,
\\
\ifMag\nn7>\else\nn8>\fi
X'_m(u;x;y)\,=\prod_{1\le j<m}} \def\tprjm{\!\tprod_{1\le j<m}\!\!(u-x_j)\}\Prod\limits} \let\Int\int \def\int{\Int\limits_{m<k\le n}\!(u-y_k)\,,
\endgather
$$
and for any \)$\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ set
$$
\gather
\\
\ifMag\cnn-2.2>\else\cnn-1.9>\fi
r_\la(\eta)\,=\,\pron\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1}^{\om_{m,\la}}\,{1-\eta\over1-\eta^s}\;,
\\
\nn8>
P_\la(t;x;y;\eta)\,=\,
r_\la(\eta)\)\sum_{\,\si\in\Sl\!}\]\Bigl({}\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\>X_{\la_a}(t_{\si_a};x;y)\!
\prod_{1\le a<b\le\ell}{t_{\si_a}\!-\eta\)t_{\si_b}\over t_{\si_a}\!-t_{\si_b}}\Bigr)\,,
\Tag{P}
\\
\ald
\nn8>
P'_\la(t;x;y;\eta)\,=\,
r_\la(\eta)\)\sum_{\,\si\in\Sl\!}\]\Bigl({}\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\>X'_{\la_a}(t_{\si_a};x;y)\!
\prod_{1\le a<b\le\ell}{\eta\)t_{\si_a}\!-t_{\si_b}\over t_{\si_a}\!-t_{\si_b}}\Bigr)\,.
\Tag{P'}
\endgather
$$
\ifMag\else\vvv-.2>\fi
Notice that
$$
\gather
\\
\cnn-2.1>
X_m(u;x;y)\,=\,u\>X'_m(u;y;x)
\\
\nn1>
\Text{and}
\nn-6>
P_\la(t;x;y;\eta)\,=\,
\eta^{\)\ell(\ell-1)/2\,-\!\sum_{m=1}^n} \def\pron{\prod_{m=1}^n} \def\prmn{\prod_{m=1}^{n-1}\!\om_{m,\la}(\om_{m,\la}-1)/2}\>
t_1\ldots t_\ell\,P'_\la(t;y;x;\eta^{-1}} \def\_#1{_{\Rlap{#1}})\,.
\\
\cnn.2>
\endgather
$$
Notice also that both $P_\la$ and $P'_\la$ are \pol/s in all the \indet/s
involved.
\Th{id}
Let $\,x_j=\eta^{\)\ell-1}y_i\,$ for some $\)i<j$. Then
$$
\gather
\sum_{\,\la\in\Plij\!}
c_\la\"{i,j}(x;y;\eta)\,P_\la(t;x;y;\eta)\,=\,0
\Tag{id1}
\\
\nn6>
\Text{where $\;\Plij\]=\lb\>\la=(\la_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\la_\ell)\ |\ } \def\nin{\not\in
j\ge\la_1\lsym\ge\la_\ell\ge i\>\rb\,$ and}
\ifMag\nn6>\else\nn2>\fi
{\align
c_\la\"{i,j}(x;y;\eta)\,=\,
(-1)\vpb{\om_{i,\la}}\,\eta\vpb{\,\om_{j,\la}(\om_{j,\la}-1)/2}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<j}\!\!\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{k,\la}-1}(x_k-\eta^s y_k)\,\x{}&
\ifMag\kern-1.6em\fi
\\
\nn4>
{}\x\,\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\]\Bigl({}\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<\la_a\!}\!(\eta^{\)\ell-a}\)y_i-x_k)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\la_a\]<m<j}(\eta^{\)\ell-a}\)y_i-y_m)\Bigr)\,& .
\ifMag\kern-1.6em\fi
\endalign}
\endgather
$$
\par\endproclaim
\nt
The theorem is proved in the next section.
\Ex
Identity \(id1) for $i=1$, $j=n=2$ is \eqv/ to identity \(Ji).
\sk@@p\medskipamount
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
Multiplied by a certain \pol/ in $x,y,\eta$ the identity \(id1) becomes more
transparent and understandable, see \(id2).
\mmgood
\sk@@p\medskipamount
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
All over the paper we assume integers $\ell$ and $n$ to be fixed. There are
two natural embeddings of $\Cal P_{\ell,n-1}$ into $\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ given by either
\>$\la\map\la$ \,or $\>\la\map\la'=(\la_1+1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\la_\ell+1)$. Indicating for
a while dependence of \pol/s defined by \(P) on $n$ explicitly, that is,
writing $P_{n,\la}$ instead of $P_\la$, we have
\ifMag\vv-.2>\fi
$$
\align
P_{n,\la}(t;x;y;\eta)\,&{}=\,P_{n-1,\la}(t;x\"n;y\"n;\eta)\>\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\,(t_a-x_n)\,,
\\
\nn7>
&{}=\,P_{n-1,\la'}(t;x\"1;y\"1;\eta)\>\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\,(t_a-y_1)\,,
\endalign
$$
where $\>x\"n=(x_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_{n-1})$, $\>y\"n=(y_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox y_{n-1})$,
\vv.1>
$\>x\"1=(x_2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_n)$, $\>y\"1=(y_2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox y_n)$. Therefore, the claim of Theorem
\[id] for $i=1$, $j=n$ implies the claim of the theorem for general $i<j$.
\sk@@p\medskipamount

\Sect[Pf]{Proof of Theorem \[id]}
In this section we think $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\yn$, $\eta$ being complex \var/s rather than
\indet/s and consider them as parameters which \fn/s of $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$ can in addition
depend on.
\vsk.2>
Let $f,g$ be \pol/s in $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$. We define below their scalar product
$\bra f,g\ket_{\]S}\:} \def\Of{_\Om\:$. Let
\ifMag\vv.1>\fi
$$
S(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,=\,\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\,\pron\,(t_a-x_m)\>(t_a-y_m)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\tsize{a,b=1\atop a\ne b}}^\ell\}{t_a-\eta\)t_b\over t_a-t_b}\;.
$$
\ifMag\vvv.1>\fi
If $|\eta\)|>1$ and $|x_m|<1$, $|\)y_m|>1$ for all $m=1\lc n} \def\aell{a=1\lc\ell$, then we set
\ifMag\vv.5>\else\vvn.3>\fi
$$
\bra f,g\ket_{\]S}\:} \def\Of{_\Om\:\,=\,{1\over(2\pi i)^\ell\>\ell\)!}\;
\int_{\Bbb T^\ell\]}{f(t)\,g(t)\over S(t)}\;\dtt
\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore
$$
\ifMag\vvv.2>\fi
\vvn-.2>
where $\,\topsmash{\dtt=\}\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell dt_a/t_a}\,$ and
$\,{\Bbb T^\ell=\lb\)t\in\C^{\>\ell}\ |\ } \def\nin{\not\in\ |t_1|=1\,,\alb\ {\ldots\ ,}\alb\ |t_\ell|=1\)\rb}$ each circle
oriented counterclockwise. Then one can show that $\bra f,g\ket_{\]S}\:} \def\Of{_\Om\:$ is
a \raf/ of $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\yn$, $\eta$.
\vsk.3>
The \pol/s defined by \(P) and \(P') are biorthogonal \wrt/ the introduced
scalar product.
\Th{PP}
\back\Cite{TV1, Theorem C.9\)}
$\ \bra P'_\la\),P_\mu\ket_{\]S}\:} \def\Of{_\Om\:\)=\)N_\la^{-1}} \def\_#1{_{\Rlap{#1}}\>\dl_{\la\)\mu}\,$ where
\ifMag\vv.3>\else\vv.1>\fi
$$
N_\la\,=\,\pron\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1}^{\om_{m,\la}}\,
{(1-\eta^s)\>(x_m-\eta^{s-1}y_m)\over 1-\eta}\;.
\Tag{N}
$$
\par\endproclaim
\Pf.
The proof is based on Lemmas \[Pxy] and \[ResI]. The first equality in \(<>)
implies that $\bra P'_\la\),P_\mu\ket_{\]S}\:} \def\Of{_\Om\:\)=0$ unless $\la\le\mu$, while
the second equality in \(<>) gives $\bra P'_\la\),P_\mu\ket_{\]S}\:} \def\Of{_\Om\:\)=0$
unless $\la\ge\mu$ and
$\bra P'_\la\),P_\la\ket_{\]S}\:} \def\Of{_\Om\:\)={P_\la(\yt\la)}\>{P'_\la(\yt\la)}$.
The rest of the proof is straightforward.
\mmgood
\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}\enddemo
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
Notations here and in \Cite{TV1} are not always the same though we try to keep
them consistent whenever possible. The identification of labels should be
mentioned: a partition $\la$ here corresponds to a label
$\frak l=(\om_{1,\la}\)\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\om_{n,\la})$ in \Cite{TV1}, the \pol/ $P_\la$ being a
numerator of the \tri/ weight \fn/ $w_\frak l$ in \Cite{TV1} up to a simple factor.
\sk@@p\medskipamount
For any $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ introduce points $x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\la\,,\,\yt\la\in\C^{\>\ell}\}$ as follows:
$$
\gather
x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\la\,=\,(\eta^{1-\om_{\)1,\la}}x_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_1,\)\eta^{\)1-\om_{2,\la}}x_2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox
x_2,\ \ldots\ ,\)\eta^{\)1-\om_{n,\la}}x_n\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_n)\,,
\\
\nn6>
\yt\la\,=\,(\eta^{\)\om_{1,\la}-1}\)y_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox y_1,\)\eta^{\)\om_{2,\la}-1}\)y_2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox
y_2,\ \ldots\ ,\)\eta^{\)\om_{n,\la}-1}\)y_n\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox y_n)\,.
\endgather
$$
\vvv.2>
For partitions $\la\),\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ say that $\la\ge\mu$ \>if \)$\la_a\ge\mu_a$
for any $\aell$.
\Lm{Pxy}
\bls1.1\bls
$P_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\mu)=0$ \)and $P'_\la(\yt\mu)=0$ unless $\la\ge\mu$.
$P_\la(\yt\mu)=0$ \)and $P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\mu)=0$ unless $\la\le\mu$.
Besides, only the terms in \(P), \(P') corresponding to the identity \perm/
contribute into the values $P_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\la)$, $P_\la(\yt\la)$, $P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\la)$,
$P'_\la(\yt\la)$.
\par\endproclaim
\nt
The proof is straightforward.
\vsk.5>} \def\setparindent{\edef\Parindent{\the\parindent}
\ifMag\else\nt\fi
For a \fn/ $f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})$ and a point ${t^{\sss\star}} \def\dtt{(dt/t)^\ell\}=(t^{\sss\star}} \def\dtt{(dt/t)^\ell_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\oxt^{\sss\star}} \def\dtt{(dt/t)^\ell_\ell)}$ define
\vvm.2>
a multiple residue $\mathop{\roman{Res}\>}\limits\bigl(f(t)\>\dtt\)\bigr)\vst{t=t^{\sss\star}} \def\dtt{(dt/t)^\ell}\]$%
\ifMag\else\nobreak\hskip0pt minus20pt\alb\fi\ by
$$
\gather
\\
\ifMag\cnn->\else\cnn-1.6>\fi
\mathop{\roman{Res}\>}\limits\bigl(f(t)\>\dtt\)\bigr)\vst{t=t^{\sss\star}} \def\dtt{(dt/t)^\ell}=\,\mathop{\roman{Res}\>}\limits\bigl(\,\ldots\,
\mathop{\roman{Res}\>}\limits\bigl(f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\>(dt_\ell/t_\ell)\bigr)\vst{t_\ell=t^{\sss\star}} \def\dtt{(dt/t)^\ell_\ell}\ldots\,
(dt_1/t_1)\bigr)\vst{t_1=t^{\sss\star}} \def\dtt{(dt/t)^\ell_1}
\\
\ald
\nn4>
\Text{and set}
\nn1>
\mathop{\xt\]\italic{Res}\)}\limits(f)\,=\sum_{\,\la\in\Pln\!}\mathop{\roman{Res}\>}\limits\bigl(f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\>\dtt\)\bigr)\vst{t=x\>\tright\)\la}\,,
\Tag{xRes}
\\
\nn8>
\mathop{\yt\]\italic{Res}\)}\limits(f)\,=\sum_{\,\la\in\Pln\!}\mathop{\roman{Res}\>}\limits\bigl(f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\>\dtt\)\bigr)\vst{t=y\>\trileft\>\la}\,.
\endgather
$$
\Lm{ResI}
\back\Cite{TV1, Lemma C.8}
Let \pol/s $f,g$ be \st/ their product is a \sym/ \pol/ in $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$
of degree less than $2n$ in each of the \indet/s and divisible by
$t_1\ldots t_n$. Then
$$
\bra f,g\ket_{\]S}\:} \def\Of{_\Om\:\,=\,\mathop{\xt\]\italic{Res}\)}\limits(fg/S)\,=\,(-1)^\ell\>\mathop{\yt\]\italic{Res}\)}\limits(fg/S)\,.
\Tag{<>}
$$
\ifMag\vsk.2>\else\vsk.3>\fi
\par\endproclaim
Let $Q_\la$ be a monomial \sym/ \pol/:
$$
Q_\la(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,=\;{1\over\om_{1,\la}!\ldots\>\om_{n,\la}!}\,\)
\sum_{\,\si\in\Sl\!} t_{\si_1}^{\la_1}\}\ldots\)t_{\si_\ell}^{\la_\ell}\,.
$$
Introduce transition coefficients $A_{\la\)\mu}$, $\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$:
\ifMag\vvn.2>\fi
$$
P_\la\,=\sum_{\,\mu\in\Pln\!}\!A_{\la\)\mu}\>Q_\mu\,.
\Tag{PQ}
$$
Then by Theorem \[PP] for any $\nu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$
\ifMag\vvn.1>\fi
$$
\sum_{\,\mu\in\Pln\!}\!A_{\la\)\mu}\>\bra P'_\nu,Q_\mu\ket\,=\,N_\la^{-1}} \def\_#1{_{\Rlap{#1}}\>\dl_{\la\)\nu}\,,
$$
and inverting this relation we get
\ifMag\vv.2>\fi
$$
\sum_{\,\la\in\Pln\!}\!N_\la\>\bra P'_\la,Q_\mu\ket\,A_{\la\)\mu}\,=\,\dl_{\mu\)\nu}\,.
$$
Calculating the scalar products by Lemma \[ResI] we have
\ifMag\vv.3>\fi
$$
\sum_{\,\ka,\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}\!}\!\!M_\ka^{-1}} \def\_#1{_{\Rlap{#1}}\>
Q_\mu(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka)\>P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka)\>N_\la\>A_{\la\)\mu}\,=\,\dl_{\mu\)\nu}\,,
\Tag{MN}
$$
where
$\>M_\ka^{-1}} \def\_#1{_{\Rlap{#1}}=\}\mathop{\roman{Res}\>}\limits\bigl(S(t)\vpb{-1}\dtt\bigr)\vst{t=x\>\tright\)\ka}\!$.
Notice that $M_\ka$ is a \raf/ in $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\eta$ and a \pol/ in $\yn$.
\vsk.2>
The matrix $\bigl[\)Q_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka)\bigr]_{\ka,\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}}\!$ is invertible,
\vv.1>
see e.g.\ \(detQ), and we denote by $B$ the inverse matrix, which is a \raf/
of $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\eta$ and, trivially, does not depend on $\yn$. Finally,
the relation \(MN) is transformed to
\ifMag\vv.4>\fi
$$
\sum_{\,\la\in\Pln\!}\!P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka)\>N_\la\>A_{\la\)\mu}\,=\,
M_\ka\>B_{\ka\)\mu}\,.
$$
\Pf of Theorem \[id].
Fix $i<j$. Let $\ka\"j\]=(j\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox j)$. Assume that $y_i=\eta^{\)1-\ell}x_j$.
Then $M_{\ka\"j}=0$, and taking into account the definition \(PQ) of
$A_{\la\)\mu}$ we have an identity
\ifMag\vv-.5>\fi
$$
\sum_{\,\la\in\Pln\!}^{\vp.}\!P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j)\>N_\la\>P_\la\,=\,0\,.
\Tag{id2}
$$
\vvv-.3>
Notice that $x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j=(\eta^{\)1-\ell}x_j\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox x_j)$, so only the term
in \(P') corresponding to the identity \perm/ contribute into the value
$P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j)$, and $P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j)=0$ unless
${\la\in\Plij}\!$.
Calculating ${P'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j)\>N_\la}$ explicitly and removing all factors
which does not depend on $\la$ we get the identity \(id1).
Theorem \[id] is proved.
\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}\enddemo
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
Let us mention two determinant formulae though they are not actually used
in the proof:
\ifMag
\vv-.2>
$$
\align
\quad\det\bigl[Q_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\mu)\bigr]_{\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}}=\,
\eta\vpb{-n(n+1)/2\)\cdot\!\!\tsize{n+\ell-1\choose n+1}}
\pron\,\Rlap{x_m}\hp{x}{\vp{\big@{1.5}|}}^{\!\tsize{n+\ell-1\choose n}}\,\x{}\!\] &
\Tag{detQ}
\\
\nn4>
{}\x\,\prsl\)\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!(\eta^sx_k-x_j)\vpb{D(n,\ell,s)} &
\\
\cnn.3>
\endalign
$$
\else
\vv.4>
$$
\det\bigl[Q_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\mu)\bigr]_{\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}}=\,
\eta\vpb{-n(n+1)/2\)\cdot\!\!\tsize{n+\ell-1\choose n+1}}\,
\pron\,\Rlap{x_m}\hp{x}{\vp{\big@{1.5}|}}^{\!\tsize{n+\ell-1\choose n}}\!
\prsl\)\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!(\eta^sx_k-x_j)\vpb{D(n,\ell,s)}\kern-.3em
\Tag{detQ}
$$
\vvv.2>
\fi
where
\vv-.3>
$\ \dsize D(n,\ell,s)\,=\!\!\sum_{\tsize{\,r\in\Z_{\ge 0}} \def\Zpp{\Z_{>0}\!\atop 2r\le\ell-|s|-1}}^{}
\!\!{n+\ell-|s|-2r-3\choose n-2}$, \ see formula (A.14) in \Cite{TV1}, \ and
\ifMag\vv->\else\vv-.05>\fi
$$
\det\][A_{\la\)\mu}]\vpp{\)\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}}=\,\prod_{s=0}^{\ell-1}} \def\prsl{\prod_{s=1-\ell}^{\ell-1}\,
\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!(\eta^s y_j-x_k)\vpb{\tsize{n+\ell-s-2\choose n-1}},
\Tag{detA}
$$
see formula (A.10) in \Cite{TV1}. Formula \(detQ) is a deformation of
a \sym/ power of the Vandermonde determinant. Formula \(detA) implies that
the \pol/s $P_\la$, $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ are linear independent if $x,\)y,\)\eta$ are
generic, and there are linear relations between them if \)$x_k=\eta^{\)s}y_j$
\)for some $j<k$ and $s\in\lb\)0\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\ell-1\)\rb$. For $s=\ell-1$ there is just
one linear relation given by \(id2). Relations for $s<\ell-1$ can be obtained
in a similar manner. They also can be derived from the relation \(id2) written
for \pol/s in less number of \indet/s.
\sk@@p\medskipamount

\Sect[Uq]{Tensor products of \emod/s over $\Uqh$}
Let $q$ be a nonzero complex number which is not a root of unity.
Consider the \qg/ $U_q(\gsl)} \def\Uqh{U_q(\Tilde{\gl})$ with generators $E,\>F,\>q^H$ and relations
\vvn-.3>
$$
q^H\]E=q\)E\>q^H\,,\qquad\quad} \def\respace{\hglue-.25em\ignore q^H\]F=\q F\>q^H\,,\qquad\quad} \def\respace{\hglue-.25em\ignore
[\)E\),\]F\>]\;=\,{q^{\)2H}-q^{-2H}\over q-\q}\;,
$$
and the \qlo/ $\Uqh$ with generators $L_{ij}\"{+0},\ L_{ji}\"{-0}\!$,
${1\le j\le i\le 2}$, and $L_{ij}\"s\!$, $i,j=1,2$, $s=\pm 1,\pm 2,\ldots$,
subject to relations \(Ughr).
\par
Let $e_{ij}$, $i,j=1,2$, be the ${2{\x}2}$ matrix with the only nonzero entry
$1$ at the intersection of the \^{$i$-}th row and \^{$j$-}th column. Set
$$
\align
R(u)\,=\,{} &(u\)q-\q)\>(e_{11}\ox e_{11}+e_{22}\ox e_{22})\;+
\\
\ifMag\nn7>\else\nn6>\fi
{} +\,{} &(u-1)\>(e_{11}\ox e_{22}+e_{22}\ox e_{11})\,+\,
u\)(q-\q)\>e_{12}\ox e_{21}\,+\,(q-\q)\>e_{21}\ox e_{12}\,.
\endalign
$$
Introduce the generating series
$L^{\pm}_{ij}(u)=L_{ij}\"{\pm0}+\sum_{s=1}^\8 L_{ij}\"{\pm s}u^{\pm s}\!$.
The relations in $\Uqh$ have the form:
$$
\gather
\\
\ifMag\cnn-1.5>\else\cnn-1.3>\fi
{\align
& R(u/z)\>L^\nu\'1(u)\>L^\nu\'2(z)\,=\,
L^\nu\'2(z)\>L^\nu\'1(u)\>R(u/z)\,,\Rlap{\qquad\quad} \def\respace{\hglue-.25em\ignore\nu=\pm\,,}
\Tag{Ughr}
\\
\ifMag\nn8>\else\nn7>\fi
& R(u/z)\>L^+\'1(u)\>L^-\'2(z)\,=\,
L^-\'2(z)\>L^+\'1(u)\>R(u/z)\,,
\endalign}
\\
\ifMag\nn8>\else\nn7>\fi
L_{ii}\"{+0}L_{ii}\"{-0}\,=\,1\,,\qquad L_{ii}\"{-0}L_{ii}\"{+0}\,=\,1\,,
\Rlap{\qquad\quad} \def\respace{\hglue-.25em\ignore i=1,2\,,}
\endgather
$$
where \,$L^\nu\'1(u)=\sum_{ij}\)e_{ij}\ox 1\ox L_{ij}^\nu(u)$ \,and
\,$L^\nu\'2(u)=\sum_{ij}\)1\ox e_{ij}\ox L_{ij}^\nu(u)$.
\vsk.15>
The \qlo/ $\Uqh$ is a Hopf algebra with a coproduct
$$
\Dl:L^\pm_{ij}(u)\,\map\,\mathop{\tsize\Sum}\limits} \def\ssum{\mathop{\ssize\Sum}\limits_k\)L^\pm_{ik}(u)\ox L^\pm_{kj}(u)\,.
$$
\ifMag\vv-.8>\else\vv-.3>\fi
There is a one-parametric family of \aut/s
\mmgood
\>$\rho_z:L_{ij}^\pm(u)\,\map\,L_{ij}^\pm(u/z)$ \,and the evaluation \hom/
$\epe:\Uqh\toU_q(\gsl)} \def\Uqh{U_q(\Tilde{\gl})$:
\ifMag\else\vv-.6>\fi
$$
\epe\):\)L^\pm(u)\,\map\,{}\mp\,u^{\thi_\pm}
\pmatrix
\,u\)q^H-q^{-H}} \def\q{q\1\; & u\)F\>(q-\q)\> \\ \nn8>
\)E\>(q-\q)\; & u\)q^{-H}} \def\q{q\1-q^H
\endpmatrix\kern-1em
$$
where ${\)\thi_+=0}$, ${\)\thi_-=-1\)}$. For any \Umod/ $V$ denote by $V(z)$
the \Uhmod/ obtained from $V$ via the \hom/ ${\epe\o\rho_z}$. The module $V(z)$
is called the \emod/. In a tensor product $V_1(z_1)\lox V_n(z_n)} \def\Vozz{V_n(z_n)\lox V_1(z_1)$ of \emod/s actions of
the series $L^+(u)$ and $(-u)^nL^-(u)$ coincide, so it is enough to look
at the action of only one of them. Besides, this implies that the series
$L^+(u)$ acts as a \pol/ in $u$, so it can be evaluated for $u$ being
a complex number.
\vsk.2>
Let $V_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox V_n$ be $U_q(\gsl)} \def\Uqh{U_q(\Tilde{\gl})$ \Vmod/s with \hw/s $q^{\)\La_1}\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox q^{\)\La_n}$ and
\gv/s $v_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox v_n$, \resp/. Using commutation relations in $\Uqh$ it is rather
straightforward to obtain the following statements.
\Prop{BC}
Consider $\Uqh$ \emod/s $V_1(z_1)\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox V_n(z_n)$.
\vv.1>
Let $z_i=q^{\)2\La_i+2\La_j-2\ell}z_j$ for some $i<j$ and $\ell\in\Z_{\ge 0}} \def\Zpp{\Z_{>0}$. Then
\newbi\bitem The vector ${v_1\lox v_n} \def\voxx{v_n\lox v_1\inV_1(z_1)\lox V_n(z_n)} \def\Vozz{V_n(z_n)\lox V_1(z_1)}$ generates
\ifMag\vv.25>\else\vv.1>\fi
a \$\Uqh$-submodule, which is annihilated by the action of a product
\>$L^+_{21}(q^{\)2\La_j}z_j)\ldots L^+_{21}(q^{\)2\La_j-2\ell}z_j)$.
\vsk.3>
\bitem
\ifMag\vv.15>\else\vv.1>\fi
A vector ${\tilde v\>=L^+_{12}(q^{\)2\La_j}z_j)\ldots
L^+_{12}(q^{\)2\La_j-2\ell}z_j)\,\voxx\in\Vozz}$
is singular \wrt/ the \$\Uqh$-action, that is, \>$L^\pm_{21}(u)\,\tilde v=0$.
\vadjust{\penalty-500}} \def\vgood{\strut\vvn->\vvgood\leavevmode\hfill\break} \def\~{\leavevmode\@fn@xt~\m@n@s\@md@sh\ignore
\par\endproclaim
\Cr{BCC}
Let $z_i=q^{\)2\La_i+2\La_j-2\ell}z_j$ for some $i<j$ and $\ell\in\Z_{\ge 0}} \def\Zpp{\Z_{>0}$. Then
for arbitrary $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$
$$
L^+_{21}(q^{\)2\La_j}z_j)\ldots L^+_{21}(q^{\)2\La_j-2\ell}z_j)\>
L^+_{12}(t_1)\ldots L^+_{12}(t_\ell)\,v_1\lox v_n} \def\voxx{v_n\lox v_1\,=\,0
\Tag{BC1}
$$
in $V_1(z_1)\lox V_n(z_n)} \def\Vozz{V_n(z_n)\lox V_1(z_1)$ and
$$
L^+_{21}(t_1)\ldots L^+_{21}(t_\ell)\>L^+_{21}(q^{\)2\La_j}z_j)\ldots
L^+_{12}(q^{\)2\La_j-2\ell}z_j)\,\voxx\,=\,0
\Tag{BC2}
$$
in $\Vozz$.
\par\endproclaim
\nt
Both relations \(BC1) and \(BC2) are \eqv/ to the identitiy \(id2)
and, hence, the identity \(id1). For \(BC1) it follows from Proposition \[KBI]
and formula \(N), while for \(BC2) an analogue of Proposition \[KBI] for
the tensor product $\Vozz$ is required.
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
It is rather easy to see that Corollary \[BCC], Proposition \[KBI] and
the determinant formula \(detA) together imply Proposition \[BC].
\sk@@p\medskipamount
Let parameters $q$, $q^{\)\La_1}\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox q^{\)\La_n}$, $\zn$ used in this section
be related to the previously used parameters $\eta$, $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\yn$ as follows:
$$
\eta=q^2\,,\qquad\quad} \def\respace{\hglue-.25em\ignore x_m=\)q^{\)2\La_m}z_m\,,\qquad\quad} \def\respace{\hglue-.25em\ignore y_m=\)q^{-2\La_m}z_m\,.
\kern-2em
\mmgood
$$
\Prop{KBI}
\back\Cite{KBI},\;\Cite{TV1, Lemma 4.3},\;\Cite{TV2, Lemma 4.18}
\ifMag
$$
\gather
\\
\cnn-1.2>
{\align
L^+_{12}(t_1)\ldots L^+_{12}(t_\ell)\,v_1\lox v_n} \def\voxx{v_n\lox v_1\,=\,
(q &{}-\q)^\ell\>\tpron(-z_m)^{-\ell}\!\sum_{\,\la\in\Pln\!}\!P_\la(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,\x{}
\\
\nn6>
{}\x\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\! &
q^{\)\La_j\om_{k,\la}-\)\La_k\om_{j,\la}-\>\om_{j,\la}\om_{k,\la}}\,F\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n\,.
\endalign}
\\
\nn15>
\Lline{L^+_{21}(t_1)\ldots L^+_{21}(t_\ell)\,F\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n\,={}}
\\
\nn8>
\Rline{{}=\tpron(-z_m)^{\)\om_{m,\la}-\)\ell}\,\)
P'_{\la}(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,\pron\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1}^{\om_{m,\la}}\,
{(q^s-q^{-s})\>(q^{\)2\La_m-s+1}-q^{-2\La_m+s-1})\over q-\q}\;\x{}\!}
\\
\nn6>
\Rline{{}\x\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!
q^{\)\La_k\om_{j,\la}-\)\La_j\om_{k,\la}-\>\om_{j,\la}\om_{k,\la}}\,v_1\lox v_n} \def\voxx{v_n\lox v_1\,.\!}
\\
\cnn->
\endgather
$$
\else
$$
\gather
{\align
L^+_{12}(t_1)\ldots L^+_{12}(t_\ell)\,v_1\lox v_n} \def\voxx{v_n\lox v_1\, &{}=\,
(q-\q)^\ell\>\tpron(-z_m)^{-\ell}\sum_{\,\la\in\Pln\!}\!P_\la(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,\x{}
\\
\nn6>
& {}\>\x\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!
q^{\)\La_j\om_{k,\la}-\)\La_k\om_{j,\la}-\>\om_{j,\la}\om_{k,\la}}\,F\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n\,.
\endalign}
\\
\nn12>
\alignedat2
L^+_{21}(t_1)\ldots L^+_{21} & (t_\ell)\,F\vpb{\,\om_{1,\la}}v_1\lox F\vpb{\,\om_{n,\la}}v_n\,={} &&
\\
\nn7>
& {}=\tpron(-z_m)^{\)\om_{m,\la}-\)\ell}\,
P'_{\la}(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,\pron\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1}^{\om_{m,\la}}\,
{(q^s-q^{-s})\>(q^{\)2\La_m-s+1}-q^{-2\La_m+s-1})\over q-\q}\;\x{} &&
\\
\nn6>
&& \Llap{{}\x\,\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!
q^{\)\La_k\om_{j,\la}-\)\La_j\om_{k,\la}-\>\om_{j,\la}\om_{k,\la}}\,v_1\lox v_n} \def\voxx{v_n\lox v_1\,.} &
\endalignedat
\\
\cnn->
\endgather
$$
\fi
\par\endproclaim

\Sect[Ei]{Elliptic identities}
In this section we extend Theorem \[id] to the elliptic case, see Theorem
\[idp]. The obtained identities have the same relation to \rep/s of the \eqg/
$E_{\rho,\gm}(\gsl)$ as the identities \(id1) have to \rep/s of the \qlo/ $\Uqh$.
\vsk.5>} \def\setparindent{\edef\Parindent{\the\parindent}
Fix a nonzero complex number $p$ \st/ ${|p|<1}$. Let
\ifMag
\vv.1>
$$
(u)_{\kern-.02em\l@inf}\>=\,(u;p)_{\kern-.02em\l@inf}\>=\,\topsmash{\mathop{\tsize\Prod}\limits_{s=0}^\8(1-p^su)}
$$
\vvv-.3>
\else
$(u)_{\kern-.02em\l@inf}\]=(u;p)_{\kern-.02em\l@inf}\]=\topsmash{\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^\8(1-p^su)}$
\vv-.4>
\fi
and let ${\tht(u)=\tht(u;p)=(u)_{\kern-.02em\l@inf}(p\)u^{-1}} \def\_#1{_{\Rlap{#1}})_{\kern-.02em\l@inf}(p)_{\kern-.02em\l@inf}}$ be the Jacobi theta-\fn/.
\vsk.3>
In addition to $t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1}$, $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$, $\yn$ and $\eta$ introduce one more \var/
$\al$ called the \em{dynamical parameter}.
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
The parameter $\al$ is related to the dynamical \var/ in the \eqg/ $E_{\rho,\gm}(\gsl)$.
\sk@@p\medskipamount
For any $m=1\lc n} \def\aell{a=1\lc\ell$ set
$$
\gather
\\
\cnn-1.4>
Z_m(u;x;y;\al)\,=\,\tht\bigl(\al_m^{-1}} \def\_#1{_{\Rlap{#1}} u/x_m)\}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<m}\!\tht(u/y_j)\}\Prod\limits} \let\Int\int \def\int{\Int\limits_{m<k\le n}\!\tht(u/x_k)\,,
\\
\nn9>
Z'_m(u;x;y;\al)\,=\,\tht\bigl(\al_m\)u/y_m)\}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<m}\!\tht(u/x_j)\}\Prod\limits} \let\Int\int \def\int{\Int\limits_{m<k\le n}\!\tht(u/y_k)\,,
\endgather
$$
\ifMag\vvv.2>\fi
where $\>\smash{\al_m=\al\tprjm x_j/y_j}$, \;and for any \)$\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ set
$$
\gather
\\
\ifMag\cnn-1.2>\else\cnn-1.3>\fi
\rho_\la(\eta)\,=\,\pron\,
\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=1}^{\om_{m,\la}}\,{\tht(\eta)\over\tht(\eta^s)}\;,
\\
\ifMag\ald\nn7>\else\nn8>\fi
\Xj_\la(t;x;y;\eta\);\al)\,=\,\rho_\la(\eta)\)\sum_{\,\si\in\Sl\!}\]\Bigl({}
\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\>Z_{\la_a}(t_{\si_a};x;y;\al\)\eta^{\)2a-2\ell})\!\prod_{1\le a<b\le\ell}\!
{\tht(\eta\)t_{\si_b}/t_{\si_a})\over\tht(t_{\si_b}/t_{\si_a})\!}\,\Bigr)\,,
\ifMag\kern-3em\fi
\Tag{T}
\\
\nn8>
\Xj'_\la(t;x;y;\eta\);\al)\,=\,\rho_\la(\eta)\)\sum_{\,\si\in\Sl\!}\]\Bigl({}
\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\>Z'_{\la_a}(t_{\si_a};x;y;\al\)\eta^{\)2\ell-2a})\!\prod_{1\le a<b\le\ell}\!
{\tht(\eta\)t_{\si_a}/t_{\si_b})\over\tht(t_{\si_a}/t_{\si_b})\!}\,\Bigr)\,.
\ifMag\kern-3em\fi
\Tag{T'}
\endgather
$$
\vvv-.4>
Notice that
$$
\gather
\\
\cnn-2.3>
Z_m(u;x;y;\al)\,=\,Z'_m(u;y;x;\al^{-1}} \def\_#1{_{\Rlap{#1}})
\\
\nn2>
\Text{and}
\nn-8>
\Xj_\la(t;x;y;\eta\);\al)\,=\,
\eta^{\)\ell(\ell-1)/2\,-\!\sum_{m=1}^n} \def\pron{\prod_{m=1}^n} \def\prmn{\prod_{m=1}^{n-1}\!\om_{m,\la}(\om_{m,\la}-1)/2}\>
\Xj'_\la(t;y;x;\eta^{-1}} \def\_#1{_{\Rlap{#1}};\al^{-1}} \def\_#1{_{\Rlap{#1}})\,.
\endgather
$$
\Th{idp}
Let $\,x_j=\eta^{\)\ell-1}y_i\,$ for some $\)i<j$. Then
\ifMag
$$
\gather
\sum_{\,\la\in\Plij\!}
C_\la\"{i,j}(x;y;\eta\);\al)\,\Xj_\la(t;x;y;\eta\);\al)\,=\,0
\Tag{idp1}
\\
\nn6>
\Text{where $\;\Plij\]=\lb\>\la=(\la_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\la_\ell)\ |\ } \def\nin{\not\in
j\ge\la_1\lsym\ge\la_\ell\ge i\>\rb\,$,}
\nn9>
\alignedat2
C_\la\"{i,j} & \Rlap{(x;y;\eta\);\al)\,={}} &&
\\
\nn1>
{}=\,{}&&
(\al_i\)x_i/y_i)\vpb{\om_{i,\la}}\,\eta\vpb{-\om_{i,\la}(\om_{i,\la}-1)}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<j}\!\!\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{k,\la}-1}{\tht(\eta^{-s}x_k/y_k)\over
\tht(\eta^s\al_{k,\la}^{-1}} \def\_#1{_{\Rlap{#1}})\>\tht(\eta^{1-s-\om_{k,\la}}\al_{k,\la}\>x_k/y_k)}\;
\x{}&
\\
\nn7>
&&\Llap{{}\x{}}\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{i,\la}-1}
{1\over\tht(\eta^{1-s-\om_{i,\la}}\al_{i,\la}\>x_i/y_i)}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{j,\la}-1}{1\over\tht(\eta^s\al_{j,\la}^{-1}} \def\_#1{_{\Rlap{#1}})}\;
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\,a=\om_{j,\la}+1\!}^{\,\ell-\om_{i,\la}\!}\!
\tht(\al_{\la_a}\:\eta^{a-\ell}\)y_i/y_{\la_a}\:)\;\x{}&
\\
\nn5>
&&\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\]\Bigl({}\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<\la_a\!}\!\tht(\eta^{\)\ell-a}\)y_i/x_k)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\la_a\]<m<j}\tht(\eta^{\)\ell-a}\)y_i/y_m)\Bigr)\,&,
\endalignedat
\endgather
$$
\else
$$
\gather
\sum_{\,\la\in\Plij\!}
C_\la\"{i,j}(x;y;\eta\);\al)\,\Xj_\la(t;x;y;\eta\);\al)\,=\,0
\Tag{idp1}
\\
\nn7>
\Text{where $\;\Plij\]=\lb\>\la=(\la_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\la_\ell)\ |\ } \def\nin{\not\in
j\ge\la_1\lsym\ge\la_\ell\ge i\>\rb\,$,}
\nn8>
{\align
C_\la\"{i,j}(x;y;\eta\);\al)\,=\,
(\al_i\)x_i/y_i)\vpb{\om_{i,\la}}\,\eta\vpb{-\om_{i,\la}(\om_{i,\la}-1)}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<j}\!\!\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{k,\la}-1}{\tht(\eta^{-s}x_k/y_k)\over
\tht(\eta^s\al_{k,\la}^{-1}} \def\_#1{_{\Rlap{#1}})\>\tht(\eta^{1-s-\om_{k,\la}}\al_{k,\la}\>x_k/y_k)}
\,\x{}&
\\
\nn6>
{}\x\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{i,\la}-1}
{1\over\tht(\eta^{1-s-\om_{i,\la}}\al_{i,\la}\>x_i/y_i)}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{j,\la}-1}{1\over\tht(\eta^s\al_{j,\la}^{-1}} \def\_#1{_{\Rlap{#1}})}
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\,a=\om_{j,\la}+1\!}^{\,\ell-\om_{i,\la}\!}\!
\tht(\al_{\la_a}\:\eta^{a-\ell}\)y_i/y_{\la_a}\:)\;\x{}&
\\
\nn5>
{}\x\,\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\]\Bigl({}\Prod\limits} \let\Int\int \def\int{\Int\limits_{i<k<\la_a\!}\!\tht(\eta^{\)\ell-a}\)y_i/x_k)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\la_a\]<m<j}\tht(\eta^{\)\ell-a}\)y_i/y_m)\Bigr)\,& ,
\endalign}
\\
\cnn-.2>
\endgather
$$
\fi
$\al_{k,\la}=\al\!\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k}^{\vp.}\!\eta^{-2\)\om_{j,\la}}\)x_j/y_j$
\,\)and \;$\al_k=\al\!\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k}\!x_j/y_j$.
\ifMag\vv.3>\else\vv.5>\fi
\par\endproclaim
\Ex
Identity \(idp1) for $i=1$, $j=n=2$ takes the form
$$
\alignat2
\\
\ifMag\cnn-1.3>\else\cnn-1.4>\fi
& \sum_{k=0}^\ell\,(-1)^k\>\tht(\eta^{\)2k}\bt)\,
\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{k-1}\,{\eta^s\>\tht(\eta^{\)\ell-s})\>\tht(\eta^s\bt)\over
\tht(\eta^{s+1})\>\tht(\eta^{s+\ell+1}\bt)}\;\x{} &&
\ifMag\kern-1.5em\else\kern-.5em\fi
\Tag{idp2}
\\
\nn12>
&{}\x\sum_{\,\si\in\Sl\!}\]\Bigl(\]\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le a\le k}\tht(t_{\si_a})\>
\tht(\eta^{\)2-2a-\ell}t_{\si_a}/\]\bt)\;\Prod\limits} \let\Int\int \def\int{\Int\limits_{k<b\le\ell}
\tht(\eta^{\)1-\ell}t_{\si_b})\>\tht(\eta^{\)1-2b}t_{\si_b}/\]\bt)\;\x{} &&
\ifMag\kern-1.5em\else\kern-.5em\fi
\\
\nn1>
&&\Llap{{}\x\prod_{1\le a<b\le\ell}
{\tht(\eta\)t_{\si_b}/t_{\si_a})\over\tht(t_{\si_b}/t_{\si_a})}
\Bigr)\,}{}\}=\,0\,.\ifMag\kern-1.5em\else\kern-.5em\fi
\\
\ifMag\noalign{\goodbm}\fi
\cnn-.1>
\endalignat
$$
Here $\>\bt=\eta^{\)1-2\ell}\al\>x_1/y_1$. If $p\to 0$ and then either
$\bt\to 0$ or $\bt\to\8$, then \(idp2) transforms into \(Ji).
\sk@@p\medskipamount
\Pf of Theorem \[idp].
The proof of Theorem \[idp] is very similar to the proof of Theorem \[id].
We descripe the main steps below. We do not indicate dependence on $x_1\lc x_n} \def\yn{y_1\lc y_n} \def\zn{z_1\lc z_n$,
$\yn$, $\eta$, $\al$ explicitly.
\goodbreak
\vsk.5>} \def\setparindent{\edef\Parindent{\the\parindent}
Denote by $\Bbb E$ the space of \fn/s $f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})$ \hol/ outside the coordinate
hyperplanes $t_a=0$, $\aell$, and \st/
\ifMag\vv-.2>\fi
$$
\gather
f(t_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox p\)t_a\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox t_\ell)\,=\,
t_a^{-2n}\!\tpron(x_m\>y_m)\,f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,.
\\
\nn1>
\Text{Let}
\nn-6>
\Om(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,=\,\prod_{a=1}^\ell} \def\tpral{\tprod_{a=1}^\ell\,\pron\,\tht(t_a/x_m)\>\tht(t_a/y_m)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{\tsize{a,b=1\atop a\ne b}}^\ell
{\tht(\eta\)t_a/t_b)\over\tht(t_a/t_b)}\;.
\\
\cnn-.5>
\endgather
$$
There is an analogue of Lemma \[ResI].
\Lm{Res}
\back\Cite{TV1, Lemma C.11}
For any \fn/ $f\in\Bbb E$ we have
$$
\mathop{\xt\]\italic{Res}\)}\limits(f/\Om)\,=\,(-1)^\ell\>\mathop{\yt\]\italic{Res}\)}\limits(f/\Om)\,.
$$
\par\endproclaim
Define a scalar product $\bra f,g\ket\Of=\mathop{\xt\]\italic{Res}\)}\limits(fg/\Om)$.
The \fn/s \(T) and \(T') are biorthogonal \wrt/ the introduced scalar product.
\Th{XX}
\back\Cite{TV1, Theorem C.9\)}
$\ \bra \Xj'_\la\),\Xj_\mu\ket\Of\)=\)D_\la^{-1}} \def\_#1{_{\Rlap{#1}}\>\dl_{\la\)\mu}\,$ where
\ifMag\vv.1>\fi
$$
D_\la\,=\,(-1)^\ell\pron\!\Prod\limits} \let\Int\int \def\int{\Int\limits_{s=0}^{\om_{m,\la}-1}
{(p)^3_\8\>\tht(\eta^{s+1})\>\tht(\eta^{-s}x_m/y_m)\over\tht(\eta)\>
\tht(\eta^s\al_{m,\la}^{-1}} \def\_#1{_{\Rlap{#1}})\>\tht(\eta^{1-s-\om_{m,\la}}\al_{m,\la}\>x_m/y_m)}
\Tag{D}
$$
and $\,{\al_{m,\la}=\al\!\prod_{1\le j<m}} \def\tprjm{\!\tprod_{1\le j<m}\!^{\vp.}\!\eta^{-2\)\om_{\la,j}}\)x_j/y_j}$.
\ifMag\vv>\fi
\par\endproclaim
Let $\thi_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox \thi_n$ be the following \fn/s in one \var/:
$$
\thi_m(u)\,=\,u^{m-1}\>\tht\bigl(-\)p^{\)m-1}\)\eta^{\)\ell-1}\)\al^{-1}} \def\_#1{_{\Rlap{#1}}\!\]
\tpron\!x_m\>(-u)^n\);\)p^n\bigr)\,(p^n;p^n)^{-1}} \def\_#1{_{\Rlap{#1}}_\8\,(p;p)^n_\8\,.
\Tag{thi}
$$
They are linearly independent, since $\thi_m(\eps u)=\eps^{m-1}\>\thi(u)$
where $\eps=e^{2\pi i/n}\!$. For any $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ set
$$
\Tho_\la(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})\,=\;{1\over\om_{1,\la}!\ldots\>\om_{n,\la}!}\,\)
\sum_{\,\si\in\Sl\!}\thi_{\la_1}(t_{\si_1})\ldots\)\thi_{\la_n}(t_{\si_\ell})\,.
\Tag{Tho}
$$
The \fn/s $\Tho_\la$, $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$, are linearly independent, and
the \fn/s $\Xj_\mu$ are linear combinations of the \fn/s $\Tho_\la$,
see Proposition \[XT].
\goodbm
\par
We proceed further like in Section \[:Pf] and under the assumption of
Theorem \[idp]\): $y_i=\eta^{\)1-\ell}x_j$ for some $i<j$, we get an identity
$$
\sum_{\,\la\in\Pln\!}^{\vp.}\!\Xj'_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\ka\"j)\>D_\la\>\Xj_\la\,=\,0\,,
$$
and then the identity \(idp1). Theorem \[idp] is proved.
\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}\enddemo
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
There are elliptic analogues of determinant formulae \(detQ) and \(detA),
see formulae (B.11) and (B.8) in \Cite{TV1}\):
$$
\align
\\
\ifMag\cnn-.9>\else\cnn-1.4>\fi
\det\bigl[\Tho_\la(x\mathbin{\smash\tright\]}} \def\yt{y\mathbin{\]\smash\trileft}\mu)\bigr]_{\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}} &{}=\,K\>
\eta\vpb{n(1-n)/2\)\cdot\!\!\tsize{n+\ell-1\choose n+1}}\,
\pron\,(-x_m)\vpb{(m-1)\tsize{n+\ell-1\choose n}}\,\x{} \kern-1em
\Tagg{detT}
\\
\ifMag\nn9>\else\nn8>\fi
& {}\>\x\,\prod_{s=0}^{\ell-1}} \def\prsl{\prod_{s=1-\ell}^{\ell-1}\,\tht(\eta^s\)\al^{-1}} \def\_#1{_{\Rlap{#1}})\vpb{\tsize{n+s-1\choose n-1}}
\}\prsl\)\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!\tht(\eta^sx_j/x_k)\vpb{D(n,\ell,s)}
\ifMag\kern-2.5em\fi
\\
\ifMag\cnn.5>\else\cnn.3>\fi
\endalign
$$
where \ $\dsize K\,=\,\Bigl[\>\Rph{(p)}{(p)_{\kern-.02em\l@inf}}\vpb{n^2-1}\>\prmn\>
\Bigl(\>{\tht(e^{2\pi i\)m/n})\over e^{2\pi i\)m/n}\}-1}\)\Bigr)
^{\!\!\raise 3pt\mbox{\ssize m-n}}\>\Bigr]^{\tsize{n+\ell-1\choose n}}\!$,\quad
and
$$
\align
\det\][A^{\sscr{e\]l\}l}}_{\la\)\mu}]\vpp{\)\la,\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}}& {}=\,K^{-1}} \def\_#1{_{\Rlap{#1}}\}\prsl\)\prmn\>
\tht({\tsize\eta^{s+\ell-1}\al^{-1}} \def\_#1{_{\Rlap{#1}}\!\!\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j\le m}\!y_j/x_j})
\vpb{d(n,m,\ell,s)}\,\x{} \kern-1em
\Tagg{detAe}
\\
\nn10>
& \>{}\x\,\pron\,y\_m\vpb{(m-n){\tsize{n+\ell-1\choose n}}}\;\prod_{s=0}^{\ell-1}} \def\prsl{\prod_{s=1-\ell}^{\ell-1}\)
\Prod\limits} \let\Int\int \def\int{\Int\limits_{1\le j<k\le n}\!\tht(\eta^s\)y_j/x_k)
\vpb{\tsize{n+\ell-s-2\choose n-1}} \ifMag\kern-1.5em\fi
\\
\cnn.5>
\endalign
$$
where \ $\dsize\Xj_\la\,=\sum_{\,\mu\in\Pln\!}\!A^{\sscr{e\]l\}l}}_{\la\)\mu}\>\Tho_\mu$ \ and
\ $\dsize d(n,m,\ell,s)\,=
\sum_{\tsize{\vp( i,j\ge 0\atop\tsize{\vp( i+j<\ell\atop i-j=s\vp(}}}
{m-1+i\choose m-1}\,{n-m-1+j\choose n-m-1}$.
\sk@@p\medskipamount
\demo{\sl Remark}} \def\Ex{\demo{\bf Example}
Consider a limit $p\to 0$ in all formulae of this section, which essentially
amounts to replacing the theta-\fn/ $\tht(u)$ by the linear \fn/ $1-u$.
The \fn/s $\thi_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\thi_n$ tend to power \fn/s as $p\to 0$:
$$
\thi_1(u)\to 1+\eta^{\)\ell-1}\)\al^{-1}} \def\_#1{_{\Rlap{#1}}\!\]\tpron\!x_m\>(-u)^n\,,\qquad\quad} \def\respace{\hglue-.25em\ignore
\thi_m(u)\to u^{m-1}\,,\qquad m=2\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox n\,.
$$
The limit ${p\to 0}$ of the identities \(idp1) and the determinant formulae
\(detT) and \(detAe) deliver one-parametric deformations of the identities
\(id1) and the determinant formulae \(detQ) and \(detA), which can be recovered
in the limit either $\al\to 0$ or $\al\to\8$. The limit $\al\to 0$ is
straightforward while the limit $\al\to\8$ produces formulae \(id1), \(detQ),
\(detA) up to a certain change of \var/s.
\sk@@p\medskipamount
\Prop{XT}
For any $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ the \fn/ $\Xj_\la$, \cf. \(T), is a linear combination
of the \fn/s $\Tho_\mu$, $\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$, \cf. \(Tho).
\par\endproclaim
\Pf.
Let $\Cal E$ be a space of \fn/s $f(u)$ \hol/ for $u\ne 0$ and \st/
$$
f(p\)u)\,=\,\al\)\eta^{1-\ell}\]\tpron\}x_m\,(-u)^n\,f(u)\,.
$$
Let $\Cal E_\ell$ be a space of \sym/ \fn/s $f(t_1\lc t_\ell} \def\twll{t_2\lc t_\ell} \def\telm{t_1\lc t_{\ell-1})$ which considered as \fn/s
of one \var/ $t_a$ belong to $\Cal E$ for any $\aell$. In particular, $\Cal E_1=\Cal E$.
\par
It is clear that $\dim\Cal E=n$, say by Fourier series. A basis in $\Cal E$ is given
by the \fn/s $\thi_1\lsym,} \def\lcd{\lsym\cdot} \def\lx{\lsym\x} \def\lox{\lsym\ox\thi_n$. Hence, the \fn/s $\Tho_\mu$, $\mu\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$, form
a basis in the space $\Cal E_\ell$.
\par
Since for any $\la\in\Pc_{\ell,n}} \def\Plij{\Pc_\ell^{i,j}$ the \fn/ $\Xj_\la$ belongs to the space $\Cal E_\ell$,
the proposition follows.
\hbox{}\nobreak\hfill\nobreak{\m@th$\,\square$}\enddemo

\myRefs
\widest{VV}
\parskip.1\bls

\ref\Key J
\cl@s@\@p@n\b@b@x} \def\paper{\cl@s@\@p@n\p@p@rb@x\p@p@rf@nt\ignore N\&Jing
\paper Quantum Kac-Moody algebras and vertex \rep/s
\cl@s@\@p@n\j@@rb@x} \def\yr{\cl@s@\@p@n\y@@rb@x \LMP/ \cl@s@\@p@n\v@lb@x\v@lf@nt\ignore 44 \yr 1998 \cl@s@\ifp@g@\@p@n\z@\else\p@g@strue\@p@n\p@g@b@x\fi 261--271
\endref

\ref\Key KBI
\cl@s@\@p@n\b@b@x} \def\paper{\cl@s@\@p@n\p@p@rb@x\p@p@rf@nt\ignore \Kor/, N\&M\&Bogolyubov and A\&G\&Izergin
\cl@s@\@p@n\b@@kb@x\b@@kf@nt\ignore Quantum inverse scattering method and correlation \fn/s
\yr 1993 \cl@s@\@p@n\p@blb@x} \def\publaddr{\cl@s@\@p@n\p@bl@db@x \CUP/
\endref

\ref\Key T
\cl@s@\@p@n\b@b@x} \def\paper{\cl@s@\@p@n\p@p@rb@x\p@p@rf@nt\ignore \VT/
\paper Bilinear identity for \$q$-\hint/s
\cl@s@\@p@n\j@@rb@x} \def\yr{\cl@s@\@p@n\y@@rb@x Preprint \yr 1997 \cl@s@\ifp@g@\@p@n\z@\else\p@g@strue\@p@n\p@g@b@x\fi 1--24 \cl@s@\@p@n\inf@b@x} \def\paperinfo{\cl@s@\@p@n\p@p@nf@b@x to appear in Osaka J. Math.
\endref

\ref\Key TV1
\cl@s@\@p@n\b@b@x} \def\paper{\cl@s@\@p@n\p@p@rb@x\p@p@rf@nt\ignore \VT/ and \Varch/
\paper Geometry of \$q$-\hgeom/ \fn/s, \qaff/s and \eqg/s
\cl@s@\@p@n\j@@rb@x} \def\yr{\cl@s@\@p@n\y@@rb@x Ast\'erisque \cl@s@\@p@n\v@lb@x\v@lf@nt\ignore 246 \yr 1997 \cl@s@\ifp@g@\@p@n\z@\else\p@g@strue\@p@n\p@g@b@x\fi 1--135
\endref

\ref\Key TV2
\cl@s@\@p@n\b@b@x} \def\paper{\cl@s@\@p@n\p@p@rb@x\p@p@rf@nt\ignore \VT/ and \Varch/
\paper Geometry of \$q$-\hgeom/ \fn/s as a bridge between Yangians and
\qaff/s \cl@s@\@p@n\j@@rb@x} \def\yr{\cl@s@\@p@n\y@@rb@x \Inv/ \yr 1997 \cl@s@\@p@n\v@lb@x\v@lf@nt\ignore 128 \cl@s@\@p@n\is@b@x\iss@f@nt\ignore 3 \cl@s@\ifp@g@\@p@n\z@\else\p@g@strue\@p@n\p@g@b@x\fi 501--588
\endref

\endgraf\goodbreak\endgroup

\bye

redpajama_set_name: RedPajamaArXiv


Steve Levin has been an angel investor, advisor, and start-up executive. Steve has founded and developed successful businesses across a broad spectrum of products and services with technology at its core. With a strong background in technology, business model innovation, development, and analysis, Steve's ability to deliver business-building breakthroughs is among the best.
From aerospace to corporate relocation to health insurance, marketing services and real estate development, Steve's experiences have allowed him to see both the key differences and common threads necessary for commercial success across many industries. Steve has also helped some of the world's largest companies, including General Electric and Procter & Gamble, develop and launch new business innovation ventures in marketing, software and business services. He's also seen success in founding agricultural, education, manufacturing, and advanced materials ventures, including award-winning Sirrus.
Steve lives in Cincinnati, Ohio with his wife Donna and teenage daughter Hannah. Steve majored in both in chemical engineering and computer science at the University of Oklahoma.


text:
Prime Minister welcomes foreign investors
Prime Minister Nguyen Xuan Phuc hosted receptions for Mike Spanos, CEO of the Asia, Middle East, and North Africa unit of the US-based PepsiCo, Inc.; and Sarath Ratanavadi, founder and CEO of Thailand's Gulf Energy Development, in Hanoi on August 14.
VNA Tuesday, August 14, 2018 20:33
US firms appreciate Vietnam's investment environment
Vietnamese firms face uphill task to export farm produce to Thailand
Vietnamese, Thai publishers seek partnerships
Tuesday, May 22, 2018 18:40
Government always creates optimal conditions for US firms, says PM
Prime Minister Nguyen Xuan Phuc (R) welcomes Mike Spanos, CEO of the Asia, Middle East, and North Africa unit of the US-based PepsiCo, Inc. (Photo: VNA)
Hanoi (VNA) – Prime Minister Nguyen Xuan Phuc hosted receptions for Mike Spanos, CEO of the Asia, Middle East, and North Africa unit of the US-based PepsiCo, Inc.; and Sarath Ratanavadi, founder and CEO of Thailand's Gulf Energy Development, in Hanoi on August 14.
The PM congratulated PepsiCo – an American multinational food, snack, and beverage corporation – on its successful operations in Vietnam in recent times.
He spoke highly of the cooperation between PepsiCo Vietnam and the Ministry of Agriculture and Rural Development by their founding a public-private partnership group on fruits and vegetables in 2010.
This helped create a sustainable supply chain based on collaboration, promoting trade development, as well as opening up opportunities to raise living standards for hundreds of farm households – contributing to the sustainable development of cultivation in Vietnam.
PM Phuc hailed the US firm for expanding its cultivation area of potato in Vietnam's Central Highlands province of Lam Dong from 55 hectares to over 450ha, while also supporting farmers with cultivation skills that can help them increase their output three times the average level.
He expressed hope that PepsiCo Vietnam will continue to study processing in order to create high added-value products made from artichoke, a specialty in the Central Highlands city of Da Lat, and bring its products to the company's distribution chains.
The leader suggested PepsiCo expand cooperation with the Ministry of Agriculture and Rural Development and Vietnamese farm households to create new products that make use of Vietnam's natural advantages, such as with potato, artichoke, and tea.
Spanos said Vietnam is one of the company's top 20 priority markets across the globe, and eighth in the Asian-Middle Eastern-North African region.
PepsiCo has invested over 500 million USD in Vietnam over the past five years, attracting about 3,000 workers. It is currently implementing cooperation programmes with Vietnamese farmers and supporting over 580 farm households.
The company is keen to do long-term business in Vietnam in the field of food and beverages, among others, the official said.
While receiving Sarath Ratanavadi, PM Phuc reaffirmed the Vietnamese Government has always created the best conditions for foreign businesses to invest in the country. This is one of the reasons why many Thai firms have chosen to enter the Vietnamese market, which currently operate 492 projects worth 10 billion USD, he noted.
Prime Minister Nguyen Xuan Phuc (R) receives Sarath Ratanavadi, founder and CEO of Thailand's Gulf Energy Development (Photo: VNA)
The PM underlined Vietnam's policy of discouraging investment in coal-fired thermal power generation, but welcoming investment in renewable and clean energy.
Vietnam needs more abundant energy to meet the country's economic development, he said, adding that this is a good opportunity for the Thai company.
Sarath Ratanavadi said his business has started to invest in Vietnam through a solar power project with a capacity of 100MW.
The company plans to continue investing in the field of energy to increase electricity for Vietnam, as well as draw more investors, he said, adding that it is also able to provide water for the industrial sector. –VNA
Prime Minister Nguyen Xuan Phuc US-based PepsiCo Inc. hailand's Gulf Energy Development beverage food public-private partnership group fruits vegetables farm households vietnamplus Vietnam News Agency Related stories Thailand Related stories United States
Greetings for 70th anniversary of Vietnam-China diplomatic ties
Party General Secretary and State President Nguyen Phu Trong has exchanged greetings with his Chinese counterpart Xi Jinping on the occasion of the 70th anniversary of the VN-China diplomatic ties.
Party economic commission urged to raise strategic proposals
HCM City leader receives Japanese city governor
Permanent Vice Chairman of the Ho Chi Minh City People's Committee Le Thanh Liem hosted a reception for Nakamura Hiro, Governor of Yatsushiro city, the Japanese prefecture of Kumamoto, on January 17.
Friendship and cooperation have been the mainstream of Vietnam-China relations over the past seven decades, Deputy Prime Minister and Foreign Minister Pham Binh Minh affirmed in his article on the occasion of the 70th anniversary of Vietnam-China diplomatic relationship.
Vietnamese Ambassador meets Ukrainian friendship official
Vietnamese Ambassador to Ukraine Nguyen Anh Tuan held a working session with Chairman of the Ukraine - Vietnam Friendship Parliamentarians' Group Bragar Evgeni Vadimovich in Kiev on January 15.
Congratulations to new Prime Minister of Russia
Prime Minister Nguyen Xuan Phuc on January 16 sent a message of congratulations to the new Prime Minister of Russia Mikhail Mishustin.
Int'l, NGOs meet ahead of Lunar New Year festival
A gathering of diplomatic representative agencies, international and foreign non-governmental organisations was held in Hanoi on January 16, ahead of the upcoming Lunar New Year festival.

redpajama_set_name: RedPajamaCommonCrawl


Mary Beth joined the Water4 team in 2017 as the Financial Stewardship Manager. Mary Beth and her husband John, Water4 Business Development Manager, started working with Water4 in 2015 as Financial Stewardship trainers, before joining the team full time. Before moving to Oklahoma City, Mary Beth lived in Richmond, Virginia and served as the Mid South Regional Director for Perspectives and prior to that was an art teacher for 29 years. Mary Beth and John have discovered that they serve the Lord better together and they love it! Together they have five grown children and five grandchildren.
Mary Beth is passionate about sharing Christ's love through teaching and mentoring. Her favorite part is watching "lightbulbs" go on in people's eyes as they grow, learn and accomplish things they thought were impossible.


text:
Burton TMD was a traction maintenance depot located in Burton upon Trent, Staffordshire, England.  The depot was situated on the Midland Main Line and was near Burton-on-Trent station.

The depot code was BU.

History
Before its closure in 1968, Class 02, 03, 08 and 25 locomotives could be seen at the depot.

References

1840 establishments in England
1968 disestablishments in England
 Railway depots in England
Rail transport in Staffordshire

redpajama_set_name: RedPajamaWikipedia


This post is excerpted from the Connectedness chapter of Cultivating Attention: the Paradoxical Secret of Team Success.
Agile software development includes a wide range of practices. The common characteristic of these practices is an incremental and iterative approach: do the work rapidly and in small batches, get feedback early and often, and repeat. At its founding in 2001, the Agile movement tied together many existing methodologies, and it has had a profound influence on many more methodologies since then. But at the heart of Agile is the concept of connectedness. The first phrase of the Agile Manifesto, the document that defined and continues to shape Agile work in all its flavors, is "Individuals and interactions over processes and tools." A bit later the Manifesto raises the importance of connectedness still further: "Customer collaboration over contract negotiation." These are simple words, butpowerful in their implications. Your relationships with the people you work with—and the people you are doing the work for—are more important to successful software development than how you do the work. In short: connectedness matters.
The abstract principles of the Manifesto are borne out inthe day-to-day of technology delivery. Incremental and iterative development emerged in response to the need for results that are fast, responsive, and flexible (it's not called "Agile" for nothing). In the chaos and swirl of a complex delivery project that's constantly changing, it is ridiculously easy to get out of synch with the teammate in the cube next to you, let alone a customer on the other side of the planet. As the Manifesto suggests, all the processes and communication tools in the world can't outperform a healthy habit of human connectedness.
As Agile has matured, it has helped spawn a number of important movements. Two of the most widely used and critical are SAFe (Scaled Agile Framework) and DevOps. SAFe, as the name suggests, takes the concepts of Agile into large enterprise contexts. Heavy-duty processes to coordinate extended teams with hundreds of members loom large in SAFe. If anything, the greater complexity and potential for miscommunication on a large project makes developing the habit of connectedness in an explicit, deliberate way even more important. The DevOps movement (a term that mashes up "development" and "IT operations") emerged to expand the scope of close collaboration across organizational boundaries to include operations, information security, testing,architecture, data management, and more. Given the increased complexity and intensity of these interactions, there's all the more need for cultivating the habit of connectedness.
Make it a practice, individually or as a team, to Expand Your Circle of Concern.The simple act of bringing to mind those individuals and teams you work with elsewhere in the building, across the country or around the world—with an attitude of concern—will increase your connectedness. At the same time, it will improve the quality of your work and help you deliver better results.
There are many different ways to expand your circle of concern. From my base in Seattle I was working with a team in Calgary, Canada,and could see that my understanding of their world had its limits, and those limits were impacting the Agile development work we were doing together. So, I paid a visit. At it happened, I arrived during the week of the Calgary Stampede,a huge midsummer rodeo and festival that (as I thought) perfectly captured the spirit of the city. Thinking to build some solidarity, I went to the Stampede,watched some cattle roping, and bought a black cowboy hat. At lunch with the team the next day, I discovered I was the only owner of a cowboy hat at the table: for that group, the Stampede was not really the place to go. They had a good laugh at my expense—but my attempt to build a connection strengthened our relationship.


text:
Home » Media Center » Benefits Seen All Round From ATO Payment Initiative
Media Source: https://insidesmallbusiness.com.au/
Date Published: September 4, 2016
Known as the Single Touch Payroll (STP), the initiative is set to roil out from July 1 next year. It recognises the payroll process as a fundamental business activity involving key employee data. As well as providing a basis to reduce compliance costs for employers, says the ATO, STP also securely leverages the data to also benefit employees.
"The streamlining of business payroll data and super payments should reduce costs for SMEs," says Sharp. "It will eliminate a lot of paperwork the Pay as You Go (PAYG) tax system generates for employers, who have to manually track their employee PAYG payments."
As the initiative makes the payment of wages more efficient and transparent, there will be flow-on effects for superannuation guarantee payments. "The government is looking to open the option to introduce real-time payments of superannuation at the time of making payroll payments, which is a sensible approach," says Sharp. "As money is invested sooner on behalf of the employee, it creates the potential to boost each individual's retirement savings."
"It may also help to reduce or even eliminate the incidence of non-compliance. Around $3 billion in annual superannuation payments is currently not paid, and this streamlined approach will help to track, monitor and reduce the incidence of non-compliance."
"What we see with STP is an efficiency bonus for small employers," says Sharp. "The government has also announced a one-off $100 tax offset for SMEs generating less than $2 million that buy approved reporting software."
Spring, 2016 Inside Small Business


A major concern for those foreigners coming to Thailand is the acquisition of a long term Thai visa. Even amongst the expatriate community in Thailand, the Thai visa issue can be a constant concern. This concern can be particularly acute because the Thai Immigration regulations regarding the issuance and extension of Thai visas are in a constant state of flux. Therefore, it can be difficult for foreigners to do research on Thai visa rules because much information quickly goes stale. This research problem is compounded by the fact that each Thai visa category has its own rules regarding issuance and extension.
If the applicant is not in the Kingdom of Thailand, then the submission of the Thai visa application will depend upon the applicant's country of origin. For many, obtaining the preferred Thai visa category will prove difficult if the proper documentation is not prepared before submission at the Thai Embassy or Consulate. In recent years, the Royal Thai Immigration Police have made entry regulations more stringent. Meanwhile, the Thai Ministry of Foreign Affairs has made it more difficult for applicants to receive multiple tourist visas at Embassies and Consulates in countries such as Laos, Cambodia, Malaysia, Singapore, Myanmar (Burma), Hong Kong, Vietnam, and the Philippines. Finally, the ultimate decision in Thai visa matters is left to the officer reviewing the application. It is not uncommon to see a Thai visa application rejected because the forms are not properly filled out or the applicant is light on documentation. It would seem that professional assistance may be a necessity when it comes to applying for visas to Thailand.
Some foreign nationals opt to retain the services of a firm to assist with the compilation of a Thailand visa application. In many cases, retaining a Thai firm to compile the Thailand visa application can facilitate faster visa approval. It is also true that oftentimes assistance may be necessary for those who wish to travel to the Kingdom of Thailand and remain for a prolonged period. There are many firms that offer Thai visa application assistance for as much as four to five hundred US Dollars. We at Integrity Legal feel that these types of fees are outrageous. In an effort to provide competent legal services at reasonable prices, Integrity Legal offers our Thai visa application service for 4710 Thai baht. This fee includes shipping and handling as well as application preparation. Not included in this fee is the application fee that must be paid directly to the Royal Thai Embassy or Consulate. This visa processing fee must be paid at the time the application is submitted.


text:
Mutual trust key to peaceful int'l relations
Author : JIANG SHIXUE Source : Chinese Social Sciences Today 2019-06-18
The contemporary world is facing a great change unseen in a hundred years. Though peace and development remain the themes of our era, the world is facing significantly increasing instability and uncertainty; various challenges faced by humanity are becoming increasingly severe. In his speech addressed in the United Nations Office at Geneva on Jan 18, 2017, President Xi Jinping asked, "What happened to the world, and what should we do?" These questions are shared by people all around the world.
In addressing the various challenges facing humanity, various plans have been proposed. Admittedly, remarkable achievements have been made in global governance in past decades, including the signing of the Paris Agreement on Climate Change, the breakthroughs made in the Doha Round negotiations, progress in the joint fight against terrorism, and the suppression of cybercrime. All of these are manifestations of effective cooperation among nations. However, compared with the severity of the challenges faced, and with, particularly, the expectations of the people in all nations, these achievements of global governance are hardly satisfying. Currently, people from the United Nations and from many non-governmental organizations, as well as leaders, scholars and civilians from every nation, all realize the necessity and importance of improving global governance. There is a voice calling for further reformation of the system of global governance.
Trust is not only the precondition for the harmonious coexistence of people, but also the basis for the peaceful coexistence of nations. Only on the basis of mutual trust can a global partnership with a shared future be established.
Addressing the trust deficit requires all nations to treat each other with respect, consideration and mutual understanding. At present, mutual trust and cooperation in the international community have been eroded by fierce international competition and ceaseless geopolitical gameplay. Against this backdrop, it is urgent that we prioritize mutual respect. In international relations, upholding mutual respect means seeking common ground while reserving disagreement. It means respecting the rights of the people in all countries to choose their own road of development. It means stopping imposing one's own will on other people and stopping using one's strength to bully the weak, let alone to intervene in the internal affairs of other countries.
A right approach to balancing righteousness and interests is an important idea for promoting trust. Chinese culture emphasizes that "a virtuous person prioritizes his principal of righteousness." This idea of prioritizing righteousness will help strengthen the mutual trust between different people and nations. China upholds this approach to righteousness and interests in international relations. China advocates the democratization of international relations and opposes the idea of developing oneself at the cost of other's interests. This idea is gaining increasing support from other nations. China has proposed the establishment of a community of shared future for mankind and the construction of a new type of international relations. The nation also promotes the joint construction of the Belt and Road. All these ideas and practices enrich multilateralism under the new circumstances, which will contribute greatly to enhancing the mutual trust among nations.
Honoring international commitments and abiding by the international rules is significant for advancing mutual trust. As a responsible major country, China has always been a builder of world peace, a contributor to global development and a defender of international order. China has been strictly abiding by the international rules and performing its international obligations. At the same time, China has taken an active role in drafting international rules. The reason why China proposes new mechanisms and initiatives is neither simply to start new ones nor directed against any certain country. China aims more for cooperation, mutual benefit and common development by supplementing and improving the existing international system.
In addressing the trust deficit, it is also necessary to strengthen the communication and dialogue among different civilizations. Every civilization has its own unique charm and profound tradition, which is also part of the spiritual treasure of humanity. Hence, different civilizations should learn from each other and advance side by side rather than despise and treat each other with arrogance and prejudice. Civilizations are colorful because of communication and rich because of mutual learning. The mutual learning among civilizations will deepen mutual understanding and recognition, which will guide people in different nations to understand, know, trust and respect each other. Mutual communication and learning among civilizations should become a bridge of friendship for people in all nations. It will also provide momentum for the progress of humanity and become a way to maintain peace in the world. This will help reach greater achievements in global governance.
This article was edited and translated from People's Daily. Jiang Shixue is a research fellow from the Chinese Academy of Social Sciences.
(Edited by CHEN ALONG)


text:
Q: How to use 'root = ' in Obnam? Using obnam 1.4 how do I get both /etc and /var in the same root to enable me to back them up please?
I've tried 'root = /etc, /var' which backs up /etc but not /var, and I've googled without success.
This is to be used on Debian 7, under Linux.

A: Have a try with obnam 1.5 or later, these work fine for me.
You should also make sure that obnam has the right to access those paths (i.e. the user executing obnam has)

redpajama_set_name: RedPajamaStackExchange


text:
local meta = FindMetaTable ("Player")
if (!meta) then
    Msg ("Failed to load Wire Player Extension for Target Finder!")
    return
end

// Variable to track lock count
meta.m_TargetLocks = {}

//-----------------------------------------------------------------------------
// Name: Player:TargetFinderLock()
// Desc: Called by target finder when it locks onto this player.
//-----------------------------------------------------------------------------

function meta:TargetFinderLock (ent)
    if (!table.HasValue (self.m_TargetLocks, ent)) then
        table.insert (self.m_TargetLocks, ent)
    end
end

//-----------------------------------------------------------------------------
// Name: Player:TargetFinderUnlock()
// Desc: Called by target finder when it unlocks this player
//-----------------------------------------------------------------------------

function meta:TargetFinderUnlock (ent)
    if (table.HasValue (self.m_TargetLocks, ent)) then
        for k,v in ipairs (self.m_TargetLocks) do
            if (v == ent) then
                table.remove (self.m_TargetLocks, k)
                return
            end
        end
    end
end

//-----------------------------------------------------------------------------
// Name: Player:IsLockedOnto ()
// Desc: Returns true if this player is a target, false otherwise.
//-----------------------------------------------------------------------------

function meta:IsLockedOnto ()
    return #self.m_TargetLocks > 0
end

redpajama_set_name: RedPajamaGithub


text:
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