%% symbols.sty %% Math definitions for the Book of Involutions %%% CHANGES OF SOME STANDARD DEFINITIONS %% This defines \dots in text as LaTeX's \textellipsis %% which seems to be the same as plain's \dots in text. \newcommand{\AMSdots}{} \let\AMSdots\dots \renewcommand{\dots}{\relax\ifmmode\let\next\AMSdots \else\let\next\textellipsis\fi\next} %% This makes \dots before \cdot to lower dots \newcommand{\OLDcdot}{} \let\OLDcdot\cdot \def\cdot{\OLDcdot} %% \perp a binary operator \mathchardef\perp"223F %% This gives \sqrt[3]{xxx} the same before-spacing as \sqrt{xxx} \providecommand{\sqrtsign}{\radical "270370\relax} % for older latex versions \def\r@@t#1#2{% \setbox\z@\hbox{$\m@th#1\sqrtsign{#2}$}% \dimen@\ht\z@ \advance\dimen@-\dp\z@ \mkern4mu\raise.6\dimen@\copy\rootbox \mkern-10mu\box\z@} %%% END OF CHANGES OF SOME STANDARD DEFINITIONS %% Some blackboard-letters: \L_ = "letter_" \DeclareMathSymbol\LA{\mathalpha}{AMSb}{"41} \DeclareMathSymbol\LC{\mathalpha}{AMSb}{"43} \DeclareMathSymbol\LF{\mathalpha}{AMSb}{"46} \DeclareMathSymbol\LH{\mathalpha}{AMSb}{"48} \DeclareMathSymbol\LN{\mathalpha}{AMSb}{"4E} \DeclareMathSymbol\LP{\mathalpha}{AMSb}{"50} \DeclareMathSymbol\LQ{\mathalpha}{AMSb}{"51} \DeclareMathSymbol\LR{\mathalpha}{AMSb}{"52} \DeclareMathSymbol\LS{\mathalpha}{AMSb}{"53} \DeclareMathSymbol\LZ{\mathalpha}{AMSb}{"5A} \newcommand{\LQZ}{\LQ/\LZ} %% category-font %% this loads cmssi, the Companion calls it \mathsfsl, we call it \cat \DeclareMathAlphabet{\cat}{OT1}{cmss}{m}{sl} %% For the AMS's automatic \dots selection, %% one has to be careful with defining or renaming some %% binary operators. %% \newcommand{\tens}{\otimes} is wrong, but \let\tens\otimes is OK. %% also \newcommand{\tens}{\DOTSB\otimes} is OK. %% Math operators %% generic rule: \rm and \bf symbols should be mathop's, %% but not bold greek letters %% for lonely sub- and superscript this is not important %% %% \id will not be defined as mathop %% \DeclareMathOperator{\funC}{\mathbf{C}} % some functors \DeclareMathOperator{\funD}{\mathbf{D}} \DeclareMathOperator{\funEnd}{\mathbf{End}} \DeclareMathOperator{\funN}{\mathbf{N}} \DeclareMathOperator{\funM}{\mathbf{M}} \DeclareMathOperator{\funP}{\mathbf{P}} \DeclareMathOperator{\funS}{\mathbf{S}} \DeclareMathOperator{\funX}{\mathbf{X}} \DeclareMathOperator{\seqi}{\mathbf{i}} % a sequence \DeclareMathOperator{\Ad}{Ad} % adjoint representation \DeclareMathOperator{\ad}{ad} % inner derivation \DeclareMathOperator{\an}{an} % anisotropic \DeclareMathOperator{\alg}{alg} \DeclareMathOperator{\Alt}{Alt} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\CayD}{{\it CD}} % Cayley-Dickson process \DeclareMathOperator{\card}{card} \DeclareMathOperator{\car}{char} % \char is a TeX-command \newcommand{\Cay}{\mathfrak{C}} \DeclareMathOperator{\const}{const} % subscript \DeclareMathOperator{\cores}{cor} % \cor for corollary \DeclareMathOperator{\Der}{Der} \DeclareMathOperator{\Dyn}{Dyn} \newcommand{\dia}{} \let\dia\diamond \DeclareMathOperator{\diag}{diag} % operator and subscript \newcommand{\dir}{} \let\dir\oplus \DeclareMathOperator{\disc}{disc} \DeclareMathOperator{\End}{End} \newcommand{\equaldet}{\mathrm{Det}} % superscript in chap. 8-9 \newcommand{\equalnorm}{\mathrm{Norm}} % superscript in chap. 8-9 \newcommand{\et}{\cat{Et}} \DeclareMathOperator{\ett}{et} % unfortunate choices \DeclareMathOperator{\expon}{exp} % exponent of algebra \newcommand{\Fs}{F_{\sep}} \newcommand{\galpha}{\boldsymbol{\alpha}} % usual letter \DeclareMathOperator{\gAlt}{\mathbf{Alt}} \DeclareMathOperator{\gAut}{\mathbf{Aut}} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Fix}{Fix} %new Knus 4.3.97 \DeclareMathOperator{\gal}{\cat{Gal}} \DeclareMathOperator{\GO}{GO} \newcommand{\gbeta}{\boldsymbol{\beta}} \DeclareMathOperator{\gG}{\mathbf{G}} \DeclareMathOperator{\gGa}{\mathbf{G}_a} \newcommand{\gGamma}{\boldsymbol{\Gamma}} \DeclareMathOperator{\gGL}{\mathbf{GL}} \newcommand{\gGLm}{\gGL_m} \newcommand{\gGLn}{\gGL_n} \DeclareMathOperator{\gGm}{\mathbf{G}_m} \newcommand{\gGmF}{\gG_{\mathrm{m},F}} \newcommand{\gGmK}{\gG_{\mathrm{m},K}} \newcommand{\gGmL}{\gG_{\mathrm{m},L}} \newcommand{\gGmZ}{\gG_{\mathrm{m},Z}} \DeclareMathOperator{\gGO}{\mathbf{GO}} \DeclareMathOperator{\gGSp}{\mathbf{GSp}} \DeclareMathOperator{\gGU}{\mathbf{GU}} %\DeclareMathOperator{\gISO}{\mathbf{Isom}} \DeclareMathOperator{\gISO}{\mathbf{Iso}} \DeclareMathOperator{\gI}{\mathbf{I}} \DeclareMathOperator{\gE}{\mathbf{E}} \DeclareMathOperator{\gi}{\mathbf{i}} \DeclareMathOperator{\gJ}{\mathbf{J}} \DeclareMathOperator{\gj}{\mathbf{j}} \DeclareMathOperator{\GL}{GL} \newcommand{\gmu}{\boldsymbol{\mu}} \DeclareMathOperator{\gN}{\mathbf{N}} \newcommand{\gOmega}{\boldsymbol{\Omega}} \DeclareMathOperator{\gOrth}{\mathbf{O}} \DeclareMathOperator{\gONE}{\mathbf{1}} \DeclareMathOperator{\gO}{\mathbf{O}} \newcommand{\gOplus}{\gO^+} \DeclareMathOperator{\GU}{GU} \DeclareMathOperator{\E}{E} \DeclareMathOperator{\gP}{\mathbf{P}} \DeclareMathOperator{\gPGL}{\mathbf{PGL}} \DeclareMathOperator{\gPGO}{\mathbf{PGO}} \DeclareMathOperator{\gPGSp}{\mathbf{PGSp}} \DeclareMathOperator{\gPGU}{\mathbf{PGU}} \DeclareMathOperator{\gQ}{\mathbf{Q}} \DeclareMathOperator{\Gr}{Gr} \DeclareMathOperator{\gSim}{\mathbf{Sim}} \DeclareMathOperator{\gSL}{\mathbf{SL}} \DeclareMathOperator{\gSpin}{\mathbf{Spin}} \DeclareMathOperator{\gSp}{\mathbf{Sp}} \DeclareMathOperator{\GSp}{GSp} \DeclareMathOperator{\gSU}{\mathbf{SU}} \DeclareMathOperator{\gS}{\mathbf{S}} \DeclareMathOperator{\gT}{\mathbf{T}} \DeclareMathOperator{\gUn}{\mathbf{U}} \DeclareMathOperator{\gU}{\mathbf{U}} \DeclareMathOperator{\gu}{\mathfrak{g}} \DeclareMathOperator{\gV}{\mathbf{V}} \DeclareMathOperator{\Hom}{Hom} %%\DeclareMathOperator{\Id}{Id} \newcommand{\Id}{{\rm Id}} % not a mathoperator \DeclareMathOperator{\im}{im} % Image \DeclareMathOperator{\Ind}{Ind} % induced module \DeclareMathOperator{\ind}{ind} % index \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Inv}{Inv} % cohomological invariants \newcommand{\inv}{^{-1}} \DeclareMathOperator{\ISO}{Isom} \DeclareMathOperator{\isomet}{Iso} \DeclareMathOperator{\K}{K} % K-theory \DeclareMathOperator{\LIE}{Lie} \newcommand{\Lie}{\mathfrak{L}} % unfortunate choices \newcommand{\Lorth}{\mathfrak{o}} \DeclareMathOperator{\M}{M} % matrices \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\mult}{mult} \DeclareMathOperator{\Norm}{\mathfrak{n}} \DeclareMathOperator{\nil}{nil} % set of nilpotent elements \DeclareMathOperator{\Nrd}{Nrd} \DeclareMathOperator{\Nrp}{Nrp} \DeclareMathOperator{\op}{op} % superscript \DeclareMathOperator{\Orth}{O} \newcommand{\orthoplus}{\DOTSB\mathbin{\stackrel{\perp}{\oplus}}} \newcommand{\bigorthoplus}{\mathop{\stackrel{\perp}{\bigoplus}}} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\Pcrd}{Prd} % ? see \Prd \DeclareMathOperator{\Pc}{Pc} % characteristic polynomial \DeclareMathOperator{\pf}{pf} % Pfaffian \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\PGO}{PGO} \DeclareMathOperator{\PGSp}{PGSp} \DeclareMathOperator{\PGU}{PGU} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Prd}{Prd} \DeclareMathOperator{\Prp}{Prp} \DeclareMathOperator{\PSim}{PSim} \DeclareMathOperator{\Quot}{Quot} \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\rdim}{rdim} % reduced rank \DeclareMathOperator{\red}{red} % subscript \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Sand}{Sand} \DeclareMathOperator{\SB}{SB} \DeclareMathOperator{\sets}{\cat{Sets}} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\SGU}{SGU} \DeclareMathOperator{\sgu}{\mathfrak{s}} \DeclareMathOperator{\Sim}{Sim} \DeclareMathOperator{\Skew}{Skew} \DeclareMathOperator{\SK}{SK} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\SN}{SN} \DeclareMathOperator{\Sn}{Sn} \DeclareMathOperator{\spelin}{\mathfrak{sl}} \DeclareMathOperator{\Spin}{Spin} \DeclareMathOperator{\Spec}{Spec} \let\Sp\undefined \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\Srd}{Srd} \DeclareMathOperator{\SU}{SU} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\SSym}{SSym} %% Since Sym corresponds to H^{1}(U), %% I thought SSym should correspond %% to H^{1}(SU) (jp) \DeclareMathOperator{\Symd}{Symd} \DeclareMathOperator{\sep}{sep} % subscript \newcommand{\tens}{} \let\tens\otimes \newcommand{\gradtens}{\DOTSB\mathbin{\widehat{\otimes}}} %% graded \tens \DeclareMathOperator{\tors}{\cat{Tors}} %? \DeclareMathOperator{\Trd}{Trd} \DeclareMathOperator{\Trp}{Trp} %\DeclareMathOperator{\Tr}{tr} % remove in the end \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\trdeg}{tr.deg} % the '.' gets an unwanted space \DeclareRobustCommand{\trdeg}{\qopname\relax o{tr.deg}} \DeclareMathOperator{\UK}{UK} \DeclareMathOperator{\USK}{USK} \DeclareMathOperator{\Un}{U} \newcommand{\unsigma}{\underline{\sigma}} %%\DeclareMathOperator{\Witt}{W} %%% TeX is not made for left superscripts. %% I don't see any other way than to deal with each case separately \DeclareMathOperator{\goneA}{\mathbf{{}^1\mkern-2muA}} \DeclareMathOperator{\gtwoA}{\mathbf{{}^2\mkern-2muA}} \newcommand{\oneA}{{}^1\!A} \newcommand{\twoA}{{}^2\!A} \newcommand{\iA}{{}^i\!A} \newcommand{\onecatA}{{}^1\mkern-2mu\cat{A}} \newcommand{\twocatA}{^2\mkern-2mu\cat{A}} \DeclareMathOperator{\gtwoC}{\mathbf{{}^2C}} \DeclareMathOperator{\goneC}{\mathbf{{}^1C}} \newcommand{\oneDfour}{{}^1\mkern-1muD_4} \newcommand{\twoDfour}{{}^2\mkern-1muD_4} \newcommand{\threeDfour}{{}^3\mkern-1muD_4} \newcommand{\sixDfour}{{}^6\mkern-1muD_4} \newcommand{\onecatOku}{{}^1\mkern-1mu\cat{Oku}} \newcommand{\twocatOku}{{}^2\mkern-1mu\cat{Oku}} \newcommand{\icatOku}{{}^i\mkern-1mu\cat{Oku}} \newcommand{\twocatSepalt}{{}^2\mkern-1mu\cat{Sepalt}} \newcommand{\nPic}{\mathop{{}_n\mathrm{Pic}}\nolimits} \newcommand{\eBr}{\mathop{{}_e\mathrm{Br}}\nolimits} \newcommand{\twoBr}{\mathop{{}_2\mathrm{Br}}\nolimits} \newcommand{\fourBr}{\mathop{{}_4\mathrm{Br}}\nolimits} \newcommand{\nBr}{\mathop{{}_n\mathrm{Br}}\nolimits} \newcommand{\leftone}{{}^1} \newcommand{\leftgf}{\mkern1mu{}^g\mkern-2muf} \newcommand{\leftag}{\mkern1mu{}^a\mkern-1mug} \newcommand{\leftiota}{{}^\iota} \newcommand{\iotaA}{{}^\iota\!A} \newcommand{\iotaa}{{}^\iota\mskip-1.5mu a} \newcommand{\iotab}{{}^\iota b} \newcommand{\iotaab}{\iotaa\mskip1mu \iotab} \newcommand{\iotac}{{}^\iota\mskip-1.5mu c} \newcommand{\iotae}{{}^\iota\mskip-1.5mu e} \newcommand{\iotaf}{{}^\iota\mskip-1.5mu f} \newcommand{\iotasigma}{{}^\iota\mskip-1mu\sigma} \newcommand{\leftgamma}{{}^{\gamma}} \newcommand{\leftgammarho}{\leftgamma\mkern-1.5mu\rho} \newcommand{\leftrho}{{}^\rho} \newcommand{\leftrhosquare}{{}^{\rho^2}} \newcommand{\leftrhoinvalpha}{{}^{\rho\inv}\!\!\alpha} \newcommand{\leftsigma}{{}^{\sigma}} \newcommand{\sigmaf}{{}^{\sigma}\mkern-2muf} \newcommand{\lefttheta}{{}^\theta} %%% some special symbols %% stand-alone Involution \newcommand{\upstar}{{}^*} \newcommand{\invo}{\mkern1mu\rule[1.3ex]{0.48em}{0.1ex}\mkern1mu} %% Pfisterforms \newcommand{\Pf}{\langle\!\langle} \newcommand{\pF}{\rangle\!\rangle} \newcommand{\pFt}{]]} %% Quaternions in char 2 %\newcommand{\Quat}{\left\Vert} %\newcommand{\quaT}{\right\Vert} %\newcommand{\Quat}{\left(\mkern-4mu\left(} %\newcommand{\quaT}{\right)\mkern-4mu\right)} \newcommand{\Quat}{(\!(} \newcommand{\quaT}{)\!)} %% empty brackets (not suitable for use in scriptstyle) \newcommand{\emptyform}{\langle\mskip2mu\kern.1em,\mskip2mu\rangle} \newcommand{\emptylie}{[\mskip3mu\kern.1em,\mskip3mu]} %% p-adic \newcommand{\p}{} \def\p-{\relax\ifdim\fontdimen1\font>0pt$p$\else$p\mkern1.5mu$\fi-} % \bmod with no left argument \def\amod{{\rm mod}\mkern5mu} % \Iff \newcommand{\Iff}{}\let\Iff\Leftrightarrow %% there should be only one \subsetneq: \let\subsetneq\subsetneqq \let\varsubsetneq\subsetneqq \let\varsubsetneqq\subsetneqq %% there should be only one \setminus: \let\setminus\smallsetminus %% : as a punctuation \newcommand{\col}{} \let\col\colon %% reverse \mapsto \newcommand{\lmapsto}{\mathrel{\leftarrow\mathrel {\vcenter{\llap{\vrule\vbox{\vphantom{$\mapstochar$}}$\mkern1mu$}}}}} %% Jordan product \jdot only one size \newcommand{\jdotfont}{} \font\jdotfont lcircle10 scaled 913 % 833=\magstep-2 913=\magstep-1 \newcommand{\osbullet}{\jdotfont\char113} % 'blob' (2pt) \newlength{\sbwd} \settowidth{\sbwd}{\osbullet} \newcommand{\csbullet}{\kern.5\sbwd\osbullet\kern-.5\sbwd} \newcommand{\jdot}{\DOTSB \mathbin{\mkern1mu\vcenter{\hbox{\csbullet}}\mkern1mu}} %% \isom : isomorphism-arrow, better version of \xrightarrow{\sim} %% not for \scriptstyle or \scriptscriptstyle \newcommand{\joinrelshort}{\mathrel{\mkern-9mu}} \newcommand{\shortlongrightarrow}{\relbar\joinrelshort\rightarrow} \newcommand{\isom}{\mathrel{\mathop{\setbox0\hbox{$\mathsurround0pt \shortlongrightarrow$}\ht0=0.7\ht0\box0}\limits ^{\sim\mkern2mu}}} %% degree of field extension, or index of groups, usage: \degree(L:F) %% \degree is convenient, but does not work if the second argument %% contains a ')'. Then use \degre{L}{F} (Vive la France!) \newcommand{\degre}[2]{\mathchoice% {[#1\mskip\thinmuskip{\mathop:}\mskip\thinmuskip#2]}% {[#1\mskip\thinmuskip{\mathop:}\mskip\thinmuskip#2]}% {[#1{\mathop:}#2]}% {[#1{\mathop:}#2]}} \newcommand{\degree}{} \def\degree(#1:#2){\degre{#1}{#2}} %% small 2x2 matrices in text \newcommand{\smatrix}[4]{\bigl(\begin{smallmatrix} #1\\#3\end{smallmatrix}\bigr)} %% Easy change to display size \newcommand{\Smatrix}[4]{\begin{pmatrix} #1\\#3\end{pmatrix}} %%% Spacing, boxes, etc. %% paranthesis in second level are \bigl,\bigr %% paranthesis in third level are \bigll,\bigrr or sometimes \left,\right \newcommand{\bigll}{} \let\bigll\bigl \newcommand{\bigrr}{} \let\bigrr\bigr %% \left,\right give more space than \bigl,\bigr %% the names \bigll(rr) make it more easy to keep control %% displayed sets: \newcommand{\bmidb}[2]{\{\,#1\mid#2\,\}} \newcommand{\Bmidb}[2]{\left\{\,#1 \mathrel{\left.\vphantom{#1}\vphantom{#2}\right|}#2\,\right\}} %% \textq(uad): an abbreviation for certain use of text in displays \newcommand{\textq}[1]{\quad\text{#1}\quad} %% Sasha's Boxes. \newcommand{\sashabox}[1]{\fbox{\upshape\begin{tabular}{c}#1\end{tabular}}} \newcommand{\sashaboxA}[6]{% \[\begin{array}{rcl} \sashabox{#1} & \longleftrightarrow & \sashabox{#2} \\ \\ #3& \mapsto & #4 \\ #5& \lmapsto & #6 \end{array}\]} \newcommand{\sashaboxB}[2]{% \[\sashabox{#1} \quad\longleftrightarrow\quad \sashabox{#2}\]} \newcommand{\sashaboxC}[2]{% \[\sashabox{#1} \quad \longleftrightarrow\quad #2\]} \newcommand{\sashaboxCa}[2]{\[ \hbox to \hsize{\sashabox{#1}\hfill$\longleftrightarrow$\hfill$#2$}\]} \newcommand{\sashaboxD}[7]{% \[\begin{array}{rclrcl} \sashabox{#1} & \to &\multicolumn{2}{c}{\sashabox{#2}} & \to & \sashabox{#3} \\ \\ #4 & \mapsto  & \mapsto & #7 \end{array}\]} \newcommand{\sashaboxJP}[8]{% \par \[\begin{array}{rcc} \sashabox{#1} & \longleftrightarrow & \sashabox{#2} \\\\ \sashabox{#3} & \longleftrightarrow & \sashabox{#4} \\\\ \sashabox{#5} & \longleftrightarrow & \sashabox{#6} \\\\ \sashabox{#7} & \longleftrightarrow & \sashabox{#8} \\ \end{array}\]} %% french << >> for Tits' preface %% taken from german.sty \newcommand{\allowhyphens}{} \newcommand{\save@sf@q}{} \def\allowhyphens{\penalty\@M \hskip\z@skip} \def\save@sf@q#1{{\ifhmode \edef\@SF{\spacefactor\the\spacefactor}\else \let\@SF\empty \fi \leavevmode #1\@SF}} \DeclareRobustCommand{\flqq}{% \ifmmode\mathrel{\hbox{\guillemotleft}}\else\guillemotleft\fi} \ProvideTextCommandDefault{\guillemotleft}{% \UseTextSymbol{OT1}\guillemotleft} \ProvideTextCommand{\guillemotleft}{OT1}{% \ifmmode \ll \else \save@sf@q{\penalty\@M \raise .27ex\hbox{$\m@th\scriptscriptstyle \ll $}% \allowhyphens}\fi} \DeclareRobustCommand{\frqq}{% \ifmmode\mathrel{\hbox{\guillemotright}}\else\guillemotright\fi} \ProvideTextCommandDefault{\guillemotright}{% \UseTextSymbol{OT1}\guillemotright} \ProvideTextCommand{\guillemotright}{OT1}{% \ifmmode \gg \else \save@sf@q{\penalty\@M \raise .27ex\hbox{$\m@th\scriptscriptstyle \gg $}% \allowhyphens}\fi} %% end of symbols.sty %%% Local Variables: %%% mode: latex %%% TeX-master: "boi" %%% End: