%% 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&#2\\#3&#4\end{smallmatrix}\bigr)}
%% Easy change to display size 
\newcommand{\Smatrix}[4]{\begin{pmatrix}
    #1&#2\\#3&#4\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]{%
\[\begin{array}{rclrcl}
      \sashabox{#1} & \to
    &\multicolumn{2}{c}{\sashabox{#2}} & \to 
    & \sashabox{#3}
 \\   \\
    #4 & \mapsto  
    &#5&#6
    & \mapsto      & #7
\end{array}\]}
\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: 
