\documentclass{beamer} % This file is a solution template for: % - Giving a talk on some subject. % - The talk is between 15min and 45min long. % - Style is ornate. \usepackage{color} \definecolor{hue}{rgb}{.202, .602, .58} \definecolor{red}{rgb}{.902, .02, .08} \definecolor{green}{rgb}{.02, .902, .08} \definecolor{blue}{rgb}{.02, .02, .908} \definecolor{orange}{rgb}{.902, .502, .08} \def\ppp{{\mathbb{P}}} \def\aaa{{\mathbb{A}}} \def\fff{{\mathbb{F}}} \def\qqq{\mathbb{Q}} \def\rrr{\mathbb{R}} \def\ccc{\mathbb{C}} \def\zzz{\mathbb{Z}} \def\nnn{\mathbb{N}} \def\Aut{{\rm{Aut}}} \def\supp{{\rm{Supp}}} \def\Stab{{\rm{Stab}}} \def\eval{{\rm{eval}}} \def\wt{{\rm{wt}}} \def\pf{\noindent {\bf Proof}:\ } \def\qed{$\Box$} % Copyright 2004 by Till Tantau . % % In principle, this file can be redistributed and/or modified under % the terms of the GNU Public License, version 2. % % However, this file is supposed to be a template to be modified % for your own needs. For this reason, if you use this file as a % template and not specifically distribute it as part of a another % package/program, I grant the extra permission to freely copy and % modify this file as you see fit and even to delete this copyright % notice. \mode { \usetheme{Warsaw} %\usecolortheme[named=Green]{structure} %% must define Green... % or ... \setbeamercovered{transparent} % or whatever (possibly just delete it) } \usepackage[english]{babel} % or whatever \usepackage[latin1]{inputenc} % or whatever \usepackage{times} \usepackage[T1]{fontenc} % Or whatever. Note that the encoding and the font should match. If T1 % does not look nice, try deleting the line with the fontenc. \title{Riemann-Roch space repns from a bad hyperelliptic} \author{D.~Joyner} % - Use the \inst{?} command only if the authors have different % affiliation. \institute[Math Dept, USNA] % (optional, but mostly needed) { Department of Mathematics\\ US Naval Academy\\ Annapolis, MD 21402} \date[May, 2005]{\color{blue} Riemann-Roch space representations from bad hyperelliptic curves: questions and computations \normalcolor\\ \ \\ {\footnotesize{Computational Algebraic Geometry Conference\\ Idaho\\ 5-27-2005}}} \subject{Talks} % This is only inserted into the PDF information catalog. Can be left % out. % If you have a file called "university-logo-filename.xxx", where xxx % is a graphic format that can be processed by latex or pdflatex, % resp., then you can add a logo as follows: % \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename} % \logo{\pgfuseimage{university-logo}} % Delete this, if you do not want the table of contents to pop up at % the beginning of each subsection: \AtBeginSubsection[] { \begin{frame} \frametitle{Outline} \tableofcontents[currentsection,currentsubsection] \end{frame} } % If you wish to uncover everything in a step-wise fashion, uncomment % the following command: %\beamerdefaultoverlayspecification{<+->} \begin{document} \begin{frame} \titlepage \end{frame} \begin{frame} \frametitle{Outline} \tableofcontents[pausesections] \pause The main results described in this talk are the result of joint work with Amy Ksir. Some of this is ``work in progress''. \end{frame} % Since this a solution template for a generic talk, very little can % be said about how it should be structured. However, the talk length % of between 15min and 45min and the theme suggest that you stick to % the following rules: % - Exactly two or three sections (other than the summary). % - At *most* three subsections per section. % - Talk about 30s to 2min per frame. So there should be between about % 15 and 30 frames, all told. \section{Introduction} \subsection[Motivation]{Motivation} \begin{frame} \frametitle{History - a classical problem.} \framesubtitle{Questions.} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item $F=\ccc$ (for now) \item $C$ a non-singular projective curve over $F$, genus $>1$, automorphism group $G$. \item $G$ acts on space of differentials on $C$ and, more generally, on the Riemann-Roch space $L(D)$ ($D$ $G$-equivariant divisor) \pause \item \color{hue} {\bf Questions}: What are these representations? \pause \item Can we compute their character? \pause \item Their multiplicities? \normalcolor \pause \item Special cases: Eichler trace formula, Weil-Chevalley formula \pause \item General (in the extreme): the Borne character formula. \end{itemize} \end{frame} \subsection[Borne's character formula]{Borne's character formula} \begin{frame} \frametitle{Borne's character formula} \framesubtitle{Character of a R-R repns, even in wild/bad case} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item $F$ ``arbitrary'', $P\in C(F)$ \item $G_P$ the subgroup of $G$ fixing $P$ \pause \item $G_P$ acts on the cotangent space at $P$ by an $F$-character. This is the {\bf ramification character} of $C$ at $P$. \pause \item {\bf ramification module} is $ \Gamma_G= \sum_{P\in C(F)_{ram}} Ind_{G_P}^G(\sum_{\ell=1}^{e_P-1}\ell\psi_P^{\ell}), $ where $e_P=|G_P|$, $\psi_P=$ramification character at $P$. \pause \item There is a unique $G$-module $\tilde{\Gamma}_G$ such that $ \Gamma_G= |G| \tilde{\Gamma}_G. $ By abuse of terminology call $\tilde{\Gamma}_G$ the {\bf ramification module} \end{itemize} \end{frame} \begin{frame} \frametitle{Borne's character formula, cont.} \framesubtitle{Character of a R-R repns, even in wild/bad case} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item {\bf reduced orbit}: $D=\frac{1}{e_P}\sum_{g\in G}g(P)$, some $P\in C(F)$ \pause \item The reduced orbits generate $Div(C)^G$. \pause \item {\bf equivariant degree}: a map $ deg_{eq}:Div(C)^G\rightarrow R(G), $ satisfying explicit conditions which can be computed if $D$ is a reduced orbit. \pause \item {\bf Borne's formula}: $D$ a $G$-equivariant nonspecial divisor $\implies$ (virtual) character of $L(D)$ is \[ [L(D)]=(1-g(C/G))[F[G]]+[deg_{eq}(D)]-[\tilde{\Gamma}_G], \] where $g(C/G)=$genus of $C/G$, $\deg_{eq}(D)=$equivariant degree of $D$, $\tilde{\Gamma}_G=$ramification module. \end{itemize} \end{frame} \subsection[Our multiplicity formula]{Our multiplicity formula} \begin{frame} \frametitle{Multiplicity formula} \framesubtitle{Rational repns case} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item Let $F=\ccc$ denote the complex numbers (for now) \pause \item For each conjugacy class $\gamma_i \in G_*$, pick a cyclic subgroup $H_i$ generated by a $g\in \gamma_i$. \pause \item Let $n_{\rho,j}=n_{\rho,j}(H)\geq 0$ denote the multiplicities $\langle \rho|_H,\psi^j\rangle_H$, (so $Res_{H}(\rho)=\sum_{j =1}^{|H|} n_{\rho,j}\psi^j$), where \begin{itemize} \item $\rho=$irreducible representation of $G$, \item $H=$cyclic subgroup of $G$, \item $\psi=$primitive character of $H$. \end{itemize} \end{itemize} \end{frame} \begin{frame} \frametitle{Multiplicity formula - cont.} \framesubtitle{Rational repns case - cont.} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item %$Div(C)^G$ is generated by orbits $r\sum_{g\in G/G_P}g(P)$ %\pause \item ${\rm supp}(D)=\cup_{i=1}^s \{g(P_i)\ |\ g\in G/G_{P_i}\}$, a disjoint union \pause \item write ${\rm supp}(D)/G=\{P_i\ |\ 1\leq i\leq s\}$, \pause \item \[ D=\sum_{P\in {\rm supp}(D)/G}r_P \sum_{g\in G/G_P} g(P), \] for some $r_P\in \zzz$. \pause \item Let $L(D)_\rho=$ the $\rho$-isotypical component of $L(D)$ \end{itemize} \end{frame} \begin{frame} \frametitle{Rational repns case - cont.} \framesubtitle{Rational repns case - cont.} \begin{itemize} \item Let $G\subset Aut(C)$, $D$ a $G$-equivariant nonspecial divisor. \pause \item Assume all irreducible characters of $G$ are rational-valued. \pause \item {\bf multiplicity formula}: \[ {\rm dim}(L(D)_\rho) ={\rm dim}(\rho)(1-g(C/G))+ \sum_{P\in {\rm supp}(D)/G} T_{P,\rho} \] \[ \ \ \ \ \ -\sum_\ell (\dim(\rho)-\dim(\rho^{H_\ell}))\frac{R_\ell}{2}, \] (in Joyner-Ksir, ``Decomposing representations of finite groups on Riemann-Roch spaces''), \end{itemize} \end{frame} \begin{frame} \frametitle{Rational repns case - cont.} \framesubtitle{Rational repns case - cont.} \begin{itemize} \item where \[ T_{P,\rho}= \left\{ \begin{array}{l} \sum_{\ell =1}^{r_P} n_{\rho,\ell }(G_P) =\sum_{\ell =1}^{r_P} \langle \psi_P^\ell,{\rm Res}_{H_\ell}^G \rho\rangle, \ \ r_P>0,\\ -\sum_{\ell =0}^{-(r_P+1)} n_{\rho,-\ell }(G_P) =\sum_{\ell =1}^{-(r_P+1)} \langle \psi_P^{-\ell},{\rm Res}_{H_\ell}^G \rho\rangle,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r_P<0, \end{array} \right. \] \begin{itemize} \item $\rho^{H_\ell}=$restriction of $\rho$ to $H_\ell$, \item $R_\ell=R(H_\ell)=$number of branch points with decomposition group conjugate to $H_\ell$, \item $\rho_1$, ...., $\rho_k$ a complete set of irreducible $\qqq [G]$-modules ($=$irreducible $\ccc [G]$-modules). \end{itemize} \end{itemize} \end{frame} \begin{frame} \frametitle{SAGE} \begin{center} \fbox{Advertisement} \end{center} \begin{itemize} \item \begin{center} For number theory and modular forms (and more), try SAGE! \pause \color{blue} SAGE 0.3 \normalcolor is released! \end{center} \pause \item \begin{center} sage.sourceforge.net \item {\footnotesize{ (permanent \color{blue} SAGE\normalcolor \ homepage now moving from William Stein's site at Harvard to his new site at Univ. Calif. San Diego) }} \end{center} \pause \item \begin{center} It is \color{orange}open source\normalcolor\ and free! \end{center} \begin{figure}[h] \begin{center} \includegraphics[height=2.5cm,width=2cm]{/home/wdj/texfiles/latex-beamer/beamer/solutions/generic-talks/sage.eps} \end{center} \end{figure} \end{itemize} \end{frame} \section[Bad curves and big automorphism groups]{Repns of big groups on R-R spaces of bad curves} \begin{frame} \frametitle{Big groups, bad curves} \framesubtitle{Big and bad} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item If $C$ is a curve defined over a field $F$, $G=Aut_F(C)$ is finite with $|G|>|C(F)|$, then we call $G$ {\bf big}. If in addition char$(F)$ divides $|G|$ then we call $C$ {\bf bad}. \pause \item {\bf Lemma}: If $G$ is large then every point of $C(F)$ is ramified for the covering $C\rightarrow C/G$. \pause \item Proof: Suppose $P\in C(F)$ is not ramified, so the stabilizer of $P$, $G_P$, is trivial. In this case, $|G\cdot P|= |G|/|G_P|= |G|$. But $G\cdot P\subset C(F)$ so $|G\cdot P|\leq |C(F)|$, a contradiction. QED \end{itemize} \color{hue} {\bf Open problem}: Find multiplicity formulas for bad curves. \normalcolor \end{frame} \subsection[A big bad hyperelliptic]{A bad hyperelliptic curve with a big automorphism group} \begin{frame} \frametitle{$y^2=x^p-x$ over $GF(p)$} \framesubtitle{$y^2=x^p-x$ over $GF(p)$} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item $p\geq 5$ be a prime, $F=GF(p)$, $C: y^2=x^p-x$ \pause \item the weighted projective model: $Y^2=X^pZ-XZ^p$ (here $(X,Y,Z)$ ($x=X/Z$, $y=Y/Z^{g+1}$) with weights $1$, $\frac{p+1}{2}$, and $1$) \pause \item automorphism group $G=\Aut_F(C)$ is a central 2-fold cover of $G_0=PSL(2,p)$, we have a short exact sequence, \[ 1\rightarrow Z \rightarrow G\rightarrow G_0\rightarrow 1, \] \[ G\cong SL(2,p),\ \ \ |SL(2,p)|=\frac{(p^2-1)(p^2-p)}{p-1} \] where $Z$ denotes the center of $G$ ($Z$ is generated by the hyperelliptic involution). \end{itemize} \end{frame} \begin{frame} \frametitle{Generators of $G$ acting on $C$} \framesubtitle{Generators of $G$} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item $G$ is generated by: \item \[ \begin{array}{cc} \gamma_1= \left\{ \begin{array}{c} x\longmapsto x,\\ y\longmapsto -y, \end{array} \right. , & \gamma_2=\gamma_2(a)= \left\{ \begin{array}{c} x\longmapsto a^2x,\\ y\longmapsto ay, \end{array} \right.\\ \gamma_3= \left\{ \begin{array}{c} x\longmapsto x+1,\\ y\longmapsto y, \end{array} \right. , & \gamma_4= \left\{ \begin{array}{c} x\longmapsto -1/x,\\ y\longmapsto y/x^{\frac{p+1}{2}}, \end{array} \right. \end{array} \] where $a\in F^\times$ is a primitive $(p-1)^{st}$ root of unity and $Z=\langle \gamma_1\rangle$. \end{itemize} \end{frame} \begin{frame} \frametitle{$C$ is bad, $G$ is big} \framesubtitle{Bad $C$, big $G$} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item $G$ acts transitively on $C(F)$, \pause \item every point in \[ C(F)=\{[1:0:0],[0:0:1],[1:0:1],...,[p-1:0:1] \} \] is ramified over the covering $C\rightarrow C/G$ \pause \item each stabilizer $G_P=\Stab_{G}(P)$ is non-trivial, $P\in C(F)$. \pause \item Let $P_1=[1:0:1]$, $H=G_{P_1}$. $H$ is a solvable group of order $p(p-1)$ generated by $\gamma_1$, $\gamma_2(a)$ and $\gamma_3$ (think of $H$ as a ``Borel subgroup'' of $G$) \end{itemize} \end{frame} \begin{frame} \frametitle{Riemann-Roch representation of $H$ on $L(mP_1)$} \framesubtitle{Repn of $H$ on $L(mP_1)$} % - A title should summarize the slide in an understandable fashion % for anyone how does not follow everything on the slide itself. \begin{itemize} \item for each $m\geq 1$, the Riemann-Roch space of $D=mP_1$ has a basis: $ x^iy^j,\ \ \ \ \ \ 0\leq i\leq p-1,\ j\geq 0,\ 2i+pj\leq m. $ \pause \item The semisimplification $\rho_{ss}$ of the representation $\rho$ of $H$ acting on $L(D)$ is the direct sum of one-dimensional representations of $H$. \pause \item %{\footnotesize{ {\tiny{ \[ \gamma_3: \left( \begin{array}{c} 1\\ x\\ \vdots \\ x^ry^s \end{array} \right) \longmapsto \left( \begin{array}{c} 1\\ x+1\\ \vdots \\ (x+1)^ry^s \end{array} \right) = \left( \begin{array}{cccc} 1 & 0 & ...& 0\\ 1 & 1 & ...& \vdots\\ \vdots &\ddots & &0 \\ 0 & ... & r & 1 \end{array} \right) \left( \begin{array}{c} 1\\ x\\ \vdots \\ x^ry^s \end{array} \right), \] }} where the non-zero terms in bottom row of the matrix representation of $\gamma_3$ consist of the binomial coefficients $\frac{r!}{(r-j)!j!}$, $0\leq j\leq r$. \end{itemize} \end{frame} \begin{frame} \frametitle{Background on the function field $K=F(C)=F(x,y)$} Properties of the function field $K=F(C)=F(x,y)$ \pause \begin{itemize} \item genus $g=\frac{p-1}{2}$ \pause \item $[K:F(y)]=p$ \pause \item As an $F$-vector space \[ F(x,y)=F(y)\oplus xF(y)\oplus ... x^{p-1}F(y). \] \pause \item $K/F(y)$ is Galois and \[ \begin{array}{ccc} Gal(K/F(y)) & =& F\\ \sigma &\mapsto & a\\ \sigma(x)=x+a& & \end{array} \] \end{itemize} \end{frame} \begin{frame} \frametitle{Background on the function field $K=F(C)=F(x,y)$, cont.} \begin{itemize} \item The pole $P_\infty$ of $y$ in $F(y)=F(\ppp^1)$, has a unique totally ramified extension $Q_\infty$=``point at infinity on $C$'' \pause \item $e(Q_\infty/P_\infty)=p$ and $Q_\infty$ is a place of $C$ of degree $1$. \pause \item $(x)_\infty =2Q_\infty$, $(y)_\infty = pQ_\infty$ \pause \item \[ L(rQ_\infty) = Span[\ x^iy^j\ |\ 2i+pj \leq r, 0\leq i, 0\leq j \leq p-1\ ]. \] \end{itemize} \end{frame} \subsection{$G$-equivariant divisors and their RR spaces} \begin{frame} \frametitle{$G$-equvariant divisors} \begin{itemize} \item Let $ D_F =\sum_{P\in C(F)}P, $ so $\deg(D_F)=|C(F)|=p+1=2g+2$ and therefore $rD_F$ is non-special for each $r\geq 1$. \pause \item \[ \begin{array}{ll} \dim L(rD_F)&=\deg(rD_F)-g+1\\ &=r(p+1)-\frac{p+1}{2}+1\\ &=(2r-1)g+2r+1 \end{array} \] \pause \item $r\geq 1\implies$ \[ \dim L((r+1)D_F)/L(rD_F) =p+1=2g, \] \end{itemize} \end{frame} \begin{frame} \frametitle{Irreducible $G$-modules} \begin{itemize} \item \color{hue} {\bf Questions}: : Is $L(D_F)$ an irreducible $G$-module? Is $L((r+1)D_F)/L(rD_F)$ an irreducible $G$-module? \pause \normalcolor \item {\bf Answer}: No! No! \pause \item Irred $F[G]$-modules: Denote the irred degree $n$ $G$-module by $V_n$, $V_n=\{\sum_i a_iX^iY^{n-i-1}\}$, $n=1$, $2$, ..., $p$. (These are all of them.) \pause \item Therefore, no $G$-module of dimension $p+1$ can be irreducible. \end{itemize} \end{frame} \begin{frame} \frametitle{$L(D_F)$ as a $G$-module} Let $ W={\rm Span}\{\ \frac{yx^k}{x^p-x} \ \ |\ 0\leq k\leq \frac{p+1}{2}\}. $ \begin{itemize} \item $\ $ \item $\dim W=\frac{p+3}{2}$. \pause \item $W$ remains invariant under the action of $\gamma_1$, $\gamma_2(a)$, and $\gamma_3$ \pause \item $\gamma_4:\frac{yx^k}{x^p-x}\longmapsto -\frac{yx^{\frac{p+1}{2}-k}}{x^p-x}, $ \pause \item Therefore $W$ is a $G$-module. \end{itemize} \end{frame} \begin{frame} \frametitle{$L(D_F)$ as a $G$-module, cont.} \begin{itemize} \item {\bf Lemma}: $L(D_F)=W\oplus {\mathbf 1}$, as $G$-modules. \pause \item Proof: $W$ is a $G$-module and by definition, $W\subset L(D_F)$. $L(D_F)$ contains the constant functions on $C$, so $W\oplus {\mathbf 1}\subset L(D_F)$. Since $D_F$ is non-special, Riemann-Roch $\implies$ $\dim L(D_F)=1+\dim W$. QED \pause \item \color{hue} {\bf Questions}: Is there a similar theorem for $L(rD_F)$? \pause \item {\bf Question}: How does $W$ decompose? \normalcolor \end{itemize} \end{frame} \begin{frame} \frametitle{The character of $W$} \begin{itemize} \item {\small{ \[ {\rm tr\, }\rho(\gamma_2(a)) =\sum_{k=0}^{d} a^{2k-1} =a^{-1}\frac{a^{2d+2}-1}{a^2-1} \] \[ \ \ \ \ \ \ =a^{-1}\frac{a^{p+3}-1}{a^2-1} =a^{-1}\frac{a^{4}-1}{a^2-1}=a+a^{-1}, \] }} \pause \item \[ {\rm tr\, }\rho(\gamma_3)=\deg\, \rho=\frac{p+3}{2} \] \pause \item \[ {\rm tr\, }\rho(\gamma_4)=0. \] \end{itemize} \end{frame} \begin{frame} \frametitle{The character of $W$, cont.} \begin{itemize} \item $t=\left( \begin{array}{cc} a & 0\\ 0 & a^{-1} \end{array} \right)\implies$ \pause \item \[ {\rm tr\, }V_n(t) =\sum_{i=0}^{n-1} a^{2i-n} =a^{1-n}\frac{a^{2n}-1}{a^2-1} =\frac{a^{n}-a^{-n}}{a-a^{-1}}, \] when $n\geq 1$. \pause \item When $n=2$ this agrees with ${\rm tr\, }\rho(\gamma_2(a))$. \pause \item When $n=\frac{p-1}{2}$, ${\rm tr\, }V_n(t)=0$. \end{itemize} \end{frame} \begin{frame} \frametitle{The character of $W$, cont.} \begin{itemize} \item For {\it modular} representations, two representations can have the same character yet not only be inequivalent but have different decompositions. \pause \item same character $\implies$ their corresponding multiplicities must be $\equiv \pmod p$. \pause \item $\ $ \item {\bf Conjecture}: $W=V_2\oplus V_{\frac{p-1}{2}}$. \end{itemize} \end{frame} \begin{frame} \frametitle{GUAVA} \begin{center} \fbox{Advertisement} \end{center} \begin{itemize} \item \begin{center} For error-correcting codes, try \color{green} GUAVA! \normalcolor \pause \color{green} GUAVA 2.3 \normalcolor just released! \end{center} \pause \item \color{green} GUAVA! \normalcolor home: www.gap-system.org/Packages/guava.html \pause \item $\ $ \item \begin{center} It is \color{orange}open source\normalcolor\ and free! \end{center} \item \begin{figure}[h] \begin{center} \includegraphics[height=1cm,width=2cm]{/home/wdj/texfiles/latex-beamer/beamer/solutions/generic-talks/guava.eps} \end{center} \end{figure} \end{itemize} \end{frame} \subsection{A really cool example} \begin{frame} \frametitle{Cool example - $y^2=x^7-x$} \begin{itemize} \item Let $F=GF(7)$, $C$: $y^2=x^7-x$, . This has genus $3$. \pause \item There are $8$ $F$-rational points: {\footnotesize{ \[ C(F)=\{ P_1=[1:0:0], P_2=[0:0:1], P_3=[1:0:1], ..., P_8=[6:0:1]\}. \] }} \pause \item $H=Stab(P_1,G)=$group of order \pause \color{red} $42$ \normalcolor \pause generated by $\gamma_1$, $\gamma_2$, $\gamma_3$ \item ($H/center=$unique non-abelian group of order $21$) \end{itemize} \end{frame} \begin{frame} \frametitle{A cool example, cont.} \begin{itemize} \item For each $m\geq 1$, the Riemann-Roch space $L(mP_1)$ has a basis consisting of monomials, \[ x^iy^j,\ \ \ \ \ \ 0\leq i\leq 6,\ j\geq 0,\ 2i+7j\leq m. \] \pause \item Let $D=5P_1$, $S=C(F)-\{P_1\}$, and let \[ C_D(S)=\{(f(P_2), ...,f(P_8))\ |\ f\in L(D)_F\}. \] \pause \item This $C_D$ is a $[7,3,5]$ code over $F$ and the evaluation map $f\longmapsto (f(P_2), ...,f(P_8))$, $f\in L(D)$, is injective. \end{itemize} \end{frame} \begin{frame} \frametitle{A cool example, cont.} \begin{itemize} \item $H$ fixes $D$ and preserves $S$, so it acts on $C_D(S)$ via \[ g:(f(P_2), ...,f(P_8))\longmapsto (f(g^{-1}P_2), ...,f(g^{-1}P_8)),\ \ \ \ \ \ g\in H. \] \pause \item $P=$permutation automorphism group of this code. \pause \begin{center} $|P|=$\pause \color{red} $42$ \normalcolor. \end{center} \pause {\small{ Note for Douglas Adams fans: Recall the ultimate answer to the universe is \pause \color{red} $42$ \normalcolor.}} \pause \item The (permutation) action of $G$ on this code implies that there is a homomorphism $\psi :H_1\rightarrow P$. \pause \color{hue} {\bf Questions}: What is the kernel of this map? Is it trivial?? \pause \item {\bf Answer}: Big kernel - non-trivial! \normalcolor \end{itemize} \end{frame} \begin{frame} \frametitle{A cool example, cont.} \begin{itemize} \item If $S=\{P_2,...,P_8\}\leftrightarrow\{1,2,...,7\}$ then \item \[ \gamma_1 \leftrightarrow (2,7)(3,6)(4,5)=g_1, \] \[ \gamma_2 \leftrightarrow (2,5,3)(4,6,7)=g_2, \] \[ \gamma_3 \leftrightarrow (1,2,...7)=g_3. \] \pause \item $N={\rm ker}(\psi)=$non-abelian normal subgroup of $H\cong \langle g_1,g_2,g_3\rangle$ of order $21$, $N=\langle g_2,g_3\rangle$ \end{itemize} \end{frame} \begin{frame} \frametitle{A cool example, cont.} \begin{itemize} \item \color{hue} {\bf Questions}: What is the character of $\rho:H\rightarrow Aut_F(L(5P_1))$? \pause \normalcolor \item The character table (over $\ccc$) of $N$ is \begin{center} \begin{tabular}{c|ccccc} Class & 1& 2& 3& 4& 5\\ Size & 1& 7& 7& 3& 3\\ Order & 1& 3& 3& 7& 7\\ \hline p = 7 & 1 & 2 & 3 & 1 & 1\\ \hline $\chi_1$ & 1 & 1 & 1 & 1 & 1\\ $\chi_2$ & 1 & $\omega$ &$-1-\omega$& 1 & 1\\ $\chi_3$ & 1 &$-1-\omega$ & $\omega$ & 1 & 1\\ $\chi_4$ & 3 & 0 & 0 & $\zeta$ & $\zeta^3$\\ $\chi_5$ & 3 & 0 & 0 & $\zeta^3$ & $\zeta$\\ \end{tabular} \end{center} \noindent where $\omega=$cube root of unity ($\zeta$ is unimportant here) \end{itemize} \end{frame} \begin{frame} \frametitle{A cool example, cont.} \begin{itemize} \item The (Brauer) character table (over $F$) of $N$ is \begin{center} \begin{tabular}{|c|ccccc|} \hline $\chi_{1a}$ & 1 & 1 & 1 & 1 & 1 \\ $\chi_{1b}$ & 1 & $\omega^2$ & $\omega$ & 1 & 1 \\ $\chi_{1c}$ & 1 & $\omega$ & $\omega^2$ & 1 & 1 \\ \hline \end{tabular} \end{center} \noindent (last two conjuacy classes are irregular mod $7$) \pause \item The character table of $N$ $\implies$ \[ {\rm tr}\rho=\chi_{1a}+\chi_{1b}+\chi_{1c}. \] \pause \item \color{hue} {\bf Open Problem}: Generalize this $p=7$ example. \normalcolor \pause \item \color{hue} {\bf Discussion topic}: Does $42$ have cosmic significance? \normalcolor \end{itemize} \end{frame} \begin{frame} \frametitle{The end} \begin{center} \fbox{The end} \end{center} \begin{center} Thank you! \end{center} %\vskip .2in % latex-beamer.sourceforge.net \end{frame} \end{document}