% Bill Cook -- AMS Bard Meeting
% October 9, 2005 3:30pm
% 


\documentclass[pdf,azure,slideColor,colorBG]{prosper}
%\usepackage{graphics}
%\Logo{\includegraphics[height=1in]{ncsu_math.eps}}

%\usepackage[dvips]{color}
%\definecolor{yellow}{named}{yellow}


\def \<{\langle} 
\def \>{\rangle}

% title page information
\title{Aff{}ine Lie Algebras,\\ Vertex Operator Algebras,\\ and Multisum Identities}
\subtitle{Technical Slides}
\author{William J. Cook\\
        Rutgers University\\
	Department of Mathematics\\
	New Brunswick, NJ}
\email{wjcook@math.rutgers.edu}
\institution{
joint work with\\
Kailash Misra and Haisheng Li
}

% text at the bottom of each slide. if none given, uses talk title
\slideCaption{\textit{Technical Slides}}

\begin{document}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}

\newcommand{\g}{\mathfrak{g}}
\newcommand{\h}{\mathfrak{h}}

% make the title slide
\maketitle

\begin{slide}{Outline}

\vspace{0.3in}

\begin{itemize}
\item highest-weight integrable modules for aff{}ine Lie algebras and their VOA-module structure
\item VOA-module isomorphisms and recurrence relations
\item a character formula for $L(\Lambda_0)$ 
\item principal characters and multisum identities
\end{itemize}

\end{slide}

\overlays{4}{
\begin{slide}{Notation:}

\FromSlide{1} 
\begin{itemize}
\item Let $\g$ be a simple Lie algebra (over $\mathbb{C}$) of {\yellow rank $\ell$}.
\FromSlide{2} 
\item Let $C=(a_{ij})_{1 \leq i,j \leq \ell}$ be the {\yellow Cartan matrix} of $\g$.
\FromSlide{3} 
\item F{}ix a set of simple roots. Def{}ine the {\yellow root lattice} to be the $\Z$-span of these roots:
      \[ Q = \Z \alpha_1 + \Z \alpha_2 + ... + \Z \alpha_{\ell}. \]
\FromSlide{4} 
\item Def{}ine the {\yellow weight lattice} to be the $\Z$-span of the {\yellow fundamental weights}:
      \[ P = \Z \lambda_1 + \Z \lambda_2 + ... + \Z \lambda_{\ell}. \]
\end{itemize}

\end{slide}
}

\overlays{4}{
\begin{slide}{Notation:}

\FromSlide{1}
\begin{itemize}
\item F{}ix {\yellow Chevalley generators} $\{ E_i, F_i, H_i \,|\, 1 \leq i \leq \ell \}$.
\FromSlide{2} 
\item The $H_i$ ($1 \leq i \leq \ell$) span a {\yellow Cartan subalgebra} (CSA) denoted by $\h$. 
      (Note: $\dim(\h)=\ell$.)
\FromSlide{3} 
\item Let $\< \cdot, \cdot \> : \g \times \g \rightarrow \C$ be the {\yellow normalized nondegenerate
      symmetric invariant bilinear form}.
\FromSlide{4}
\item We have the root space decomposition 
                \[ \g = \h \oplus \bigoplus\limits_{\alpha \in \Delta} \g_{\alpha} \]
      where {\yellow $\Delta$ is the set of roots}.
\end{itemize}

\end{slide}
}

\overlays{4}{
\begin{slide}{Notation:}

\FromSlide{1}
\begin{itemize}
\item By def{}inition, $\{ H_1,...,H_{\ell} \}$ and $\{ \lambda_1,...,\lambda_{\ell} \}$ are dual bases 
      (i.e. $\lambda_i(H_j)=\delta_{i,j}$). 
\FromSlide{2}
\item Def{}ine the {\yellow fundamental coweights}, $\{ H^{(1)}, ..., H^{(\ell)} \} \subseteq \h$, 
      to be the basis dual to $\{ \alpha_1,...,\alpha_{\ell} \}$. 
\FromSlide{3}
\item Def{}ine the {\yellow coroot lattice}:
      \[ Q^{\vee} = \Z H_1 + \Z H_2 + ... + \Z H_{\ell}. \]
\FromSlide{4}
\item Def{}ine the {\yellow coweight lattice}:
      \[ P^{\vee} = \Z H^{(1)} + \Z H^{(2)} + ... + \Z H^{(\ell)}. \]
\end{itemize}

\end{slide}
}

\overlays{3}{
\begin{slide}{Notation:}

\FromSlide{1}
\begin{itemize}
\item Let ${\yellow \rho^{\vee}} = \sum\limits_{i=1}^{\ell} H^{(i)}$.
\FromSlide{2}
\item We have that $\langle H_i, H^{(j)} \rangle = \frac{2}{\< \alpha_i, \alpha_i \>}$.
\FromSlide{3}
\item The coroots and coweights are related as follows: 
\[ H_j = \sum\limits_{i=1}^{\ell} a_{ij}H^{(i)} \] 
      where $C = (a_{ij})$ is the Cartan matrix of $\g$.
\end{itemize}

\end{slide}
}

\begin{slide}{The Aff{}inization of $\g$}

Def{}ine {\yellow $\hat{\g}$} as follows:
\[ \hat{\g} = \g \otimes \mathbb{C}[t,t^{-1}] \oplus \C c \]
where $c$ is central and
\[ [a \otimes t^m, b \otimes t^n] = [a,b] \otimes t^{m+n} + m \< a,b \> \delta_{m+n,0} c \]
for every $a,b \in \g$ and $m,n \in \Z$.

Extend the map $d = 1 \otimes t \frac{\mathrm{d}}{\mathrm{d}t}$ to all of $\hat{\g}$ by letting $d(c)=0$.
Let $\tilde{\g} = \hat{\g} \rtimes \C d$ be the semi-direct product Lie algebra.

$\tilde{\g}$ is the {\yellow (untwisted) aff{}ine Lie algebra} associated with $\g$.

\end{slide}

\overlays{4}{
\begin{slide}{Chevalley Generators for $\tilde{\g}$}

\FromSlide{1}
\begin{itemize}
\item Let $\theta = \sum\limits_{i=1}^{\ell} a_i \alpha_i$ be the {\yellow highest (long) root} of $\g$. 
\FromSlide{2}
\item Let $E_\theta$ and $F_\theta$ be elements of the root spaces $\g_\theta$ and $\g_{-\theta}$
      respectively such that $\< E_\theta, F_\theta \> = 1$. Set $H_\theta = [E_\theta, F_\theta]$.
\FromSlide{3}
\item Let $e_i = E_i \otimes 1$, $f_i = F_i \otimes 1$, and $h_i = H_i \otimes 1$ for $1 \leq i \leq \ell$. 
\item Let $e_0 = F_\theta \otimes t$, $f_0 = E_\theta \otimes t^{-1}$, and $h_0 = [e_0, f_0]$. 
\end{itemize}

\FromSlide{4}
Then, $\{ e_i, f_i, h_i \,|\, 0 \leq i \leq \ell \}$ are {\yellow Chevalley generators} for $\tilde{\g}$. Also, 
$\tilde{\h} = \h \oplus \C c \oplus \C d$ is our CSA.

\end{slide}
}

\overlays{2}{
\begin{slide}{The Homogeneous Gradation}
\FromSlide{2}
Thus we have the following decomposition of $\hat{\g}$:
\[ \hat{\g} = \coprod\limits_{n \in \Z} \hat{\g}_{(n)} \qquad \mbox{where} \qquad
   \hat{\g}_{(n)} = \left\{ \begin{array}{ll} \g \oplus \C c & n=0 \\
                                              \g \otimes t^{-n} & n \not= 0 \end{array} \right. \]
\FromSlide{2}
We have the following graded subalgebras of $\hat{\g}$:
\[ \hat{\g}_{(\pm)} = \coprod\limits_{n > 0} \hat{\g}_{(\pm n)} 
                    = \coprod\limits_{n > 0} \g \otimes t^{\mp n} \]
and
\[ \hat{\g}_{(\leq 0)} = \coprod\limits_{n \leq 0} \hat{\g}_{(n)} 
                       = \hat{\g}_{(-)} \oplus \hat{\g}_{(0)} 
                       = \hat{\g}_{(-)} \oplus \g \oplus \C c. \]
\end{slide}
}

\overlays{2}{
\begin{slide}{The Principal Gradation}

\FromSlide{1}
The second gradation we shall consider is the principal gradation. 

The {\yellow principal gradation} is given by the derivation $d_P$ which we def{}ine on the Chevalley
generators of $\hat{\g}$ and extend using the Leibnitz rule. For $0 \leq i \leq \ell$, let
\begin{itemize}
\item $d_P(e_i)=e_i$,
\item $d_P(f_i)=-f_i$, and
\item $d_P(h_i)=0$.
\end{itemize}  

\FromSlide{2}
{\yellow Lemma:} On $\hat{\g}$,
\[ d_P = (\mathrm{ht}(\theta)+1)d + \mathrm{ad}(\rho^{\vee})  \]
where $\theta$ is the highest long root of $\g$.

\end{slide}
}

\overlays{3}{
\begin{slide}{From $\g$ to $\tilde{\g}$}

\FromSlide{1}
Given $\lambda \in \h^{\ast}$ and $k \in \C$, we can def{}ine a linear functional 
${\yellow (k,\lambda)} \in \tilde{\h}^{\ast}$ by the following:
\begin{itemize} 
\item For all $h \in \h$, let $(k, \lambda)(h) = \lambda(h)$. 
\item On $\C c$, let $(k, \lambda)(c) = k$.
\item On $\C d$, let $(k, \lambda)(d) = 0$. 
\end{itemize}

\FromSlide{2}
Recall that $\lambda_i$ ($1 \leq i \leq \ell$) are the fundamental weights of $\g$. For convenience, let
$\lambda_0 = 0$. Then, def{}ine {\yellow $k \Lambda_i = (k,\lambda_i)$} for $0 \leq i \leq \ell$.

\FromSlide{3}
\vskip 0.2in
{\yellow $\Lambda_0 = (1,\lambda_0) = (1,0)$}, $\Lambda_1 = (1,\lambda_1)$, ..., $\Lambda_\ell = (1,\lambda_\ell)$
are the {\yellow fundamental weights} for $\tilde{\g}$.

\end{slide}
}

\overlays{5}{
\begin{slide}{The Verma Module $V(k,\lambda)$}

\FromSlide{1}
F{}ix a complex number $k \in \C$ and let $U$ be the irreducible f{}inite dimensional $\g$-module with 
highest-weight $\lambda \in P_+$. Extend the $\g$-module, $U$, to a $\hat{\g}_{(\leq 0)}$-module $U_k$
 as follows:
\FromSlide{2}
{\small
\begin{itemize}
\item Let $U_k = U$ (as a $\C$-vector space).
\FromSlide{3}
\item Let $\hat{\g}_{(-)}$ act trivially on $U_k$.
\FromSlide{4}
\item Let $c$ act as the scalar $k 1_U$ (i.e. $c \cdot u = ku$ for all $u \in U$).
\end{itemize}
}

\FromSlide{5}
Def{}ine the {\yellow Verma module} (of level $k$ and highest-weight $(k,\lambda)$) as follows:
\[ V(k,\lambda) = U(\hat{\g}) \otimes_{U(\hat{\g}_{(\leq 0)})} U_k. \]

\end{slide}
}

\overlays{3}{
\begin{slide}{The Simple VOA $L(k \Lambda_0)$}

\FromSlide{1}
{\yellow Theorem:} Suppose that $k \not= -h^{\vee}$ {\yellow the dual Coxeter number}. Then,

\begin{itemize}
\item $V(k \Lambda_0)=V(k,0)$ is a vertex operator algebra,
\FromSlide{2}
\item $N$ (the maximal proper submodule) is an ideal of $V(k \Lambda_0)$, and 
\item $L(k \Lambda_0) \,\,(\, = V(k \Lambda_0)/N \,)$ is a simple vertex operator algebra.
\end{itemize}   

\FromSlide{3}
{\yellow Theorem:} If $k \not= -h^{\vee}$ and $\lambda \in P_+$, 
then $L(k,\lambda)$ ($=V(k,\lambda)/\mbox{(maximal submodule)}$ is an irreducible $V(k\Lambda_0)$-module.
   
\end{slide}
}

\overlays{3}{
\begin{slide}{Li's Theorem}

\FromSlide{1}
For all that follows, let $k \in \Z_{>0}$.

\vspace{0.1in}

\FromSlide{2}
Let $H \in P^{\vee}$. Def{}ine 
\[ \Delta(H,x) = x^{H_0} \mathrm{exp} \left( \sum\limits_{k=1}^{\infty}
                                  \frac{H_k}{-k}(-x)^{-k} \right). \]
\vspace{0.1in}

\FromSlide{3}
{\yellow Theorem [Li]:} If $L(k,\lambda)$ is an irreducible $L(k \Lambda_0)$-module and $H \in Q^{\vee}$, then
\begin{center} $(L(k,\lambda),Y(\cdot,x))$\\
{\small is isomorphic to}\\ 
$(L(k,\lambda),Y({\yellow \Delta(H,x)} \cdot,x))$ \end{center}
as an $L(k \Lambda_0)$-module.

\end{slide}
}

\begin{slide}{The Automorphism's Effect}

{\yellow Notation:} For $v \in L(k \Lambda_0)$, let 

\[ Y(v,x) = \sum\limits_{m \in \Z} v(m)x^{-m-1}, \]

\[ Y(\Delta(H,x)v,x) = \sum\limits_{m \in \Z} v_{(H)}(m)x^{-m-1} \]

and 

\[ Y(\Delta(H,x)\omega,x) = \sum\limits_{m \in \Z} L_{(H)}(m)x^{-m-2}. \]

\end{slide}


\begin{slide}{The Automorphism's Effect}

{\yellow Lemma:} For $1 \leq i,j \leq \ell$,

\[ (H^{(j)})_{(H_i)}(0) = H^{(j)}(0) + \frac{2}{\langle \alpha_i, \alpha_i \rangle} \delta_{i,j} k \]

\begin{eqnarray*}
L_{(H_i)}(0) & = & L(0) + H_i(0) + \frac{2}{\langle \alpha_i, \alpha_i \rangle} \delta_{i,j} k \\
 & = & L(0) + \sum\limits_{j=1}^{\ell} a_{ji}H^{(j)}(0) + \frac{2}{\langle \alpha_i, \alpha_i \rangle} 
       \delta_{i,j} k 
\end{eqnarray*}

\end{slide}

\begin{slide}{Characters}

{\yellow Def{}inition:} Let $L(k,\lambda)$ be an irreducible $L(k \Lambda_0)$-module.

\vspace{0.2in}

The (full) {\yellow character} of $L(k,\lambda)$ is given by
\[ \chi_{L(k,\lambda)}(x_1,x_2,...,x_{\ell};q) = \qquad \qquad \qquad \qquad \qquad \]
\[ \mathrm{tr}_{L(k,\lambda)} x_1^{H^{(1)}(0)} 
       x_2^{H^{(2)}(0)} ... x_{\ell}^{H^{(\ell)}(0)} q^{L(0)}. \]
       
\end{slide}

\begin{slide}{The Main Theorem}

\vspace{0.5in}

{\yellow Main Theorem:} If $L(k,\lambda)$ is an irreducible $L(k\Lambda_0)$-module,
then for each $1 \leq i \leq \ell$ we have that

\vspace{0.3in}

\[ \chi_{L(k,\lambda)}(x_1,x_2,...,x_{\ell};q) = \qquad \qquad \qquad \qquad \qquad \] 

\[ (x_i q)^{\frac{2k}{\langle \alpha_i, \alpha_i \rangle}} 
   \chi_{L(k,\lambda)}(x_1q^{a_{1i}},x_2q^{a_{2i}},...,x_{\ell}q^{a_{\ell i}};q). \]

\end{slide}

\overlays{2}{
\begin{slide}{Recurrence Relations}

\FromSlide{1}
\vspace{0.1in}

{\yellow Notation:} Let 
\[ \mathbf{n} = (n_1,n_2,...,n_{\ell}) \in \Z^{\ell} \]
 and 
\[ \mathbf{x}^{\mathbf{n}} = x_1^{n_1}x_2^{n_2}...x_{\ell}^{n_{\ell}}. \]

\FromSlide{2}
\vspace{0.3in}

Def{}ine coeff{}icients ${\yellow A(\mathbf{n};q)} \in q^{-\mu} \C [[q]]$ as follows:

\[ \chi_{L(k,\lambda)}(x_1,x_2,...,x_{\ell};q) = \sum\limits_{\mathbf{n} \in \Z^{\ell}} 
    A(\mathbf{n};q) \mathbf{x}^{\mathbf{n}} \]

{\small Note: $L(0)=-d+\mu$ for some $\mu \in \C$.}
\end{slide}
}

\begin{slide}{Recurrence Relations}

\vspace{0.2in}

{\yellow Corollary:} For $1 \leq i \leq \ell$ and $(n_1,...,n_{\ell}) \in \Z^{\ell}$,

\vspace{0.3in}

{\small \[ A(n_1,...,n_{\ell};q) = \qquad \qquad \qquad \qquad \qquad \]

\[ A(n_1,...,n_{i-1},n_i - \frac{2k}{\< \alpha_i, \alpha_i \>}, n_{i+1}, ..., n_{\ell};q)
                           q^{-\frac{2k}{\< \alpha_i, \alpha_i \>} + \sum_{j=1}^{\ell} a_{ji}n_j}. \] }

\end{slide}

\begin{slide}{A Special Case}

Let us specialize to level {\yellow $k=1$} and $\g$ of {\yellow $(ADE)$-type} (simply laced).
Then $\< \alpha_i, \alpha_i \> = 2$ for all $1 \leq i \leq \ell$. So we have 
the following recurrence relations:

\vspace{0.3in}

For $1 \leq i \leq \ell$ and $(n_1,n_2,...,n_{\ell}) \in \Z^{\ell}$,

\vspace{0.2in}

\[ A(n_1,...,n_{\ell};q) = \qquad \qquad \qquad \qquad \qquad \qquad \qquad \] 

\[ A(n_1,...,n_{i-1},n_i {\yellow - 1}, n_{i+1}, ..., n_{\ell};q)
                           q^{-1 + \sum_{j=1}^{\ell} a_{ji}n_j} \]

\end{slide}

\overlays{2}{
\begin{slide}{A Sum Formula}

\FromSlide{1}
Using the lattice construction of $L(\Lambda_0) = V_Q$ we find that 
\begin{eqnarray*}
A(0,...,0;q) & = & \dim_q S(\mathfrak{h}\otimes t^{-1}\mathbb{C}[t^{-1}]) \\
             & = & \prod\limits_{j=1}^{\infty} (1-q^j)^{-\ell}. 
\end{eqnarray*}

\FromSlide{2}
{\yellow Theorem:} Let $\g$ be a simple Lie algebra of $(ADE)$-type.
 
 \[ \chi_{L(\Lambda_0)}(\mathbf{x};q) = \prod\limits_{j=1}^{\infty} (1-q^j)^{-\ell}
    \sum\limits_{\mathbf{n} \in \Z^{\ell}} q^{\frac{1}{2} \mathbf{n} C \mathbf{n}^t} \mathbf{x}^{\mathbf{n}} \]

\end{slide}
}


\overlays{2}{
\begin{slide}{Characters}

\FromSlide{1}
Recall that ($\mathrm{ht}(\theta) =$ height of the highest long root)
\begin{eqnarray*}
d_P & = & (\mathrm{ht}(\theta)+1)d + ad(\rho^{\vee}) \\
    & = & (\mathrm{ht}(\theta)+1)(-L(0)+\mu) + ad(\rho^{\vee})
\end{eqnarray*}

\FromSlide{2}
{\yellow Def{}inition:} Let $L(k,\lambda)$ be an irreducible $L(k \Lambda_0)$-module. 

\vspace{0.1in}

The {\yellow principal character} of $L(k,\lambda)$ is given by
\begin{eqnarray*}
\chi^P_{L(k,\lambda)}(q) & = & \mathrm{tr}_{L(k,\lambda)} q^{-d_P} \\
  & = & q^{-(\mathrm{ht}(\theta)+1)\mu} \chi_{L(k,\lambda)}(q^{-1},...q^{-1};q^{\mathrm{ht}(\theta)+1}).
\end{eqnarray*}

\end{slide}
}

\overlays{2}{
\begin{slide}{The Principal Character}

\FromSlide{1}
\vspace{0.2in}

{\yellow Corollary:} Let $\g$ be of $(ADE)$-type (i.e. simply laced) and $k=1$. Then,

\vspace{0.2in}
$\chi_{L(\Lambda_0)}^P(q) =$\\
\[ \prod_{j \geq 1} (1-q^{({\rm ht}(\theta)+1)j})^{-\ell}
   \sum_{{\bf n}\in \Z^\ell} q^{\frac{{\rm ht}(\theta)+1}{2} {\bf n} C {\bf n}^t - \sum_{i=1}^\ell n_i} \]

\vspace{0.3in}

\FromSlide{2}
Thus we have obtained a multisum formula for the principal character. Now using the previously known formulas,
we obtain families of multisum identities...

\end{slide}
}

\begin{slide}{Type $A_{\ell}$}

\vspace{0.2in}

{\yellow Theorem:} {\small If $C=(a_{ij})$ is the Cartan matrix of type $A_{\ell}$ ($\ell \geq 1$),}

\vspace{0.2in}

\[
 \prod_{j \geq 1} \frac{(1-q^{(\ell+1)j})^{(\ell+1)}}{(1-q^j)} = 
 \sum_{{\bf n} \in \Z^{\ell}} q^{\frac{\ell+1}{2} {\bf n}C {\bf n}^{t} - \sum_{i=1}^{\ell} n_i}. 
\]

\end{slide}

\begin{slide}{Example: Type $A_1$}

\vspace{0.2in}

{\yellow Example:} For type $A$ and rank $\ell=1$, $C = (2)$. So we have, 

\vspace{0.1in}

\[ \prod_{j \geq 1} \frac{(1-q^{2j})^2}{(1-q^j)} = \sum_{n \in \Z} q^{ 2n^2 - n}. \]

\vspace{0.1in}

This is a well known formula credited to Gauss.      
   
\end{slide}

\begin{slide}{Example: Type $A_2$}

\vspace{0.2in}

{\yellow Example:} For type $A$ and rank $\ell=2$, 
\[ C = \left( \begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array} \right). \] 
So we have, 

\vspace{0.1in}

\[ \prod_{j \geq 1} \frac{(1-q^{3j})^{3}}{(1-q^j)} = 
   \sum_{n_1,n_2 \in \Z} q^{3(n_1^2 - n_1 n_2 + n_2^2) - (n_1 + n_2)}. \]
   
\end{slide}


\begin{slide}{Type $D_{\ell}$}

\vspace{0.2in}

{\yellow Theorem:} {\small If $C=(a_{ij})$ is the Cartan matrix of type $D_{\ell}$ ($\ell \geq 4$),}

\vspace{0.2in}

\begin{eqnarray*}
 & \prod_{j \geq 1} \frac{(1-q^{2(\ell-1)j})^{\ell}}{(1-q^{2j-1})(1-q^{(\ell-1)(2j-1)})} & \\
 & & \\
 & = & \\
 & & \\
 & \sum_{{\bf n} \in \Z^{\ell}} q^{(\ell-1) {\bf n}C {\bf n}^{t} - \sum_{i=1}^{\ell} n_i}. &
\end{eqnarray*}

\end{slide}

\begin{slide}{Example: Type $D_4$}

\vspace{0.2in}

{\yellow Example:} For type $D$ and rank $\ell=4$, 
\[ C = \left( \begin{array}{cccc}  2 & -1 &  0 &  0 \\ 
                                  -1 &  2 & -1 & -1 \\
				   0 & -1 &  2 &  0 \\
				   0 & -1 &  0 &  2 \end{array} \right). \] 
\end{slide}

\begin{slide}{Example: Type $D_4$}

\vspace{0.2in}

{\yellow Example:} For type $D$ and rank $\ell=4$, 

\vspace{0.1in}

So we have, 

\vspace{0.1in}

\begin{center}
\[ \prod_{j \geq 1} \frac{(1-q^{6j})^{4}}{(1-q^{(2j-1)})(1-q^{(6j-3)})} \]
$\mathbf{=}$
\[ \sum_{ n_1,...,n_4 \in \Z} 
   q^{6(n_1^2 + n_2^2 + n_3^2 + n_4^2 - n_1n_2 - n_2n_3 - n_2n_4) - (n_1 + n_2 + n_3 + n_4)}. \]
\end{center}

\end{slide}

\begin{slide}{Type $E_6$}

\vspace{0.2in}

{\yellow Theorem:} {\small If $C=(a_{ij})$ is the Cartan matrix of type $E_6$, then}

\vspace{0.1in}

\begin{center}
\[ \varphi(q^{12})^{6} \prod_{j \equiv \pm 1, \pm 4, \pm 5 \,({\rm mod 12})} (1-q^j)^{-1} \]
 $\mathbf{=}$
\[ \sum_{{\bf n} \in \Z^6} q^{6 {\bf n}C {\bf n}^{t} - \sum_{i=1}^6 n_i} \]
\end{center}

\vspace{0.1in}

{\small where $\varphi(q) = \prod_{j \geq 1} (1-q^j)$ is the Euler product function.}

\end{slide}

\begin{slide}{Type $E_7$}

\vspace{0.2in}

{\yellow Theorem:} {\small If $C=(a_{ij})$ is the Cartan matrix of type $E_7$, then}

\vspace{0.1in}

\begin{center}
\[ \varphi(q^{18})^{7} \prod_{j \equiv \pm 1, \pm 5, \pm 7, 9 \,({\rm mod 18})} (1-q^j)^{-1} \]
 $\mathbf{=}$
\[ \sum_{{\bf n} \in \Z^7} q^{9 {\bf n}C {\bf n}^{t} - \sum_{i=1}^7 n_i} \]
\end{center}

\vspace{0.1in}

{\small where $\varphi(q) = \prod_{j \geq 1} (1-q^j)$ is the Euler product function.}

\end{slide}

\begin{slide}{Type $E_8$}

\vspace{0.2in}

{\yellow Theorem:} {\small If $C=(a_{ij})$ is the Cartan matrix of type $E_8$, then}

\vspace{0.1in}

\begin{center}
\[ \varphi(q^{30})^{8} \prod_{j \equiv \pm 1, \pm 7, \pm 11, \pm 13 \,({\rm mod 30})} (1-q^j)^{-1} \]
 $\mathbf{=}$
\[ \sum_{{\bf n} \in \Z^8} q^{15 {\bf n}C {\bf n}^{t} - \sum_{i=1}^8 n_i} \]
\end{center}

\vspace{0.1in}

{\small where $\varphi(q) = \prod_{j \geq 1} (1-q^j)$ is the Euler product function.}

\end{slide}
\end{document}
