 (1,-2*0.86603);
	 \coordinate (-2omega2-omega1) at (0.5,-3*0.86603);
	 
	\draw[thin] ($1.5*(-omega1)$) to  ($1.5*(omega1)$);
	\draw[thin] ($1.5*(-omega2)$) to ($1.5*(omega2)$);
	\draw[thin] ($1.5*(-omega1+omega2)$) to ($1.5*(-omega2+omega1)$);
	
	\draw[fill=black]
	(omega1) circle (.04) node[anchor=south west, xshift=4, yshift=-7] {$\omega_1$}
	(-omega1+omega2) circle (.04) node[anchor=north east, yshift=0] {$s_1\omega_1$}
	(-omega2) circle (.04) node[anchor=north west, yshift=7] {$-\omega_2$};

	\draw[-latex, out=160, in=60, thin, blue]  (omega1) to node[above, xshift=3]{$\tilde{f}_1$} (-omega1+omega2);
	\draw[-latex, out=-60, in=180, thin, blue]  (-omega1+omega2) to node[below, xshift=5, yshift=-2]{$\tilde{f}_2$} (-omega2);
	\draw[-latex, out=40, in=-40, thin, red]  (-omega2) to node[right, yshift=8]{$\tilde{f}_0$} (omega1);
\end{tikzpicture}
}
\qquad\qquad
&
\resizebox{4cm}{!}{
\begin{tikzpicture}[scale=0.95,every node/.style={minimum size=1cm}]
\tikzstyle{every node}=[font=\small]
	 \coordinate (omega2) at (-0.5,0.86603);
	 \coordinate (2omega2) at  (-1,2*0.86603);
	 \coordinate (2omega2+omega1) at (-0.5, 3*0.86603);
	 \coordinate (omega2+2omega1) at (0.5,3*0.86603);
	 \coordinate (2omega1) at (1,2*0.86603);
	 \coordinate (omega1) at (0.5,0.86603);
	 \coordinate (-omega1) at (-0.5,-0.86603);
	 \coordinate (-omega1+omega2) at (-1,0);
	 \coordinate (-omega2+omega1) at (1,0);
	 \coordinate (-omega2) at  (0.5,-0.86603);
	 \coordinate (-2omega1-omega2) at (-0.5,-3*0.86603);
	 \coordinate (-2omega1) at (-1, -2*0.86603);
	 \coordinate (-2omega2) at (1,-2*0.86603);
	 \coordinate (-2omega2-omega1) at (0.5,-3*0.86603);
	 
	\draw[thin] ($1.5*(-omega1)$) to  ($1.5*(omega1)$);
	\draw[thin] ($1.5*(-omega2)$) to ($1.5*(omega2)$);
	\draw[thin] ($1.5*(-omega1+omega2)$) to ($1.5*(-omega2+omega1)$);
	
	\draw[fill=black]
	(omega2) circle (.04) node[anchor=south east, xshift=0, yshift=-7] {$\omega_2$}
	(-omega2+omega1) circle (.04) node[anchor=south west, yshift=3] {$s_2\omega_2$}
	(-omega1) circle (.04) node[anchor=north west,xshift=-5,yshift=0] {$-\omega_1$};

	\draw[-latex, out=0, in=120, thin, blue]  (omega2) to node[above, xshift=-3, yshift=2]{$\tilde{f}_2$} (-omega2+omega1);
	\draw[-latex, out=-90, in=0, thin, blue]  (-omega2+omega1) to node[below, xshift=10, yshift=7]{$\tilde{f}_1$} (-omega1);
	\draw[-latex, out=120, in=-120, thin, red]  (-omega1) to node[left, yshift=-8]{$\tilde{f}_0$} (omega2);
\end{tikzpicture}
}
\\
\hbox{$B^{\mathrm{fin}}(\omega_1)$}
&\hbox{$B^{\mathrm{fin}}(\omega_2)$} 
\end{matrix}
$$




\subsection{The extremal weight module $L(\omega_1+\omega_2)$}

To construct the crystal $B(\omega_1+\omega_2)$ use 
$$B(\omega_1)\otimes B(\omega_2) 
= \{ p_{v\omega_1+k\delta}\otimes p_{w\omega_2+\ell\delta}
\ |\ v\in \{1, s_1, s_2s_1\}, w\in \{1, s_2, s_1s_2\}, k\in \mathbb{Z}, \ell\in \mathbb{Z}\}$$
with the tensor product action for crystals given by (see, for example, \cite[Prop.\ 5.7]{Ra06})
$$\tilde f_i(p_1\otimes p_2) = \begin{cases}
\tilde f_i p_1 \otimes p_2, &\hbox{if $d_i^+(p_1)>d_i^-(p_2)$,} \\
p_1 \otimes \tilde f_i p_2, &\hbox{if $d_i^+(p_1)\le d_i^-(p_2)$,} 
\end{cases}
\quad
\tilde e_i(p_1\otimes p_2) = \begin{cases}
\tilde e_i p_1 \otimes p_2, &\hbox{if $d_i^+(p_1)\ge d_i^-(p_2)$,} \\
p_1 \otimes \tilde e_i p_2, &\hbox{if $d_i^+(p_1)< d_i^-(p_2)$.} 
\end{cases}
$$
where $d_i^{\pm}(p)$ are determined by
$$\tilde f_i^{d_i^+(p)} p \ne 0 \ \hbox{and}\ 
\tilde f_i^{d_i^+(p)+1} p = 0,
\qquad\hbox{and}\qquad
\tilde e_i^{d_i^-(p)} p \ne 0 \ \hbox{and}\ 
\tilde e_i^{d_i^-(p)+1} p = 0.
$$
The crystal $B(\omega_1+\omega_2)$ is realized as a subset of 
$B(\omega_1)\otimes B(\omega_2)$ via the crystal embedding
$$
\begin{matrix}
B(\omega_1+\omega_2) &\hookrightarrow &B(\omega_1)\otimes B(\omega_2) \\
b_{\omega_1+\omega_2} &\longmapsto &p_{\omega_1}\otimes p_{\omega_2}
\end{matrix}
$$
By Theorem \ref{levelzerostructure}, $B(\omega_1+\omega_2)$ is
connected and is generated by $b_{\omega_1+\omega_2}$.  The crystal
$B(\omega_1+\omega_2)$ is pictured and its character is computed in Plate D.


\section{Alcove walks for affine flag varieties}\label{affineflags}

\subsection{The affine Kac-Moody group $G$}

The most visible form of the loop group is  $G = \mathring{G}(\mathbb{C}((\epsilon)))$, 
where $\mathbb{C}((\epsilon))$ is the field of formal power series in a variable $\epsilon$
and $\mathring{G}$ is a reductive algebraic group.  The favourite example is 
when
$$\hbox{when}\quad\mathring{G} = GL_n
\qquad\hbox{and the loop group is}\quad
G=GL_n(\mathbb{C}((\epsilon))),
$$
the group of $n\times n$ invertible matrices 
with entries in $\mathbb{C}((\epsilon))$.
A slightly more extended, and extremely powerful, point of view is to let 
$G$ be the Kac-Moody group whose Lie algebra is the affine Lie algebra
$\mathfrak{g}$ of Section \ref{affLiealgebra}, so that 
$G$ is a central extension of a semidirect product (where the semidirect
