# Coxeter Theory

> From https://math.ucr.edu/home/baez/twf_dynkin.pdf#page=1&zoom=160,-141,556

## Coxeter groups and diagrams

:::{.remark}
Main ideas:

- Elliptic subdiagrams of rank $r$ correspond to codimension $r$ faces of a polytope $P$
- Parabolic subdiagrams (of rank $n-1$) correspond to cusps of $P$
:::

:::{.remark title="A summary of hyperbolic Coxeter diagram conventions"}
Regarding this as a group of reflections in hyperplanes, we have the following interpretations:

\input{snippets/coxeter-vinberg-conventions}

Note that generally 

- $\cos(\angle(H_i, H_j)) = -(h_i, h_j)$ when $\abs{ (h_i, h_j) } < 1$ and 
- $\cosh(\rho(H_i, H_j)) = -(h_i, h_j)$ when $\abs{ (h_i, h_j) } > 1$.

Here $\rho(H_i, H_j)$ is the length of a common perpendicular to $H_i$ and $H_j$.

Moreover,

- $H_i \transverse H_j \iff \abs{w_{ij}} = \abs{(h_i, h_j)} < 1$,
- $H_i \parallel H_j \iff \abs{w_{ij}} = \abs{(h_i, h_j)} = 1$,
- $H_i \diverge H_j \iff \abs{w_{ij}} = \abs{(h_i, h_j)} > 1$.


For a hyperbolic Coxeter polytope $P$ bounded by hyperplanes $\ts{H_1,\cdots, H_n}$, one constructs the Gram matrix $G(P) = (g_{ij}) \in \Mat_{1\leq i,j\leq n}$ defined by
\[
g_{ij} = 
\begin{cases}
1 & i=j \\
-\cos(\pi/m_{ij}) & H_i\transverse H_j, \angle(H_i, H_j) = \pi/m_{ij} \\
-\cosh(\rho(H_i, H_j)) & H_i \diverge H_j \\
-1 & H_i \parallel H_j
\end{cases}
.\]
Note that $g_{ij} = (h_i, h_j)$ is the Gram matrix of the corresponding intersection form.


:::

:::{.definition title="Coxeter groups"}
A group $W$ is a \textbf{Coxeter group} if it has a presentation of the following form:
\[
W = \bracket{r_1, \cdots, r_n \mid (r_i r_j)^{m_{ij}}  \, \, \forall 1\leq i,j\leq n}\qquad m_{ij} \in \ZZ_{\geq 1} \union \ts{\infty}
\]
where

- $m_{ii} = 1$ for all $i$,
- $m_{ij} \geq 2$ for $i\neq j$, and
- $m_{i,j} = \infty$ means there is no relation imposed.

If $S = \ts{r_1, \cdots, r_n}$ is a fixed generating set, we call the pair $(W, S)$ a \textbf{Coxeter system}.
:::

:::{.definition title="Coxeter diagrams"}
Given a Coxeter system $(W, S)$, the \textbf{the pre-Coxeter diagram} of $(W, S)$ is weighted undirected graph with a single vertex $v_i$ for each $r_i\in S$, and for each pair $i\neq j$, an edge $e_{ij}$ of weight $w_{ij} \da m_{ij}$ connecting $v_i$ to $v_j$. Note that this yields a complete\footnote{Recall that a graph is \textbf{complete} if every vertex is adjacent to every other vertex.} graph on $\abs{S}$ vertices. The \textbf{Coxeter diagram} $D(W)$ of $(W, S)$ is the partially weighted graph obtained from the pre-Coxeter diagram by the following modifications:

- Edges $e_{ij}$ of weight $w_{ij} = 2$ are deleted.
- Edges $e_{ij}$ of weight $w_{ij} \geq 7$ are labeled with their weights.
- Edges $e_{ij}$ of $w_{ij} = 3,4,5,6$ follow one of two conventions: they are either replaced with an $(w_{ij}-2)\dash$fold multi-edge, or are unmodified and retain their label of $w_{ij}$.
- Edges $e_{ij}$ of weight $w_{ij} = \infty$ are replaced by bold/thick edges.

:::

:::{.remark title="Facts about Coxeter diagrams"}
We summarize several facts about the full Coxeter diagram:

- Vertices $v_i$ and $v_j$ are non-adjacent if and only if $w_{ij} = 2$,
- Vertices $v_i$ and $v_j$ are adjacent if and only if $w_{ij} \geq 3$,
- Edge weights are suppressed for small weights $w_{ij} \leq 6$, and explicitly included for every $w_{ij} \geq 7$.
:::

:::{.remark title="How to read a group presentation from a Coxeter diagram"}
One can recover the presentation of a Coxeter group from any Coxeter diagram. Explicitly, given a diagram $D$, one constructs a group $W$ such that $D = D(W)$ in the following way: first one transforms the Coxeter diagram into a pre-Coxeter diagram by adding weight 2 edges between every pair of non-adjacent vertices, forming a complete graph. One then replaces double/triple/quadruple edges with weight 4/5/6 edges respectively. Finally, reads the group presentation off of the weighted adjacency matrix of the resulting graph. Explicitly, the group $W$ will have a generator for every vertex and a relation $(r_ir_j)^{ w_{ij} }$ for each edge $e_{ij}$ of weight $w_{ij}$.d
:::

:::{.example title="Passing between Coxeter diagrams and Coxeter groups"}
Every Coxeter diagram is naturally associated with a weighted graph whose edge weights are all integers $m_{ij} \geq 2$, and from this presentation, one can immediately read off the group presentation.
For example, consider the following diagram and the associated weighted graph:
\[
\quad 
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=black node] (0) at (0, 2) {};
		\node [style=black node] (1) at (2, 0) {};
		\node [style=black node] (3) at (6, 0) {};
		\node [style=black node] (2) at (0, -2) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw[style=plain edge] (0) to (1);
		\draw (1) to (2) ;
		\draw [style=double edge] (2) to (0);
		\draw (1)  edge["7"] (3);
	\end{pgfonlayer}
\end{tikzpicture}
\leadsto 
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=black node, label=left:$r_1$] (0) at (0, 2) {};
		\node [style=black node, label=$r_2$] (1) at (2, 0) {};
		\node [style=black node, label=$r_3$] (3) at (6, 0) {};
		\node [style=black node, label=left:$r_4$] (2) at (0, -2) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw (0) edge["3"] (1);
		\draw (1) edge["3"] (2);
		\draw (2) edge["4"] (0);
		\draw (1) edge["7"] (3);
        \draw[bend left] (0) edge["2"] (3);
		\draw[bend right] (2) edge["2"] (3);
	\end{pgfonlayer}
\end{tikzpicture} 
\]

Reading generators and relations off of this graph, we obtain a group freely generated by $r_1,r_2,r_3,r_4$ subject to the following relations:
\[
W \da \bracket{r_1, r_2, r_3, r_4 \,
\middle\vert
\begin{array}{ll}
    r_1^2=r_2^2 =r_3^2=r_4^2=1 \\
    (r_1 r_2)^3=(r_2 r_3)^7 =(r_1 r_4)^4=(r_2 r_4)^3=1 \\
    (r_1 r_3)^2 =(r_3 r_4)^2=1
\end{array}
}
.\]
Letting $A$ be the weighted adjacency matrix of this weighted graph, we can read this group presentation directly off of the following symmetric matrix:
\[
A  =
\left[
\begin{matrix}
1 & 3 & 2 & 4 \\
3 & 1 & 7 & 3 \\
2 & 7 & 1 & 2 \\
4 & 3 & 2 & 1
\end{matrix}
\right]
\]
This matrix defines an exact sequence of $\ZZ\dash$modules
\[
0 \to \ZZ^4 \mapsvia{A} \ZZ^4 \to W \to 0
,\]
realizing $W \cong \coker A$ as a presentation of $W$ by generators and relations.
:::

## Coxeter polytopes

:::{.remark}
Recall the cosine formula for Euclidean inner product spaces: in $\EE^n$, the norm is $\norm{x} \da \sqrt{x^2} \da \sqrt{x.x}$, and we have
\[
vw = \norm{v}\norm{w} \cos(\angle(v, w)) 
= \sqrt{v^2}\sqrt{w^2} \cos(\angle(v, w)) 
= \sqrt{v^2 w^2 } \cos(\angle(v, w))
\]
For a general bilinear form, we can define
\[
\angle(v, w) \da \cos\inv\qty{vw \over \sqrt{ v^2 w^2 } }
.\]
We can thus interpret the pairing as measuring angles in the following way:
\[
vw = {\cos(\angle(v, w) \over \sqrt{v^2 w^2}}
,\]
which moreover allows one to compute intersections $vw$ from knowledge of $v^2, w^2$, and angles $\angle(v, w)$, which is precisely the data that is encoded in a Coxeter diagram.
:::

:::{.definition title="Dihedral angles between hyperplanes"}
If $H_i, H_j$ are intersecting hyperplanes in $\EE^n$, we write $H_i \transverse H_j$. We write $h_i \da H_i^\perp$ and $h_j\da H_j^\perp$ for unit normal vectors spanning their orthogonal complements, and define the \textbf{dihedral angle} between $H_i$ and $H_j$ as
\[
\angle(H_i, H_j) \da \angle(h_i, h_j)
.\] 
If $H_i$ is parallel to $H_j$,  we write $H_i \parallel H_j$ and define $\angle(H_i, H_j) = 0$. We similarly write $H_i \perp H_j$ if $\angle(H_i, H_j) = \pi/2$.
:::

:::{.remark}
Note that there is a common trick to get rid of the square root in these formulas: one writes
\[
(vw)^2 = v^2 w^2 \cos^2(\angle(v, w))
\]
For $\angle(v, w) = \pi/m_{ij}$, this gives a way to recover $m_{ij}$ from the bilinear form.
:::

:::{.definition title="Coxeter polytopes"}
Let $X \da \EE^n, \SS^n, \HH^n$ be a Euclidean, spherical, or hyperbolic geometry.
A polytope $P\subseteq X$ is \textbf{Coxeter polytope} if all dihedral angles between pairs of intersecting facets $H_i$ and $H_j$ are of the form $\pi/m_{ij}$ for $m_{ij}\in \ZZ_{\geq 2}$, and any two non-intersecting facets are parallel.
:::

:::{.remark title="Coxeter group $G_P$ of a Coxeter polytope $P$"}
Every Coxeter polytope $P$ defines a Coxeter group $G_P \leq \mathrm{Isom}(\bX)$ generated by reflections through the supporting hyperplanes $H_i$ of facets of $P$ and a corresponding Coxeter diagram $D_P$. For $\bX = \EE^n$, one constructs $G_P$ in the following way:

- A generator $r_i$ for each facet $H_i$ of $P$ with relation $r_i^2 = 1$, representing reflection through the hyperplane $H_i$,
- For any facets $H_i, H_j$ where $H_i \transverse H_j$, there is a relation $(r_i r_j)^{m_{ij}} = 1$ where $m_{ij}$ is defined by $\angle(H_i, H_j) = \pi/{m_{ij}}$.
- For non-intersecting facets $H_i \parallel H_j$, we set $m_{ij} = \infty$ and take a relation $(r_i r_j)^\infty = 1$, i.e. no relation is imposed at all.


:::

:::{.remark}
Note that $P$ is a fundamental domain for the action of $G_P$ on $\bX$. Moreover, if $G\leq \Isom(\bX)$ is any discrete finitely generated reflection group, then its fundamental domain is always a Coxeter polytope. If $\bX = \SS^n$ or $\EE^n$, Coxeter polytopes are classified and are either simplices or products of simplices respectively, and full lists can be found. 
For $\bX = \HH^n$, the general classification is an open problem. 
Poincaré classified them in $\HH^2$. 
Vinberg showed that no compact Coxeter polytopes exist in $\HH^n$ for $n\geq 30$, and no non-compact but finite volume polytopes exist for $n\geq 996$. 
These bounds are not sharp. 
Finding explicit examples of high-dimensional compact Coxeter polytopes is interesting because these can be used to explicitly construct high-dimensional hyperbolic manifolds.

:::

:::{.definition title="Volumes and covolumes of Coxeter groups/polytopes/diagrams"}
We define the \textbf{covolume} of $G_P$ as the volume of $P\cong \bX/G_P$, where the metric on the quotient is induced from the metric defining the geometry on $\bX$.
:::

:::{.remark}
We collect some facts about the corresponding Coxeter diagram $D(P)$:

- $D(P)$ has vertices $v_i$ corresponding to $H_i$, where $v_i, v_j$ are non-adjacent if and only if $H_i \perp H_j$\footnote{Recalling that edges with $m_{ij} = 2$ are deleted by convention.}, 
- Edges $e_{ij}$ are plain if $m_{ij} < \infty$ and $m_{ij} \neq 0$, so $H_i \transverse H_j$,
- Edges $e_{ij}$ are bold if $m_{ij} = \infty$, so $H_i \parallel H_j$ and $\angle(H_i, H_j) = \pi/\infty = 0$.
:::

:::{.example title="Euclidean Coxeter polytopes"}
Consider the following Coxeter diagram:
\[
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node] (0) at (0, 0) {};
		\node [style=white node] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) edge["$m$"] (1);
	\end{pgfonlayer}
    \end{tikzpicture}   
\]

This corresponds to a non-compact polytope in $\EE^2$ bounded by two hyperplanes $H_1, H_2$ through the origin (i.e. lines), one corresponding to each node, intersecting at an angle of $\pi/m$. Without loss of generality, we can take $H_1$ to be the $x\dash$axis and $H_2$ to be a line of slope $\pi/m$:

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/chamber_pi_m.png}
    \caption{Caption}
    \label{fig:chamberpim}
\end{figure}

One can note that if $m=2$, then one deletes the edge by convention to get the Coxeter diagram 
\[
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node] (0) at (0, 0) {};
		\node [style=white node] (1) at (3, 0) {};
	\end{pgfonlayer}
    \end{tikzpicture}   
\]
This is the Dynkin diagram of $A^1\times A^1$, which indeed has fundamental chamber the first quadrant. Similarly, if one takes $m=3$ on recovers the standard Dynkin diagram for $A_2$:
\[
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node] (0) at (0, 0) {};
		\node [style=white node] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) to (1);
	\end{pgfonlayer}
    \end{tikzpicture}   
\]
We get a fundamental chamber with two walls at a dihedral angle of $\pi/3$, corresponding to the dual hyperplanes of the two standard short roots$\alpha$ and $\beta$ with $\angle(\alpha, \beta) = 2\pi/3$ in Lie theory:

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/A2RootSystem.png}
    \caption{Caption}
    \label{fig:a2root}
\end{figure}

:::

:::{.example title="Affine examples"}
\dzg{Todo: weighted $\tilde A_2$ as a simplex.}
:::

:::{.remark}
Note that taking reflections of the fundamental domain $C$ by the Weyl group generates a \textbf{tiling} of the hyperbolic disc in these cases.
:::

:::{.remark title="Importance of tilings"}
Why this is important: given \textit{any} tiling of $\EE^2$ or $\HH$ the hyperbolic disc, we can place it at height one and take a cone to get an infinite-type toric variety. Alternatively, given any tiling we can construct a surface that is a union of toric pairs by interpreting every vertex of the tiling as a fan and the edges of tiles as gluing instructions.

Finally, we can interpret an $\IAS^2$ has an irregular spherical tiling, i.e. a tiling $\SS^2$ which is not necessarily generated by reflections, but one which has finitely many tiles. We then regard the tiling as a union of toric surfaces as described above. 
:::

:::{.definition title="The Gram matrix of a Euclidean Coxeter polytope"}
Let $P\subset \EE^n$ be a Euclidean Coxeter polytope, not necessarily compact. One defines the \textbf{Gram matrix} $G(P)$ of $P$ as
\[
G(P)_{ij} = 
\begin{cases}
1 & i = j \\
-\cos\qty{\pi \over m_{ij}} & 
H_i \transverse H_j,
\quad \angle(H_i, H_j) = \pi/m_{ij} \\
-1 &  
H_i \parallel H_j,
\quad \angle(H_i, H_j) = \pi/\infty = 0
\end{cases}
.\]
:::


## Hyperbolic Coxeter polytopes

%https://hal.science/hal-03345221/file/Survey_Discrete_Cox_Gp_V3_arxiv.pdf#page=5&zoom=auto,-147,687

:::{.remark}
See the section on hyperbolic geometry for a description of $\HH^n\da \ts{x\in \EE^{n, 1} \mid x^2 = -1, x_0 > 0}$ and terminology (space/time/light-like vectors). As a convention, $\HH^n$ means the interior of $\overline{\HH^n} \da \HH^n\union \partial \HH^n$ where $\partial \HH^n$ is the boundary at infinity consisting of ideal points.

Some unsorted notes:

- The distance $\rho$ on $\HH^n$ is defined such that $\rho(v, w) \da \mathrm{arccosh}(vw)$. 
    
- In $\HH^n$ Vinberg defines the dihedral angle as $\angle(f_i, f_j) \da \pi - \angle(f_i^\perp, f_i^\perp)$. 

- The diagram $E_{10}$ describes a polytope in $\HH^9$.
:::

:::{.remark title="Hyperplane incidence relations in hyperbolic spaces"}
In hyperbolic geometry ($\HH^2$ to simplify), there are two types of parallelism: asymptotically parallel (converging) lines, or ultraparallel (diverging) lines. Both are characterized by sharing a common orthogonal line, however, asymptotically parallel lines have a common perpendicular in $\partial \overline{\HH^2}$ going through their ideal point of intersection, while ultraparallel lines share a common perpendicular at a point in the interior $\HH^2$. By the ultraparallel theorem, $H_i, H_j$ are ultraparallel if and only if  $H_i\intersect H_j = \emptyset$ in $\overline{\HH^2}$.

\begin{figure}[H]
    \centering
\includegraphics[width=0.9\textwidth]{figures/ultraparallel.jpg}
    \caption{The two types of parallelism in hyperbolic space, visualized in the ball model and half-plane model respectively.}
    \label{fig:ultraparallel}
\end{figure}

Thus given a pair of hyperplanes $H_i$ and $H_j$, there are thus three possibilities for their incidence relations:

- $H_i, H_j$ are not parallel and thus intersect in $\HH^n$. We write $H_i \transverse H_j$ and define $\angle(H_i, H_j)$ as the usual dihedral angle.
- $H_i, H_j$ are asymptotically parallel/converging and thus intersect in an ideal point in $\partial \HH^n$. We write $H_i \parallel H_j$ and define $\angle(H_i, H_j) = {\pi \over \infty} = 0$.
- $H_i, H_j$ are ultraparallel/diverging and do not intersect in $\overline{\HH^n}$. We write $H_i \diverge H_j$.
:::

:::{.remark title="Hyperbolic distance between hyperplanes"}
Note that in the last case above, $\angle(H_i, H_j)$ is undefined but there is a minimal distance $\rho(H_i, H_j)$ between the two hyperplanes. By geometric axioms, if $H_i \intersect H_j = \emptyset$ then there is a unique geodesic $L_{ij}$ that is simultaneously orthogonal to both $H_i$ and $H_j$, intersecting them at points $p_i$ and $p_j$. One then defines $\rho(H_i, H_j)$ as the length of a geodesic segment along $L_{ij}$ with endpoints at $p_i$ and $p_j$.
:::

:::{.remark title="Extending Coxeter diagrams for hyperbolic polytopes"}
Following Vinberg, one can extend the notion of a Coxeter diagram to a weighted graph with positive weights $w_{ij} > 0$ where all $w_{ij}\in (0, 1)$ can be written in the form $w_{ij} = \cos\qty{\pi \over m_{ij}}$ for some $m_{ij} \in \ZZ_{\geq 2}$ and $w_{ij} \in [1, \infty]$ can be arbitrary real (possibly infinite) numbers. 
In this convention,

- $w_{ij} = \cos\qty{\pi \over {m_{ij}}} \in (0, 1)$ get simple edges of labeled weight $m_{ij}$ (or multiedges) corresponding to $H_i \transverse H_j$  and $\angle(H_i, H_j) =\qty{\pi\over m_{ij}}$
- $w_{ij} = 1$ get \textbf{bold} unlabeled edges of weight 1 corresponding to $H_i \parallel H_j$ and $\angle(H_i, H_j) = {\pi \over \infty} = 0$.
- $w_{ij}\in (1,\infty)$ get \textbf{dotted} labeled edges of weight $w_{ij}$ (or unlabeled) corresponding to $H_i \diverge H_j$ and $w_{ij}$ corresponds to $\rho(H_i, H_j)$

More generally, given a Coxeter-Vinberg diagram set
\[
g_{ij} =  {h_i h_j \over \sqrt{h_i^2 h_j^2 }}
,\]
then one interprets

- $g_{ij} < 1 \implies g_{ij} = \cos\qty{\angle(h_i, h_j)}$ and $H_i \transverse H_j$ with $\angle(h_i, h_j) = \pi/m_{ij}$,
- $g_{ij} = 1 \implies H_i \parallel H_j$ with $\angle(h_i, h_j) = 0$,
- $g_{ij} > 1 \implies H_i \diverge H_j$.

:::

:::{.example title="Hyperbolic Coxeter polytopes"}
\dzg{Todo: $(\infty,\infty, \infty)$.}
:::

:::{.remark}
As in the Euclidean case that taking reflections of the fundamental domain $C$ by the corresponding Weyl group naturally constructs a \textbf{tiling} of $\EE^2$ in all of these cases:

- $A_1\times A_1$ tiles $\EE^2$ with 4 non-compact quadrants,
- $A_2$ tiles $\EE^2$ with 6 non-compact sectors of angle $\pi/3$,
- In general, taking $\circ \to^{m} \circ$ with $m\in \ZZ_{\geq 1}$ tiles $\EE^2$ with $2m$ non-compact sectors of angle $\pi/m$,
- $\tilde A_2$ tiles $\EE^2$ with infinitely many compact equilateral triangles of with internal angles $\pi/3$.

:::

:::{.definition title="The Gram matrix of a hyperbolic polytope"}
Let $P\subseteq \overline{\HH^n}$ be a Coxeter polytope, possibly with ideal points. The \textbf{Gram matrix} of $P$ is the matrix
\[
G(P)_{ij} = 
\begin{cases}
    1 & i=j \\
    -\cos\qty{\pi \over m_{ij}} & H_i \transverse H_j, \quad \angle(H_i, H_j) = \pi/m_{ij}, \\
    -1 & H_i \parallel H_j, 
    \quad \angle(H_i, H_j) = \pi/\infty = 0, \\
    -\cosh(\rho(H_i, H_j)) & H_i \diverge H_j,
    \quad \angle(H_i, H_j) = \pi/0 = \infty, \\
\end{cases}
.\]
:::

:::{.remark}
When labeling the Coxeter graph, one often puts $m_{ij}$ or $\cosh(\rho(H_i, H_j))$ as the labels, mixing conventions slightly. Edges of weight 2 are deleted, edges of weight 3 are unlabeled simple edges.
:::


:::{.remark}
If $P\subseteq\HH^n$ is a compact hyperbolic Coxeter polytope, the quotients $\HH^n/G_P$ are hyperbolic orbifolds. The simplest examples of such polytopes are the hyperbolic $n\dash$gons defined by integers $p_1,\cdots, p_k \geq 2$ satisfying $\sum p_i\inv < k-2$.
:::

:::{.definition title="Simple systems"}
We say $\Delta = \ts{r_i}$ is a \textbf{simple system} of generators for a polytope $P$ if $r_i r_j \geq 0$ for all $i$ and $j$, and $P$ has a facet presentation by the mirrors $H_{r_i}$. This allows one to  write
\[
P = \ts{v\in L_\RR \mid v^2 = 0,\, r_i v \geq 0}
.\]
We call $P$ a \textbf{Weyl chamber}\footnote{This is also sometimes notated $C$.}. The closure $\overline P$ is a fundamental domain for the action of $W(L)$ and the Weyl group acts simply transitively on the set of chambers.
:::

:::{.remark title="Decomposing the future orthogonal group into a Weyl and symmetry group"}
Let $W(L)$ be the reflections in all negative norm vectors. There is an identification
\[ 
\Orth^+(L) \cong W(L) \semidirect S(C),\qquad S(C) \da \Stab_{\Orth^+(L)}(C)
\] \dzg{The semidirect might be in the wrong direction here, which one is normal?}
where $C \subset \BB^n$ be a fundamental chamber of $W(L)$ with respect to some choice of a simple set of generators.
:::

:::{.definition title="Reflective lattices"}
We say $L$ is \textbf{reflective} if $W(L)\leq \Orth^+(L)$ is finite-index.
More generally, if we define $\Orth^+(L)_k$ as the subgroup generated by all $k\dash$reflections, i.e. reflections in roots $v$ with $v^2 = k$, we say $L$ is $k\dash$reflective if $W(L)$ is finite index in $\Orth^+(L)$.
:::

:::{.remark}
If $L$ as above is reflective, it is well-known $C$ is a hyperbolic Coxeter polytope of finite volume. 
:::

:::{.definition title="Vinberg-Coxeter diagrams"}
A \textbf{Vinberg-Coxeter diagram} is an extension of a Coxeter diagram with adds the following decorations:

- Black edges
- Double-circled edges
- Dotted edges
- Thick edges

It is a weighted graph with positive edge weights $w_{ij} > 0$ where we require that any $w_{ij}\in (0, 1)$ is of form $w_{ij} = \cos\qty{\pi \over m_{ij}}$ for some $m_{ij} \in \ZZ_{\geq 2}$, but we explicitly allow some $w_{ij} \in [1, \infty]$ to be real (possibly infinite) numbers. We additionally specify vertex weights $r_{i}$ for each vertex $v_i$. In this convention,

- $w_{ij} = \cos\qty{\pi \over {m_{ij}}} \in (0, 1)$ get \textbf{simple edges} of labeled weight $m_{ij}$ (or unlabeled multi-edges of multiplicity $m_{ij} - 2$ for $m_{ij} = 3,4,5,6$),
 - $w_{ij} = 1$ get \textbf{bold unlabeled edges} of weight 1
- $w_{ij}\in (1,\infty)$ get \textbf{dotted labeled edges} of weight $w_{ij}$.
:::



## Elliptic and Parabolic subdiagrams

:::{.remark}
Given these weights, one can construct the weighted adjacency matrix $A$ with $a_{ij} = w_{ij}$ if $v_i, v_j$ are adjacent and zero otherwise. 

A matrix $A$ is a \textbf{direct sum of matrices} $A_i$ if $A$ is similar via permutations of rows and columns to the block diagonal matrix whose blocks are the $A_i$.
If $A$ can not be written as a direct sum of two matrices, we say $A$ is \textbf{indecomposable}. Every matrix has a unique representation as a sum of indecomposable components. We say a Coxeter polytope is indecomposable if its Gram matrix $G_P$ is indecomposable. Any matrix $G_P$ arising from an irreducible Coxeter polytope is either positive-definite, positive-semidefinite, or indefinite. We say a diagram $D_P$ is elliptic if $G_P$ is PD, parabolic if every subdiagram is elliptic and it has at least one degenerate irreducible component.

Connected components of the diagram correspond to indecomposable sub-block matrices of $A$. A diagram is elliptic of $A$ is positive-definite, and is parabolic if any indecomposable component of $A$ is degenerate and positive-semidefinite. There are finitely many indecomposable elliptic and parabolic diagrams. If a Coxeter diagram describes a Coxeter polytope $P$, elliptic subdiagrams of codimension 1 correspond to facets of $P$. Moreover, $P$ has finite volume iff every such elliptic subdiagram can be extended in exactly 2 ways to either an elliptic subdiagram of rank $n$ or a parabolic subdiagram of rank $n-1$, corresponding to every facet of the polytope meeting each of its adjacent facets at either an interior point or an ideal point of $\HH^n$ respectively.
:::

:::{.remark}
Idea: a subdiagram is elliptic if the Gram matrix is negative definite of full rank, and parabolic if negative semidefinite of corank equal to the number of components of the diagram. 
Elliptic diagrams of rank $r$ biject with codimension $r$ faces of $C$. 
Parabolic diagrams of corank 1 correspond to ideal points of $C$. 
Vinberg's algorithm produces a simple system $\Delta$ of generators for $W(L)$ which determines a hyperbolic polytope $C$ via the corresponding Weyl chamber. 
If the algorithm terminates, $C$ is of finite volume.
:::

:::{.definition title="Ranks of subdiagrams"}
The \textbf{rank} of a subdiagram is its number of vertices minus its number of connected components.
:::

:::{.definition title="Elliptic and parabolic Coxeter subdiagrams"}
A Coxeter diagram $G$ is called \textbf{elliptic}  (resp. \textbf{parabolic}) if every connected component of $G$ is a Coxeter diagram underlying classical (resp. affine) Dynkin diagram. This is summarized in the following table; note that the classical diagrams $B_n$ and $C_n$ become identified when the arrow is omitted:

> A possible reference that mostly agrees with this: http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2005/8.pdf#page=3&zoom=180,-91,721

> Another reference, although they seem to include a mysterious G_2^m...
% https://www.maths.dur.ac.uk/users/anna.felikson/talks/HypCoxPoly17.pdf#page=8&zoom=80,48,172


\newcommand{\taone}[0]{
\begin{tikzpicture}[baseline=-0.5ex, scale=0.75, transform shape]
    \begin{pgfonlayer}{nodelayer}
        \node [style=black node, label=below:$0$] (0) at (0, 0) {};
        \node [style=black node, label=below:$1$] (1) at (1, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
}

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Elliptic &
  Parabolic \\ \midrule
$A_n\,$ \dynkin[label, labels={1,2,n-1, n},arrows=false, edge length=.75cm]A{} &
  $\tilde A_1 = I_\infty\,$ \dynkin [Coxeter,gonality=\infty, edge length=.75cm, o/.style=black,label,labels={0,1}]I{} $= \taone$ \\
 &
  $\tilde A_n\,$ \dynkin[extended, arrows=false, edge length=.75cm, o/.style=black,label,labels={0,1,2,n-1, n}]A{} \\
$B_n = C_n\,$ \dynkin[label, labels={1,2,,n-1, n},arrows=false, edge length=.75cm]B{} &
  $\tilde B_n\,$ \dynkin[extended, arrows=false, edge length=.75cm,o/.style=black,label, labels={0,1,2,3,n-2, n-1, n}]B{} \\
 &
  $\tilde C_n\,$ \dynkin[extended, arrows=false, edge length=.75cm,o/.style=black, label, labels={0,1,,,n-1,n}]C{} \\
$D_n\,$ \dynkin[label, labels={1,2,,,n-1,n},arrows=false, edge length=.75cm]D{} &
  $\tilde D_n\,$ \dynkin[label, labels={0,1,2,3,,,n-1,n},extended, arrows=false, edge length=.75cm,o/.style=black]D{} \\
$E_6\,$ \dynkin[label, arrows=false, edge length=.75cm]E6 &
  $\tilde E_6\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E6 \\
$E_7\,$ \dynkin[label, arrows=false, edge length=.75cm]E7 &
  $\tilde E_7\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E7 \\
$E_8\,$ \dynkin[label, arrows=false, edge length=.75cm]E8 &
  $\tilde E_8\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]E8 \\
$F_4\,$ \dynkin[label, arrows=false, edge length=.75cm]F4 $=$ \dynkin[label, Coxeter]{F}{4} &
  $\tilde F_4\,$ \dynkin[label, extended, arrows=false, edge length=.75cm,o/.style=black]F4 $=$ \dynkin[label, Coxeter, extended, o/.style=black]{F}{4} \\
$G_2\,$ \coxeterGtwo $=$ \dynkin[label, Coxeter,gonality=6]{G}{2} &
  $\tilde G_2\,$ \coxeterGtwoAffine $=$ \dynkin[label, Coxeter, extended, o/.style=black]{G}{2} \\
$H_3\,$ \coxeterHthree $=$ \dynkin[label, Coxeter]{H}{3} &
   \\
$H_4\,$ \coxeterHfour $=$ \dynkin[label, Coxeter]{H}{4} &
   \\ \bottomrule
\end{tabular}%
}
\caption{Classification of elliptic and parabolic subdiagrams of a Coxeter diagram}
\label{tab:elliptic-parabolic-subdiagrams}
\end{table}

:::


## Some discrepancies

Note the following discrepancies when comparing the classification of diagrams of Coxeter diagrams to the usual notions of Dynkin diagrams:

- These are not Dynkin diagrams: we forget the arrows on double, triple, etc edges.
 - \textbf{Warning}: in a \textbf{Coxeter} diagram, an edge of label $m$ always corresponds to an $(m-2)\dash$fold edge. \textbf{In a Dynkin diagram, a 3-fold edge corresponds to $m=6$. We do not use this convention in the table above!} Compare $G_2, \tilde G_2$ in the table, which have 4-fold edges corresponding to $m=6$ to the following classical diagrams for $G_2$ and $\tilde G_2$ which still correspond to $m=6$:
\[
G_2: \dynkin[label, edge length=0.75cm]G2 \qquad \tilde G_2: \dynkin[label,extended, edge length=0.75cm]G2
\]
    The reason for this discrepancy: in a \textbf{Dynkin} diagram, the edge labels $m$ must satisfy a crystallographic condition and thus $m=2,3,4,6$. Since $m=5$ is not possible, this makes the interpretation in that special case unambiguous.
- This discrepancy also occurs for $H_i$; here a triple edge truly corresponds to $m=5$.
- In the affine case, we do not distinguish the "new" node, usually denoted by a white dot labeled $0$. Compare to the usual diagram e.g. for $\tilde A_n$:
\[
\dynkin[label, edge length=0.75cm, extended]A{}
\]

:::{.remark}
Elliptic subdiagrams are a disjoint union of classical Dynkin diagrams, while parabolic subdiagrams are a disjoint union of \textit{affine} Dynkin diagrams.

Why these matter: we are working with Coxeter polytopes $P$ in a hyperbolic space, i.e. hyperbolic Coxeter polytopes. Vinberg has a general theory which says the Coxeter diagram $D$ records the combinatorics of $P$:

- Facets of $P \mapstofrom$ nodes of $D$,
- Dihedral angles between two facets of $P \mapstofrom$ edges of $D$,
- $k\dash$faces of $P \mapstofrom$ elliptic subdiagrams of $P$ of co-rank $k$,
- Ideal vertices of $P \mapstofrom$ parabolic subdiagrams of rank $k$.

Idea: in $F_2$, Type II strata are classified by maximal parabolic subdiagrams of the single Coxeter diagram, and Type III strata by elliptic subdiagrams. Dimensions of strata correspond to number of vertices in these subdiagrams, and inclusion of diagrams corresponds to degenerations (smaller diagrams correspond to "more degenerate").

:::

## Edge conventions for Coxeter diagrams


The interpretation of these Coxeter diagrams in terms of root systems:

Needs some notation from \cite{AN06}:

\[\begin{array}{ll}
V(M) &\text {the light cone } V(M)=\left\{x \in M \otimes \mathbb{R} \mid x^2>0\right\} \text { of a hyperbolic lattice } M \\
V^{+}(X) &\text {the half containing polarization of the light cone } V\left(S_X\right) \\
\mathcal{L}(S)=V^{+}\left(S_X\right) / \mathbb{R}^{+} & \text {the hyperbolic space of a surface } S \\
W^{(2)}(M) & \text{the group generated by reflections in all $f \in M$ with $f^2=-2$} \\
W^{(4)}(M) & \text{the group generated by reflections in all $(-4)$ roots of $M$} \\
W^{(2,4)}(M) & \text{the group generated by reflections in all $(-2)$ and $(-4)$ roots of $M$} \\
\mathcal{M}^{(2)} & \text {a fundamental chamber of } W^{(2)}(S) \text { in } \mathcal{L}(S) \\
\mathcal{M}^{(2,4)} & \text {a fundamental chamber of } W^{(2,4)}(S) \text { in } \mathcal{L}(S) \\
P^{(2)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-2) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
P^{(4)}\left(\mathcal{M}^{(2,4)}\right) & \text {all }(-4) \text {-roots orthogonal to } \mathcal{M}^{(2,4)}  \\
(X, \theta) &\text{a $\Kthree$ with involution $\theta$} \\
X^\theta &\text{the fixed locus of an involution} \\
P(X)_{+I} & \text {the subset of exceptional classes of }(X, \theta)  \text{ of type } \rm{I}
\end{array}\]

\newcommand{\VinbergNoEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergSimpleEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0) edge["$m$"] (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergSingleEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=plain edge] (0)to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergDoubleEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=double edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergTripleEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=triple edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergThickEdge}[0]{
\begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=$f_1$] (0) at (0, 0) {};
        \node [style=white node, label=$f_2$] (1) at (3, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw[style=thick edge] (0) to (1);
    \end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergDashedEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=dashed edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\VinbergBrokenEdge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$f_1$] (0) at (0, 0) {};
		\node [style=white node, label=$f_2$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
        \draw[style=dashed edge] (0) edge["$t$"] (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Description                                                                          & Symbol                                                                    \\ \midrule
Black vertices: $f \in P^{(4)}\left(\mathcal{M}^{(2,4)}\right)$, i.e. $f^2 = -4$     & \tikz[baseline] { \node [style=black node, label=$f$] (0) at (0, 0) {}; } \\
White vertices: $f \in P^{(2)}\left(\mathcal{M}^{(2,4)}\right)$, i.e. $f^2 = -2$     & \tikz[baseline] { \node [style=white node, label=$f$] (0) at (0, 0) {}; } \\
Double-circled vertices: $f \in P(X)_{+I}$, i.e. the class of a rational component of $X^\theta$. &
  \tikz[baseline] { \node [style=doubled node, label=$f$] (0) at (0, 0) {}; } \\
No edge: $f_1\neq f_2 \in P\left(\mathcal{M}^{(2,4)}\right)$ with $f_1 \cdot f_2=0$, so $\angle(f_1 f_2) = \pi/2$ & \VinbergNoEdge                                                            \\
Simple edges of weight $m$, or $m-2$ simple edges when $m$ is small: ${ 2 f_1 f_2 \over  \sqrt{f_1^2 f_2^2} } = 2 \cos \frac{\pi}{m}$, so $\angle(f_1 f_2) = \pi/m$ &
  \VinbergSimpleEdge \\
Thick edges: $\frac{2 f_1 \cdot f_2}{\sqrt{f_1^2 f_2^2}}=2$                      & \VinbergThickEdge                                                         \\
Broken edges of weight $t$: ?                                                          & \VinbergBrokenEdge                                                        \\ \bottomrule
\end{tabular}%
}
\caption{Edge conventions for Coxeter diagrams}
\label{tab:edge-notation}
\end{table}


Edge conventions for Coxeter polytopes: nodes correspond to facets $f_i, f_j$ of $P$ and edges record relations in $G_P$.

\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}ll@{}}
\toprule
Description                                         & Diagram            \\ \midrule
$\angle(f_i f_j) = \pi/2$                           & \VinbergNoEdge     \\
$\angle(f_i f_j) = \pi/m$                           & \VinbergSimpleEdge      \\
$\angle(f_i f_j) = \pi/3$                           & \VinbergSingleEdge \\
$\angle(f_i f_j) = \pi/4$                           & \VinbergDoubleEdge \\
$\angle(f_i f_j) = \pi/5$                           & \VinbergTripleEdge \\
$f_i, f_j$ do not intersect                         & \VinbergDashedEdge \\
$f_i, f_j$ intersect in $\partial \overline{\HH^n}$ & \VinbergThickEdge  \\ \bottomrule
\end{tabular}%
}
\caption{}
\label{tab:vinberg-stuff}
\end{table}

## Surfaces associated with Coxeter diagrams

:::{.remark}
As described in \cite[Prop. 4.6]{AE22nonsympinv}, 
% https://arxiv.org/pdf/2208.10383.pdf#page=18&zoom=180,-45,169
for the Halphen case $S \da \HalphenInvariants$, there exists a $\Kthree$ surface with $S_X^+ = S$ with $\pi: X\to Y\da X/\iota$ where $\Nef(Y)$ can be identified with the Coxeter chamber for the full reflection group $W_r$. 

Moreover, \cite[Cor. 4.8]{AE22}
% https://arxiv.org/pdf/2208.10383.pdf#page=18&zoom=160,134,179
shows that the Coxeter diagram of $S$ can be used to write the dual graph of exceptional curves on $Y$ under the following modifications:

\newcommand{\coxetersingleedgeww}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$F_i$] (0) at (0, 0) {};
		\node [style=white node, label=$F_j$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw[style=plain edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\coxeterthickedge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$F_i$] (0) at (0, 0) {};
		\node [style=white node, label=$F_j$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw [style=thick edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\newcommand{\coxeterdoubleedge}[0]{
\begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=white node, label=$F_i$] (0) at (0, 0) {};
		\node [style=white node, label=$F_j$] (1) at (3, 0) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw [style=double edge] (0) to (1);
	\end{pgfonlayer}
\end{tikzpicture}
}

\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lcl@{}}
\toprule
Description                               & Symbol               & Description                               \\ \midrule
A single-circled white vertex                    & \tikz[baseline]{ \node [style=white node, label=$F$] at (0, 0) {}; } & $F \cong \PP^1$ with $F^2 = -1$  \\
\addlinespace[2em] A double-circled white vertex & \tikz[baseline]{\node [style=doubled node, label=$F$] at (0, 0) {};} & $F \cong \PP^1$ with $F^2 = -4$  \\
\addlinespace[2em] A black vertex                & \tikz[baseline]{\node [style=black node, label=$F$] at (0, 0) {}; }  & $F \cong \PP^1$ with $F^2 = -2$. \\
\addlinespace[2em] Any single, plain edge & \coxetersingleedgeww & $F_i, F_j \cong \PP^1$ with $F_i F_j = 1$ \\
\addlinespace[2em] Any bold edge          & \coxeterthickedge    & $F_i, F_j \cong \PP^1$ with $F_i F_j = 2$ \\
\addlinespace[2em] Any double edge        & \coxeterdoubleedge   & ??                                        \\ \bottomrule
\end{tabular}%
}
\caption{How to read a surface off of a Coxeter diagram}
\label{tab:surface_coxeter}
\end{table}

\dzg{Unclear what double edges are, need to read further.}
    
:::

## Incidence diagrams/dual complexes

## Dual complexes

:::{.definition title="Dual complex"}
Let $D$ be an RSNC divisor. The \textbf{dual complex} $\Gamma(D)$ of $D$ is the PL-homeomorphism type of the simplicial complex whose $d\dash$cells correspond with codimension $d$ strata in $D$, i.e. irreducible components of $d\dash$fold intersections $V_{i_0} \intersect \cdots \intersect V_{i_d}$.
:::

:::{.example}
Let $\cX\to \Delta$ be a semistable degeneration and let $\cX_0 = V_1 \union \cdots \union V_n$ be the smooth surfaces forming the irreducible components of the central fiber. Writing $C_{ij} \da V_i \intersect V_j$ and $p_{ijk} \da V_i \intersect V_j \intersect V_k$ for their intersections along curves and points, we call each irreducible component of $C_{ij}$ a \textbf{double curve} and the points $p_{ijk}$ \textbf{triple points}. 

Semistability ensures that the dual complex has dimension at most 3, i.e. there are at worst triple points. Thus concretely the dual complex has 

- a vertex for each component $V_i$,

- an edge from $V_i$ to $V_j$ for each double curve $C_{ij}$, and

- a 2-simplex spanning $V_i, V_j, V_k$ for each triple point $p_{ijk}$.
  
:::

:::{.remark}
For a double curve $C = C_{ij} = C_{ji}$ regarded as a curve in $V_i$ and $V_j$ respectively, Persson's triple point formula holds:
\[
C_{ij}^2 + C_{ji}^2 = - T_C
\]
where $T_C$ is the number of triple points on $C$.
:::

:::{.example}
Let $H_i \subset \PP^3$ for $0\leq i \leq 3$ be the four standard coordinate hyperplanes, i.e. $H_i = \ts{[z_1:z_2:z_3:z_4] \mid z_i =0}$ and let $D = \sum H_i$. Any 2 planes intersect in a line and any 3 planes intersect in a point, so there are ${4\choose 2} = 6$ double curves $C_{ij} \da H_i \intersect  H_j$ and ${4\choose 3} = 4$ triple points $p_{ijk} \da H_i \intersect H_j \intersect H_k$. The dual complex is the standard tetrahedron:

\includegraphics[width=0.9\textwidth]{figures/dual_complex.jpg}

:::

:::{.definition title="Incidence complex"}
Let $(X, D)$ be a RNC compactification. The \textbf{incidence complex $I(D)$ of $D$} is the simplicial complex built in the following way: let $D = \sum_i D_i$ be a decomposition into prime divisors, and take a complex $I(D)$ whose $k\dash$dimensional cells are in bijection with irreducible components of $k\dash$fold intersections of the $D_i$.
The \textbf{colored incidence complex} is $I(D)$
with an integer weight (or a coloring) attached to each 0-cell indicating the dimension of the corresponding stratum.
:::

:::{.remark}
In our case of interest, $(X, D)$ will be a Baily-Borel compactification of a moduli space where $D \da \partial \bar{X}$ is a union of boundary strata of various dimensions. Because we primarily work with hyperbolic lattices, $D$ will only contain strata of dimensions 0 and 1, i.e. points and curves. Thus $I(D)$ will reduce to a graph whose vertices are in bijection with points and curves in $D$ and whose edges record when a point $p_j$ is contained in the closure of a curve $C_i$. We can thus form the colored incidence complex $I(D)$ with two colors, taking points to be black and curves to be white.
:::

:::{.definition title="Cusp incidence diagrams"}
Let $\Omega_N$ be the period domain associated with a lattice $N$ and let $\Gamma \subseteq \Orth(N)$ be a finite-index subgroup. 
The Baily-Borel compactification $\bbcpt{ \Omega_S/\Gamma}$ is a projective variety with a boundary stratification
\[
\bbcpt{ \Omega_S/\Gamma} = \qty{\Omega_S/\Gamma} \cup \mathcal{I} \cup \mathcal{J},\qquad
\partial \bbcpt{ \Omega_S/\Gamma} = \mathcal{I} \cup \mathcal{J}
\]
where 

- $\mathcal{I}$ is a set of points referred to as \emph{$0$-cusps}, which are in bijective correspondence with $\Gamma$-orbits of primitive isotropic 2-dimensional sublattices of $N$, and 
- $\mathcal{J}$ is a set of modular curves referred to as \emph{$1$-cusps}, which are in bijective correspondence with $\Gamma$-orbits of primitive isotropic 1-dimensional sublattices of $N$.

We summarize below what information the colored incidence complex $I(\partial \bbcpt{ \Omega_S/\Gamma})$ captures:
\begin{table}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}l|l|l@{}}
\toprule
Cusp Type    & Type II, $\cJ$                 & Type III, $\cI$                             \\ \midrule
Boundary Strata     & 1-cusps/curves $C_i$    & 0-cusps/points $p_j$                  \\
Vertex type & \tikz[baseline=-0.5ex]{\node[style=white node, label=left:{$C_i$}] at (0, 0) {};} & \tikz[baseline=-0.5ex]{\node[style=black node, label=left:{$p_j$}] at (0, 0) {};} \\
Sublattice Type & Isotropic lines $[\ZZ e] \in \isoGr_1(L)/\Gamma$ & Isotropic planes $[\ZZ e \oplus \ZZ f] \in \isoGr_2(L)/\Gamma$ \\ 
Subdiagram Type & Maximal parabolic & Elliptic \\
\bottomrule
\end{tabular}%
}
\caption{Cusp types}
\label{tab:cusptypes}
\end{table}
Moreover, we draw an edge between a black and white node to denote a point $p_i$ contained in the closure of a curve $C_j$:
\[ 
\tikz[baseline=-0.5ex]{
\node[style=white node, label=$C_j$] (0) at (0, 0) {};
\node[style=black node, label=$p_i$] (1) at (3, 0) {};
\draw[style=plain edge] (0) to (1);
}
\]

:::

:::{.example}
Consider the following colored incidence diagram:

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
    \begin{pgfonlayer}{nodelayer}
        \node [style=white node, label=above:{$C_{12}$}] (1) at (0, 3) {};
        \node [style=black node, label=below:{$p_2$}] (2) at (3, 0) {};
        \node [style=white node, label=above:{$C_2$}] (3) at (3, 3) {};
        \node [style=black node, label=below:{$p_1$}] (4) at (0, 0) {};
    \end{pgfonlayer}
    \begin{pgfonlayer}{edgelayer}
        \draw (4) to (1);
        \draw (1) to (2);
        \draw (2) to (3);
    \end{pgfonlayer}
    \end{tikzpicture}
    \caption{A colored incidence diagram $I(D)$ for $D = \cI \union \cJ$.}
    \label{fig:incidence_diag_N}
\end{figure}

This represents the boundary stratification of a Baily-Borel compactification for which $\cI = \ts{p_1, p_2}$ consists of two points, $\cJ = \ts{C_{12}, C_2}$ is two curves, where $p_1, p_2\in \overline{C_{12}}$, $p_2\in \overline{C_2}$, and $p_1\not\in \overline{C_2}$. This can be represented by the following configuration of curves and points:

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
	\begin{pgfonlayer}{nodelayer}
		\node [style=none] (0) at (0, 0) {};
		\node [style=none] (1) at (4, 3) {};
		\node [style=none] (2) at (0, 4) {};
		\node [style=none] (3) at (4, 0) {};
		\node [style=black node, label=above:{$p_{2}$}] (4) at (1, 2) {};
		\node [style=black node, label=above:{$p_{1}$}] (6) at (2.75, 3) {};
	\end{pgfonlayer}
	\begin{pgfonlayer}{edgelayer}
		\draw [bend left=45] (0) edge["$C_{12}$" very near start] (1);
		\draw [bend left=60, looseness=0.50] (3) edge["$C_{2}$" near start] (2);
	\end{pgfonlayer}
    \end{tikzpicture}
    \caption{A configuration of curves and points representing $I(D)$ in \Cref{fig:incidence_diag_N}.}
    \label{fig:incidence_diag_N}
\end{figure}

:::

:::{.remark}
Each 0-cusp $p_i$ has an associated Vinberg diagram $\mathcal{D}(p_i)$ \todo{Spell out which root system this is attached to? Yes, that would be a good idea.} whose maximal parabolic subdiagrams enumerate the 1-cusps $C_{ij}$ adjacent to $p_i$ in the incidence diagram.
:::

:::{.example}

The following figure shows the Vinberg diagram for the 0-cusp ???? associated to the lattice $N \da (18, 0, 0)$:

\begin{figure}
    \centering
    \input{tikz/Vinberg1}
    \caption{Coxeter-Vinberg diagram}
    \label{fig:vin1}
\end{figure}

This has the following two maximal parabolic subdiagrams:



\begin{figure}
    \centering
    \input{tikz/Vinberg1Parabolic1}
    \caption{Caption}
    \label{fig:enter-label}
\end{figure}

\[ 
\tikzfig{Vinberg1Parabolic1}
\qquad 
\tikzfig{Vinberg1Parabolic2} 
\]

That there are exactly 2 such subdiagrams is reflected in the fact that the vertex ??? in the incidence diagram has valence 2.
:::



\end{document}