\(
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\)
Reflections and the Weyl group
Definition 49.2 Let \((L,\beta_L)\) be a nondegenerate integral lattice and \(\alpha\in L\) an anisotropic vector (that is, \(\beta_L(\alpha,\alpha)\neq 0\)) satisfying the integrality condition \[
2\,\beta_L(\alpha, L) \iscontainedin \beta_L(\alpha,\alpha)\,\ZZ
.
\] The reflection in \(\alpha\) is the isometry \[
\begin{aligned}
s_\alpha : L &\to L \\
x &\mapsto x - 2\,\frac{\beta_L(\alpha,x)}{\beta_L(\alpha,\alpha)}\,\alpha
.
\end{aligned}
\] Each reflection has determinant \(-1\), fixes the orthogonal complement \(\generators{\alpha}^{\perp L}\) pointwise, and sends \(\alpha\mapsto -\alpha\). The integrality condition is exactly what guarantees \(s_\alpha\in\Orth(L)\) rather than merely \(s_\alpha\in\Orth(L_\QQ)\). The Weyl group of \(L\) is the subgroup \[
W(L) \definedas \generators{\, s_\alpha \mid \alpha\in L \text{ anisotropic},\ s_\alpha\in\Orth(L) \,}
\leq \Orth(L)
\] generated by all such integral reflections.
Roots and \(k\)-roots
Definition 49.3 Let \(L\) be a nondegenerate integral lattice and \(v\in L\) a primitive vector. The reflection \(s_v\) is always an isometry of \(L_\QQ\), but is integral, \(s_v\in\Orth(L)\), precisely when \[
2\,\beta_L(v, L) \iscontainedin v^2\,\ZZ
.
\] For \(k>0\) the \(k\)-Weyl group \(W^k(L)\) is the subgroup generated by the integral reflections \(s_v\) with \(v^2 = -k\): \[
W^k(L) \definedas \generators{\, s_v \mid s_v\in\Orth(L),\ v^2 = -k \,}
.
\] A vector \(v\) is a \(k\)-root if \(s_v\in W^k(L)\). The full Weyl group \(W(L)\) of Definition 49.2 is generated by integral reflections in vectors of any norm, so that \(W(L) = \generators{\, W^k(L) \mid k>0 \,}\).
Definition 49.4
For a \(2\)-elementary lattice \(L\), the roots are precisely the vectors \(v\in L\) of one of the two following types:
a short root, with \(v^2 = -2\); or
a long root, with \(v^2 = -4\) and \(\div_L(v) = 2\).
Theorem 49.6 (The two root types exhaust the roots) Let \(L\) be an even \(2\)-elementary lattice and let \(v\in L\) satisfy \(v^2 < 0\). Then \(s_v\in\Orth(L)\) if and only if either \(v^2 = -2\), or \(v^2 = -4\) and \(\div_L(v) = 2\). The list of Definition 49.4 is therefore exhaustive, not merely a selection of two convenient cases.
Proof. This is (Alexeev 2022, sec. 2); that the two conditions are sufficient is Remark 49.5, and the content of the theorem is that no other norm occurs. For \(L\) \(2\)-elementary one has \(\div_L(v)\in\theset{1, 2}\), so the integrality condition \(2\div_L(v)/v^2\in\ZZ\) of Definition 49.3 restricts \(v^2\) to \(\theset{-2, -4}\), with \(\div_L(v) = 2\) forced in the second case.
Coroots and the weight space
Definition 49.7 Let \(R \definedas \generators{\Phi}_\ZZ\) be a root lattice generated by a root system \(\Phi = \theset{\alpha_1,\dots,\alpha_n}\). The dual lattice \(\dualof{R}\) is called the weight lattice of \(R\), and the dual-basis elements \(\omega_i\) determined by \((\omega_i, \alpha_j) = \delta_{ij}\) are the fundamental weights.
Hyperbolic lattices and their polytopes
Definition 49.9
Let \(L\) be a hyperbolic lattice and \(C_L^+\) one connected component of its positive cone \(C_L\). The intersection pairing induces a hyperbolic metric on \(L_\RR\), and the associated hyperbolic \(n\)-space \(\HH^n_L\) admits two standard models:
the unit hyperboloid model \[
\HH^n_L \definedas \theset{ v\in L_\RR \mid v^2 = 1,\ v\in C_L^+ }
;
\]
the projective (Klein) model \[
\BB^n_L \definedas \PP(C_L) \subset \PP(L_\RR)
,
\] the image of the positive cone in projective space.
In the projective model the boundary \(\partial\HH^n_L\) is the image \(\PP(\theset{v\in L_\RR \mid v^2 = 0})\) of the isotropic cone, geodesics are straight line segments, and angles are distorted. The Poincaré ball and upper half-space charts are the two standard conformal models, in which dihedral angles agree with the angles between the bounding hyperplanes and geodesics are circular arcs meeting the boundary sphere orthogonally.
The hyperbolic distance between points is recovered from the form: for \(v, w\in C_L^+\), \[
\cosh d\!\left([v],[w]\right) = \frac{\beta_L(v,w)}{\sqrt{v^2\,w^2}}
,
\] the right-hand side being at least \(1\) by the reverse Cauchy–Schwarz inequality on the positive cone, with equality exactly when \([v] = [w]\). On the unit hyperboloid, where \(v^2 = w^2 = 1\), this reads \(\cosh d(v,w) = \beta_L(v,w)\).
To a vector \(v\) with \(v^2 < 0\) we associate the hyperplane \(H_v \definedas v^{\perp}\intersect \HH^n_L\). Taking \(v,w\) to be unit normals of \(H_v, H_w\), the relative position of two hyperplanes is governed by the pairing \(v\cdot w \definedas \beta_L(v,w)\):
\(|v\cdot w| < 1\): the hyperplanes meet in the interior of \(\HH^n_L\), at dihedral angle \(\theta\) with \(-v\cdot w = \cos\theta\);
\(|v\cdot w| = 1\): the hyperplanes are asymptotically parallel, meeting at an ideal point of the boundary \(\partial\HH^n_L\);
\(|v\cdot w| > 1\): the hyperplanes are ultra-parallel (divergent), at distance \(d\) with \(-v\cdot w = \cosh d\).
Coxeter groups and Coxeter polytopes
Definition 49.10 A Coxeter group is a group given by a presentation \[
W = \generators{\, s_1,\dots,s_n \mid s_i^2 = e,\ (s_i s_j)^{m_{ij}} = e \,}
,
\] where \(m_{ii} = 1\) and \(m_{ij} = m_{ji}\in\theset{2,3,\dots}\union\theset{\infty}\) for \(i\neq j\), the relation \((s_i s_j)^{m_{ij}} = e\) being imposed only when \(m_{ij} < \infty\). Writing \(S = \theset{s_1,\dots,s_n}\) for the generating set, the pair \((W,S)\) is the associated Coxeter system, and \((m_{ij})\) is its Coxeter matrix (Definition 50.2).
Definition 49.11
A Coxeter polytope \(P\) in \(X = \EE^n\), \(S^n\), or \(\HH^n\) is a polytope such that:
each face lies in a hyperplane \(H_i \definedas \alpha_i^{\perp}\);
the dihedral angles between adjacent faces are \(\pi/m_{ij}\) for integers \(m_{ij}\geq 2\); and
the reflection group \(W = \generators{ s_{\alpha_i} }\) acts properly discontinuously on \(X\) with fundamental domain \(P\).
The discriminant locus and Weyl chambers
Definition 49.12 The discriminant locus \(\Delta(L)\) is the union of all mirrors, i.e. the hyperplanes orthogonal to the short roots \(\Phi_2(L) = \theset{ v\in L \mid v^2 = -2 }\): \[
\Delta(L) \definedas \Union_{v\in\Phi_2(L)} v^{\perp L}
.
\]
Definition 49.13 The Weyl chambers of \(L\) are the connected components of \[
C_L^{\circ} \definedas C_L \sm \Delta(L)
,
\] where \(C_L\) is the cone of positive-norm vectors. A simple system is a set of roots whose mirrors form the bounding hyperplanes of a single Weyl chamber, i.e. a fundamental domain for the action of the \(2\)-Weyl group \(W^2(L)\) generated by the \((-2)\)-reflections.
Coxeter–Vinberg diagrams
Definition 49.14 The Coxeter–Vinberg diagram of a hyperbolic Coxeter polytope is a graph whose vertices correspond to the simple roots \(r_i\) (equivalently, to the walls \(H_i\) of the fundamental chamber), and whose edges encode the relative position of the mirrors \(H_i, H_j\) through the normalized pairing \[
g_{ij} \definedas \frac{r_i\cdot r_j}{\sqrt{r_i^2\, r_j^2}} = \cos\!\left(\frac{\pi}{m_{ij}}\right)
.
\] Edges are drawn according to the following conventions:
omitted if \(m_{ij}\leq 2\) (orthogonal mirrors);
multiple (or labeled) lines if \(3\leq m_{ij} < \infty\), when \(H_i\intersect H_j\) lies in the interior of \(\HH^n\);
thick / bold if \(g_{ij} = 1\), when \(H_i\intersect H_j\) lies on the boundary \(\partial\HH^n\) (asymptotically parallel mirrors);
dotted if \(g_{ij} > 1\) (ultra-parallel mirrors).
For \(2\)-elementary lattices, vertices are drawn white for short roots (\(v^2 = -2\)) and black for long roots (\(v^2 = -4\)).
Elliptic and parabolic subdiagrams
Definition 49.15 Let \(G\) be a Coxeter–Vinberg diagram with vertices the simple roots \(v_i\). A subdiagram \(D\iscontainedin G\) is:
elliptic if its Gram matrix \((v_i\cdot v_j)\) is negative definite; such a subdiagram defines a finite face of the Coxeter polytope;
parabolic if the Gram matrix of each of its connected components is negative semidefinite with a one-dimensional kernel;
hyperbolic if its Gram matrix is nondegenerate of signature \((1, k-1)\), where \(k\) is the number of vertices of \(D\).
The signs follow the algebraic-geometry convention of this monograph, in which roots have negative norm; in the normalization by roots of norm \(+2\) the three conditions read positive definite, positive semidefinite of corank one, and signature \((k-1,1)\).
Definition 49.16 A parabolic subdiagram is maximal if it is not properly contained in a larger parabolic subdiagram.
Let \(L\) have signature \((2,n)\), let \(I\subset L\) be a primitive isotropic line, and let \(\overline{L}_I \definedas I^{\perp L}/I\) be the hyperbolic lattice at the \(0\)-cusp \([I]\). The maximal parabolic subdiagrams of the Coxeter–Vinberg diagram of a fundamental chamber for a reflection group acting on \(\overline{L}_I\) correspond bijectively to the primitive isotropic lines of \(\overline{L}_I\) met by that chamber, that is, to the rank-\(2\) primitive isotropic sublattices \(J\subseteq L\) containing \(I\); passing to orbits under the stabilizer of \(I\) in the arithmetic group gives the \(1\)-cusps adjacent to the \(0\)-cusp \([I]\) in the Baily–Borel compactification.
Vinberg’s algorithm
Theorem 49.18 (Vinberg’s algorithm for a fundamental chamber) Let \(\Gamma\iscontainedin\Orth(L)\) be an arithmetic reflection subgroup of a hyperbolic lattice \(L\). Vinberg’s algorithm constructs a fundamental chamber \(P\iscontainedin\HH^n\) for \(\Gamma\) as follows (Vinberg 1975).
Control vector. Choose a generic positive-norm vector \(v_0\in C_L^+\). Its stabilizer in \(\Gamma\) is the reflection group of the finite subdiagram of roots orthogonal to \(v_0\), which are used as the first walls.
Enumeration by height. Seek primitive vectors \(v\in L\) with \(v^2 < 0\), \(s_v\in\Gamma\), and mirror \(H_v\) meeting the current cone \(\tilde P\). Order the candidate roots by increasing height \[
h(v) \definedas \frac{-2\,(v_0\cdot v)}{v^2}
,
\] a quantity monotone in the hyperbolic distance from \(v_0\) to \(H_v\) (for roots of a given norm), used to order the candidate roots and process them in successive batches of increasing height.
Acceptance / pruning. A candidate root \(v\) at the current height is kept only if its mirror bounds the chamber on the side containing \(v_0\); equivalently, the inward normals of accepted walls pair non-negatively, so \(v\) is accepted iff \(v\cdot r_i \geq 0\) for every previously accepted root \(r_i\) (a mirror with some \(v\cdot r_i < 0\) would cut into the interior of \(\tilde P\)). Roots whose hyperplanes do not bound \(\tilde P\) are discarded.
Diagram. The algorithm proceeds until \(\tilde P\) becomes a convex polytope of finite volume. The Gram matrix of the accepted simple roots then determines the Coxeter–Vinberg diagram of \(P\) (Definition 49.14).
The algorithm terminates in finitely many steps if and only if \(\Gamma\) has finite covolume in \(\Orth(L)\).
Definition 49.20 (Lannér subgraph) A Lannér subgraph of a Coxeter–Vinberg diagram is a minimal non-parabolic subdiagram whose Coxeter group acts on hyperbolic space with a fundamental polyhedron of finite volume; equivalently, a diagram all of whose proper subdiagrams are elliptic but which is not itself elliptic or parabolic.
Theorem 49.21 (A sufficient criterion for completeness) Let \(\Gamma\) be a Coxeter–Vinberg diagram without dotted edges, obtained from a run of Theorem 49.18. If \(\Gamma\) contains no Lannér subgraph, and every connected parabolic subdiagram of \(\Gamma\) is contained in a maximal parabolic subdiagram of maximal rank, then \(\Gamma\) is complete: the accepted roots bound a finite-volume polytope and the algorithm may be stopped.