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50  Coxeter systems and their classification

Remark 50.1 (Orientation and sign convention) A Coxeter–Vinberg diagram (Definition 49.14) is the combinatorial record of a symmetric bilinear form, and the geometry on which the associated reflection group acts is determined by the definiteness of that form.

Simple roots are negative-definite directions of norm \(-2\), following the algebraic-geometry convention of Definition 36.2, so a spherical Coxeter system has a negative definite Gram form.

50.1 Coxeter matrices and their Gram forms

Definition 50.2 Let \(S\) be a finite set. A Coxeter matrix on \(S\) is a symmetric matrix \(M = (m_{st})_{s,t\in S}\) with \[ m_{ss} = 1, \qquad m_{st} = m_{ts} \in \theset{2,3,4,\dots}\union\theset{\infty} \quad (s\neq t) . \] The Coxeter group presented by \(M\) is the group \(W(M)\) of Definition 49.10, the relation \((s t)^{m_{st}} = e\) being imposed only for \(m_{st} < \infty\). The pair \((W(M), S)\) is a Coxeter system of rank \(\abs{S}\). Its Coxeter diagram \(\Sigma(M)\) is the graph on the vertex set \(S\) with an edge \(\theset{s,t}\) whenever \(m_{st}\geq 3\), labelled by \(m_{st}\) when \(m_{st}\geq 4\).

Definition 50.3 (The Gram form of a Coxeter matrix) Let \(M\) be a Coxeter matrix on \(S\). The Gram form of \(M\) is the symmetric bilinear form \(G(M)\) on the real vector space \(\RR^S\) with basis \(\theset{\alpha_s}_{s\in S}\) determined by \[ G(M)_{st} \definedas \beta(\alpha_s,\alpha_t) = 2\cos\!\left(\frac{\pi}{m_{st}}\right) , \] read on the diagonal as \(G(M)_{ss} = 2\cos\pi = -2\), and as \(G(M)_{st} = 2\) when \(m_{st} = \infty\). Each \(\alpha_s\) has norm \(-2\) and is therefore a short root in the sense of Definition 49.4; the reflections \(s_{\alpha_s}\) of Definition 49.2 preserve \(G(M)\), and \(s\mapsto s_{\alpha_s}\) extends to an injective homomorphism \(W(M)\to \Orth(\RR^S, G(M))\) (Bourbaki 2008; Humphreys 1990).

Remark 50.4 (Normalized pairings) Dividing by the norms recovers the quantity that labels the edges of a Coxeter–Vinberg diagram: with \(\alpha_s^2 = \alpha_t^2 = -2\), \[ g_{st} = \frac{\beta(\alpha_s,\alpha_t)}{\sqrt{\alpha_s^2\,\alpha_t^2}} = \frac{G(M)_{st}}{2} = \cos\!\left(\frac{\pi}{m_{st}}\right) , \] which is the normalization of Definition 49.14. A Coxeter matrix produces only the values \(g_{st}\leq 1\); Vinberg’s diagrams admit in addition the ultraparallel case \(g_{st} > 1\), which corresponds to no finite \(m_{st}\) and is drawn dotted.

Remark 50.5 (The values at small orders) \[ \begin{array}{c|cccccc} m & 2 & 3 & 4 & 5 & 6 & \infty \\\hline 2\cos(\pi/m) & 0 & 1 & \sqrt2 & \dfrac{1+\sqrt5}{2} & \sqrt3 & 2 \end{array} \]

Remark 50.6 (Gram and Cartan matrices) Let \(\theset{\alpha_s}_{s\in S}\) be simple roots in a lattice \(L\). Two matrices are attached to them. The Gram matrix is \(G_{st} = \beta_L(\alpha_s,\alpha_t)\); it is symmetric and records the form. The Cartan matrix is \[ A_{st} \definedas \frac{2\,\beta_L(\alpha_s,\alpha_t)}{\beta_L(\alpha_t,\alpha_t)} , \] so that \(A_{ss} = 2\) and \(s_{\alpha_t}(\alpha_s) = \alpha_s - A_{st}\alpha_t\); it records the reflections and is symmetric only when all the \(\alpha_s\) have equal norm. The two are related by a diagonal factor on the right, \[ A = G\cdot\diag\!\left(\frac{2}{G_{11}},\dots,\frac{2}{G_{nn}}\right), \qquad G = A\cdot\diag\!\left(\frac{G_{11}}{2},\dots,\frac{G_{nn}}{2}\right) , \] so \(G\) is the Cartan matrix symmetrized by the root lengths. In the simply-laced case, where every \(\alpha_s^2 = -2\), one has \(A = -G\). When two root lengths occur the two matrices record different data: for \(\alpha_s^2 = -2\) and \(\alpha_t^2 = -4\) with \(\beta_L(\alpha_s,\alpha_t) = 2\), the Cartan entries are \(A_{st} = 1\) and \(A_{ts} = 2\).

Remark 50.7 (The opposite normalization) Much of the reflection-group literature normalizes simple roots to \(\alpha_s^2 = +2\) and attaches to \(M\) the Schläfli matrix \(C_{st} = -2\cos(\pi/m_{st})\), with \(C_{ss} = 2\). Then \(C = -G(M)\), and every definiteness statement below is read with the opposite sign: a spherical system has \(C\) positive definite.

50.2 Classification by definiteness

Definition 50.8 Let \((W,S)\) be an irreducible Coxeter system of rank \(n\) with Coxeter matrix \(M\) and Gram form \(G = G(M)\). Say that \((W,S)\), or \(M\), is

  • spherical if \(G\) is negative definite;

  • euclidean if \(G\) is negative semidefinite and \(\ker G\) has rank \(1\);

  • hyperbolic if \(G\) is nondegenerate of signature \((1, n-1)\).

A Gram form of signature \((p,q)\) with \(p\geq 2\) falls under none of the three.

For subdiagrams, Definition 49.15 names the corresponding conditions elliptic, parabolic, and hyperbolic. An elliptic subdiagram is one all of whose components are spherical; a parabolic subdiagram on \(k\) vertices with \(c\) connected components has kernel of rank \(c\), hence rank \(k-c\).

Remark 50.9 By Sylvester’s law of inertia, invoked in the Lattice Theory section to define the signature, the numbers \(n_+\) and \(n_-\) of positive and negative squares in a diagonalization of \(G\) over \(\RR\), and the rank of \(\ker G\), are independent of the basis. The three conditions of Definition 50.8 are therefore properties of the form \(G\) rather than of the matrix representing it: spherical is \((n_+,n_-) = (0,n)\), euclidean is \((0,n-1)\) with a one-dimensional kernel, and hyperbolic is \((1,n-1)\).

Theorem 50.10 (The geometry of the three types)

Let \((W,S)\) be an irreducible Coxeter system of rank \(n\) with Gram form \(G\).

  1. \((W,S)\) is spherical if and only if \(W\) is finite. In that case \(W\) acts on the unit sphere of the negative definite space \((\RR^S, -G)\) as a finite reflection group whose fundamental domain is a spherical simplex (Bourbaki 2008; Humphreys 1990).

  2. \((W,S)\) is euclidean if and only if \(W\) is infinite and admits a faithful action on a euclidean space \(\EE^{\,n-1}\) as a discrete group generated by \(n\) affine reflections, with fundamental domain a euclidean simplex (Bourbaki 2008; Humphreys 1990).

  3. If \((W,S)\) is hyperbolic, then \(\RR^S\) with the form \(G\) is a model of hyperbolic \((n-1)\)-space in the sense of Definition 49.9, and the reflections \(s_{\alpha_s}\) generate a discrete subgroup of its isometry group whose fundamental domain is the Coxeter polytope \[ P \definedas \theset{\, [v] \in \HH^{\,n-1} \mid \beta(v,\alpha_s)\geq 0 \text{ for all } s\in S \,} \] of Definition 49.11 and (Vinberg 1967; Vinberg 1985).

Remark 50.11 Statement (3) asserts nothing about the volume of \(P\): the finite-volume and compact cases are separated by the criterion of Theorem 52.10.

50.3 Subdiagrams and the inheritance of signature

Definition 50.12 Let \(\Sigma\) be the Coxeter diagram of \((W,S)\) and let \(I\iscontainedin S\). The subdiagram \(\Sigma_I\) is the induced diagram on \(I\), with Coxeter matrix \(M[I,I]\) and Gram form the principal submatrix \(G[I,I]\). The rank of \(\Sigma_I\) is the rank of \(G[I,I]\). The subgroup \[ W_I \definedas \generators{\, s \mid s\in I \,} \leq W \] is the standard parabolic subgroup determined by \(I\), and \((W_I, I)\) is a Coxeter system with Coxeter matrix \(M[I,I]\) (Bourbaki 2008).

Remark 50.13 (Parabolic subgroups and parabolic subdiagrams) The two uses of parabolic are independent. The subgroup \(W_I\) is a standard parabolic subgroup for every subset \(I\iscontainedin S\), whatever the definiteness of \(G[I,I]\). The subdiagram \(\Sigma_I\) is parabolic when each of its connected components is euclidean (Definition 49.15), and then \((W_I, I)\) is a product of irreducible euclidean Coxeter systems, one for each component.

Proposition 50.14 (Restriction cannot increase the positive index) Let \(\beta\) be a symmetric bilinear form on a finite-dimensional real vector space \(V\), write \(n_\pm(\beta)\) for its numbers of positive and negative squares, and let \(V'\leq V\) be a subspace of codimension \(c\). Then \[ n_+(\beta) - c \;\leq\; n_+\!\left(\restrictionof{\beta}{V'}\right) \;\leq\; n_+(\beta) , \] and the same inequalities hold for \(n_-\).

Proof. The integer \(n_+(\beta)\) is the largest dimension of a subspace of \(V\) on which \(\beta\) is positive definite. A subspace of \(V'\) on which \(\restrictionof{\beta}{V'}\) is positive definite is such a subspace of \(V\), which gives the upper bound. If \(P\leq V\) is positive definite of dimension \(n_+(\beta)\), then \(\beta\) is positive definite on \(P\intersect V'\), and \(\dim(P\intersect V')\geq n_+(\beta) - c\), which gives the lower bound. Applying both to \(-\beta\) gives the statement for \(n_-\).

Corollary 50.15

Let \(\Sigma\) be a Coxeter diagram on \(S\) and \(J\iscontainedin I\iscontainedin S\).

  1. If \(\Sigma_I\) is elliptic then so is \(\Sigma_J\). Equivalently, a subdiagram that is not elliptic is contained in no elliptic subdiagram.

  2. If the Gram form of \(\Sigma\) has exactly one positive square — in particular if \(\Sigma\) is hyperbolic — then the Gram form of every subdiagram of \(\Sigma\) has at most one positive square, so every subdiagram is negative semidefinite or of signature \((1,q)\) with a possibly nontrivial kernel.

Proof. For (1), the restriction of a negative definite form to a subspace is negative definite. For (2), apply the upper bound of Proposition 50.14 with \(n_+(G) = 1\).

Remark 50.16 Part (1) is what makes an enumeration of the elliptic subdiagrams of a Coxeter–Vinberg diagram tractable: a subset that fails to be elliptic can be discarded together with all subsets containing it, so the search runs over the order ideal of elliptic subsets rather than over all of \(2^{S}\).

50.4 The spherical and euclidean classifications

Theorem 50.17 (Classification of spherical Coxeter systems) An irreducible Coxeter system is spherical if and only if its Coxeter diagram is one of \[ A_n\ (n\geq 1),\quad B_n\ (n\geq 2),\quad D_n\ (n\geq 4),\quad E_6,\ E_7,\ E_8,\quad F_4,\quad H_3,\quad H_4,\quad I_2(p)\ (p\geq 5) , \] and diagrams from distinct entries of this list are not isomorphic (Bourbaki 2008; Humphreys 1990).

Remark 50.18 Here \(I_2(p)\) denotes the rank-two diagram with \(m_{12} = p\), so that \(I_2(3)\cong A_2\), \(I_2(4)\cong B_2\) and \(I_2(6)\cong G_2\); the list above records each isomorphism class once. The root systems \(B_n\) and \(C_n\) of Definition 36.11 have the same Coxeter matrix and so contribute a single entry, the two being distinguished by their root lengths rather than by their Coxeter system.

Theorem 50.19 (Classification of euclidean Coxeter systems) An irreducible Coxeter system is euclidean if and only if its Coxeter diagram is one of \[ \tilde A_1,\quad \tilde A_n\ (n\geq 2),\quad \tilde B_n\ (n\geq 3),\quad \tilde C_n\ (n\geq 2),\quad \tilde D_n\ (n\geq 4),\quad \tilde E_6,\ \tilde E_7,\ \tilde E_8,\quad \tilde F_4,\quad \tilde G_2 , \] each of rank \(n+1\) and drawn in the diagram table of the Diagrams appendix (Bourbaki 2008). The diagram \(\tilde A_1\) has two vertices joined by a bond with \(m_{12} = \infty\), and \(\tilde A_n\) for \(n\geq 2\) is the cycle on \(n+1\) vertices with all bonds \(m_{ij} = 3\).

Proposition 50.20 (The radical of a euclidean diagram) Let \(\Sigma\) be an irreducible euclidean diagram on the vertex set \(I\), with Gram form \(G[I,I]\) and simple roots \(\theset{\alpha_s}_{s\in I}\). Then \(\ker G[I,I]\) is spanned by a vector \[ \delta = \Sum_{s\in I} n_s\,\alpha_s \qquad\text{with all } n_s > 0 , \] unique up to scalar, and \(\delta^2 = 0\) (Bourbaki 2008; Humphreys 1990).

Remark 50.21 The positivity of the coefficients \(n_s\) is what places \(\delta\) in the closure of a cone rather than merely in the kernel of a form, and is the reason a parabolic subdiagram of a hyperbolic Coxeter polytope determines an ideal point of the boundary rather than an arbitrary isotropic line; see Proposition 52.6. For \(\tilde E_8\) the coefficients are the marks \((1,2,3,4,6,5,4,3,2)\) of the highest root of \(E_8\).

50.5 Crystallographic Coxeter systems

Definition 50.22 A Coxeter matrix \(M\) on \(S\) is crystallographic if \(m_{st}\in\theset{2,3,4,6,\infty}\) for all \(s\neq t\).

Theorem 50.23 A Coxeter matrix \(M\) is crystallographic if and only if the group \(W(M)\) preserves a lattice of full rank in \(\RR^S\) (Bourbaki 2008; Humphreys 1990).

Example 50.24 (Rescaling to an integral form) The equal-norm normalization \(\alpha_s^2 = -2\) of Definition 50.3 does not itself make the form integral, and a crystallographic \(M\) becomes integral only after the simple roots are rescaled. For \(m_{st} = 4\) one has \(\beta(\alpha_s,\alpha_t) = 2\cos(\pi/4) = \sqrt2\); replacing \(\alpha_t\) by \(\alpha_t' \definedas \sqrt2\,\alpha_t\) gives \[ \alpha_s^2 = -2, \qquad (\alpha_t')^2 = -4, \qquad \beta(\alpha_s,\alpha_t') = 2 , \] a short root and a long root in the sense of Definition 49.4. For \(m_{st} = 6\) the same rescaling by \(\sqrt3\) gives norms \(-2\) and \(-6\) with pairing \(3\), which is the ratio occurring in \(G_2\). For \(m_{st}\in\theset{2,3}\) the entries \(0\) and \(1\) are already integral, and the simply-laced diagrams need no rescaling.

Definition 50.25 Let \(M\) be a Coxeter matrix on \(S\). Its base field is the subfield \[ K(M) \definedas \QQ\!\left(\, 2\cos(\pi/m_{st}) \;:\; s\neq t,\ m_{st} < \infty \,\right) \iscontainedin \RR , \] the smallest field over which the Gram form \(G(M)\) is defined.

Proposition 50.26 For an integer \(m\geq 2\) one has \(2\cos(\pi/m) = \zeta_{2m} + \inverseof{\zeta_{2m}}\) for a primitive \(2m\)-th root of unity \(\zeta_{2m}\), so \(\QQ(2\cos(\pi/m))\) is the maximal totally real subfield of \(\QQ(\zeta_{2m})\) and \[ \left[\QQ\!\left(2\cos(\pi/m)\right) : \QQ\right] = \tfrac12\,\#\unitsof{(\ZZ/2m\ZZ)} . \] In particular \(K(M) = \QQ\) exactly when \(m_{st}\in\theset{2,3,\infty}\) for all \(s\neq t\).

Proposition 50.27 (When the Gram form is integral) \(G(M)\) has all its entries in \(\ZZ\) if and only if \(m_{st}\in\theset{2,3,\infty}\) for every \(s\neq t\). Among the Coxeter matrices with every \(m_{st}\) finite, these are exactly the simply-laced ones.

Proof. The entries are the values of Remark 50.5, and an entry lies in \(\ZZ\) only if it lies in \(\QQ\). By Proposition 50.26 the degree of \(\QQ(2\cos(\pi/m))\) over \(\QQ\) is \(\tfrac12\#(\ZZ/2m\ZZ)^\times\), which equals \(1\) exactly when \(2m\in\theset{4,6}\), that is when \(m\in\theset{2,3}\); the value at \(m=\infty\) is \(2\). The remaining orders therefore give irrational entries, and the three admissible orders give \(0\), \(1\) and \(2\).

Example 50.28 (The non-crystallographic spherical types) Of the spherical types of Theorem 50.17, those that are not crystallographic are \(H_3\), \(H_4\), and \(I_2(p)\) for \(p\notin\theset{2,3,4,6}\).

For \(m = 5\), \[ 2\cos(\pi/5) = \varphi \definedas \frac{1+\sqrt5}{2}, \qquad \varphi^2 = \varphi + 1 , \] so \(H_3\) and \(H_4\) both have base field \(K = \QQ(\sqrt5)\), and their Gram forms have entries in the ring of integers \(\ZZ[\varphi]\) of that field. For \(m = 8\) the base field is \(\QQ(\sqrt{2+\sqrt2})\), of degree \(4\) over \(\QQ\) by Proposition 50.26.

Remark 50.29 (Galois conjugates of a Gram form) Let \(M\) be a Coxeter matrix with base field \(K = K(M)\), a totally real number field, and let \(\sigma\in\gal(K/\QQ)\). Applying \(\sigma\) entrywise to \(G(M)\) produces a symmetric \(K\)-valued form \(\sigma(G(M))\) of the same rank, with the same diagonal \(-2\). Its signature is not determined by that of \(G(M)\): the signature of a symmetric form over a totally real field is not a Galois invariant, as the rank-one forms \(\generators{1-\sqrt2}\) and \(\generators{1+\sqrt2}\) over \(\QQ(\sqrt2)\) show. Whether every nontrivial conjugate of a hyperbolic Gram form is negative definite is a condition on \(M\) that must be computed, and is the arithmeticity question recorded in the Open Problems section.

50.6 Origins

Remark 50.30 (Attribution) Coxeter introduced the matrices and diagrams now named after him and classified the finite reflection groups, obtaining the list of Theorem 50.17 with its two non-crystallographic exceptional types \(H_3\) and \(H_4\); the euclidean classification of Theorem 50.19 grew out of Weyl’s theory of root systems and the affine reflection groups. Both are given in Bourbaki (Bourbaki 2008) and in Humphreys (Humphreys 1990), and the geometric and topological theory of general Coxeter groups in Davis (Davis 2008). The hyperbolic theory — the polytope picture, the finite-volume criterion, and the algorithm of Theorem 49.18 — is due to Vinberg (Vinberg 1967; Vinberg 1975; Vinberg 1985); Lannér classified the compact hyperbolic Coxeter simplices, and Vinberg and Kaplinskaja determined the reflection groups of the odd unimodular hyperbolic lattices \(\latI_{18,1}\) and \(\latI_{19,1}\) (Vinberg and Kaplinskaja 1978), which are the ranks the moduli theory of this monograph meets.