Reflection Groups and Clifford Algebras with Signed Inner Conjugation

Introduction

A finite reflection group is generated by the reflections in the roots of a root system, and a reflection is the adjoint action of a vector, so the generators of such a group are already elements of the Clifford algebra. The group they generate there is not the reflection group but a twofold cover of it: the products of the generating vectors are versors, the sandwich action returns the reflections, and the two elements $\pm1$ of the algebra act trivially. This cover is finite whenever the form is definite and the root system is finite, and it realises the finite reflection group as a quotient of a finite group of versors.

This article develops that construction. The general theorem is that the group generated by the unit root vectors is a central extension of the reflection group by two elements, of twice the order; the corollaries are the classical binary polyhedral groups, which appear here as the even parts of the lifts, and which the corpus already meets as the unit groups of the integral orders of the quaternions. The construction is the point at which the reflection groups of Coxeter Groups and Root Systems and Classification meet the Clifford algebra, and it is the reason the 24 Hurwitz units, the 24-cell and the binary tetrahedral group are the same object seen three ways.

The Clifford algebra, the basis theorem, the even part, the volume element and the commutation of the elements of grade two with the even part are from Clifford Algebras and Clifford Algebras in Finite Dimensions; the reflection $\rho_u(v)=-uvu^{-1}$, the Clifford group, the norm $N(x)=x x^{\natural}$, the pin and spin groups as the versors of norm one, the kernel $\{\pm1\}$ of the signed inner conjugation on the even versors and the double cover are from The Clifford, Pin and Spin Groups with Signed Inner Conjugation; the versors, their length and the sandwich action are from Versors, Rotors and the Sandwich Action with Signed Inner Conjugation; the root systems, the simple roots, the reflection groups and the classification of the Coxeter groups are from Root Systems and Classification and Coxeter Groups; the binary tetrahedral, octahedral and icosahedral groups and their orders are from Biquaternion Orders and Finite Groups of Units; the Lipschitz and Hurwitz units and their lattices are from Lattices and the Quaternion Lattice. Nothing owned by those entries is re-derived. The form is positive definite on a Euclidean space $V$ of dimension $n$, and the roots are normalised to unit length unless stated otherwise.

Reflection Groups and Their Roots

Root Systems and Reflections

Definition. A root system in $V$ is a finite set $\Phi$ of nonzero vectors, the roots, closed under the reflections $\rho_\alpha$ for $\alpha\in\Phi$ and not containing a multiple of a root except its negative. The reflection group of $\Phi$ is

$$ W_\Phi=\langle\rho_\alpha:\alpha\in\Phi\rangle\subseteq O(V,q), $$

the subgroup generated by the reflections in the roots.

Theorem. $W_\Phi$ is a finite group, generated by the reflections in the simple roots, and it is a Coxeter group with the presentation determined by the angles between the simple roots.

Proof. The group permutes the finite set $\Phi$, so it is finite; the generation by the simple roots, the Coxeter presentation and the classification are in Root Systems and Classification and Coxeter Groups.

Definition. The root system is simply laced when all its roots have the same length; otherwise the roots fall into at most two length classes. In this article the roots are normalised to unit length, so that the generators of the lift are unit vectors in every case; the results below are stated for a finite root system in a positive definite space, no restriction on the lengths being needed, because a reflection is an isometry and therefore permutes the unit roots.

The Unit Roots as Vectors

Proposition. Let $\Phi$ be a root system and let $u_\alpha=\alpha/|\alpha|$ be the unit vector on a root. Then

$$ \rho_\alpha=\mathrm{Ad}^{\alpha}_{u_\alpha},\qquad (\mathrm{Ad}^{\alpha}_x)(v)=\alpha(x)\,v\,x^{-1}, $$

the reflection in the root being the signed inner conjugation action of the unit root, which for the odd element $u_\alpha$ is $-u_\alpha v u_\alpha^{-1}$.

Proof. The reflection $\rho_u(v)=-uvu^{-1}$ of The Clifford, Pin and Spin Groups with Signed Inner Conjugation is the signed inner conjugation action of the vector $u$, the grade involution $\alpha$ contributing the sign because $u$ is odd, and for $u=u_\alpha$ it is the reflection in the root $\alpha$ because the reflection in a vector depends only on the line of the vector.

So the generators of $W_\Phi$ are the images of the unit roots under the signed inner conjugation, and the subgroup of the Clifford group they generate is the natural lift of the reflection group.

The Clifford Lift

Definition

Definition. The Clifford lift of the reflection group $W_\Phi$ is the subgroup of the pin group generated by the unit roots:

$$ \Gamma_\Phi=\langle u_\alpha:\alpha\in\Phi\rangle\subseteq\mathrm{Pin}(V,q). $$

Finiteness and the Double Cover

Theorem. For a finite root system $\Phi$ in a positive definite space, the Clifford lift is finite, the signed inner conjugation restricts to a surjection

$$ \mathrm{Ad}^{\alpha}:\Gamma_\Phi\longrightarrow W_\Phi $$

with kernel $\{\pm1\}$, and $|\Gamma_\Phi|=2\,|W_\Phi|$. So $\Gamma_\Phi$ is a central extension of the reflection group by two elements: the cover is two-to-one, and its kernel is generated by $-1$.

Proof. Each generator $u_\alpha$ is an odd versor of norm one, and for a root $\beta$,

$$ \mathrm{Ad}^{\alpha}_{u_\alpha}(u_\beta)=\rho_\alpha(u_\beta)=u_{\rho_\alpha(\beta)}, $$

since the reflection permutes the roots and preserves lengths; so every element of $\Gamma_\Phi$ permutes the finite set of lines spanned by the roots. The image of $\mathrm{Ad}^{\alpha}$ is therefore a subgroup of the finite group of permutations of the roots, and it contains the reflections $\rho_\alpha$, so the image is exactly $W_\Phi$; the kernel of $\mathrm{Ad}^{\alpha}$ on the versors consists of the nonzero scalars, and a scalar in $\Gamma_\Phi$ lies in the even part, so its norm is $1$ and the scalar is $\pm1$; hence $|\Gamma_\Phi|=2|W_\Phi|$.

Remark (the extension class). The lift is a central extension of the reflection group by $\{\pm1\}$, and whether it splits depends on the root system. It splits for $A_1$, where $u_\alpha$ has order two and $\Gamma_{A_1}\cong(\mathbb{Z}/2)^2$, and for $A_2$, where $\Gamma_{A_2}\cong\mathbb{Z}/2\times S_3$; in the polyhedral cases the even part is the classical double cover of the rotation subgroup and the extension is the one carried by the binary polyhedral group. No general claim of splitting or of nonsplitting is made here, and the article uses only the order, the kernel and the even part.

Corollary. The lift of a reflection is a unit root and the lift of the identity is $\{\pm1\}$, so the cover $\Gamma_\Phi\to W_\Phi$ is two-to-one and the element $-1$ generates its kernel.

Proof. Immediate from the theorem.

The Even Part and the Rotation Subgroup

Theorem. Let $\Gamma_\Phi^0=\Gamma_\Phi\cap\mathrm{Spin}(V,q)$ be the even part of the lift. Then $\Gamma_\Phi^0$ is normal of index two in $\Gamma_\Phi$, it is the lift of the rotation subgroup,

$$ \mathrm{Ad}^{\alpha}(\Gamma_\Phi^0)=W_\Phi\cap SO(V,q), $$

of index two in $W_\Phi$, and $|\Gamma_\Phi^0|=|W_\Phi|$.

Proof. The length of a versor has the parity of the determinant of the isometry it defines, by the theorem of The Spinor Norm and the Structure of the Orthogonal Group with Inner Conjugation, so the even elements of $\Gamma_\Phi$ are exactly those mapping into $SO$, and they form the intersection with the spin group. The index of the even part is two because the generators are odd, and the orders follow from the theorem above.

Remark. The group $\Gamma_\Phi^0$ is the part of the construction that the corpus has already met: for the root system of the tetrahedron it is the binary tetrahedral group, of order twenty-four, which is the unit group of the Hurwitz order, recorded in Biquaternion Orders and Finite Groups of Units, and the vertex set of the 24-cell. The lift adds the odd elements, among which are the unit roots themselves, and so doubles the order.

Examples

One and Two Dimensions

Example (the root system $A_1$). The root system consists of $\pm e_1$ in a line. The lift is

$$ \Gamma_{A_1}=\{1,e_1,-1,-e_1\}\cong(\mathbb{Z}/2)^2, $$

of order four, since $e_1^2=q(e_1)=1$; its even part is $\{\pm1\}$, which is the lift of the trivial rotation subgroup, and the reflection $\rho_{e_1}$ is the action of $e_1$. The example shows that the unit roots have order two and that the lift of a reflection is not of order four.

Example (the root system $A_2$). The root system is the hexagonal system of six unit roots in the plane; $W_{A_2}\cong S_3$ of order six, its rotation subgroup is cyclic of order three, and the lift $\Gamma_{A_2}$ has order twelve; the product $u_1u_2$ of two adjacent unit roots has order six, so the even part $\Gamma_{A_2}^0$ is cyclic of order six and covers the cyclic rotation group twofold, and $\Gamma_{A_2}\cong\mathbb{Z}/2\times S_3$. The Clifford computation of the products of the six unit roots is the same computation as the one in Biquaternion Orders and Finite Groups of Units for the dihedral case.

The Root System $A_3$ and the Hurwitz Units

Example (the root system $A_3$). The root system $A_3$ is that of the tetrahedron and the cube, with twelve roots; $W_{A_3}\cong S_4$ of order twenty-four, of which the rotation subgroup $A_4$ has order twelve.

Theorem. The even part of the Clifford lift of $A_3$ is the binary tetrahedral group, of order twenty-four; its elements are the twenty-four Hurwitz units, and they are the vertices of the 24-cell.

$$ \Gamma_{A_3}^0\cong 2T,\qquad |\Gamma_{A_3}^0|=24=2\cdot 12 . $$

Proof. By the theorem of the even part, $|\Gamma_{A_3}^0|=|W_{A_3}|=24$, and by the identification of the lift of a reflection with a unit root the group is generated by the unit roots of $A_3$, which are unit quaternions. The unit quaternions of order twenty-four are the Hurwitz units, the binary tetrahedral group and the vertices of the 24-cell, by the cited entries, and the two groups have the same order and are generated by the same elements.

Corollary. The four descriptions of the same twenty-four elements — the even Clifford lift of the tetrahedral reflection group, the binary tetrahedral group, the unit group of the Hurwitz order and the vertex set of the 24-cell — are four routes to one group.

Proof. Immediate from the theorem and the identifications cited.

The Other Polyhedral Cases

Theorem. The lifts of the finite reflection groups of the sphere give the classical binary polyhedral groups as their even parts:

Proof. The orders of the reflection groups are those of Root Systems and Classification; the order of the even part is the order of the reflection group by the theorem on the even part; and the identification of the resulting groups with the binary polyhedral groups of the corresponding rotation groups is the classical description of the finite subgroups of the unit quaternions, recorded in Biquaternion Orders and Finite Groups of Units, which realises $2T$, $2O$ and $2I$ there. The case $B_3$ is the reflection group of the cube, of order forty-eight, whose rotation subgroup is the order twenty-four group of $A_3$; its even lift is therefore of order forty-eight and covers the rotations of the cube, which is the binary octahedral group.

Remark (the two length classes). When the root system has two root lengths, the reflections in the roots of one class alone generate a proper subgroup of $W_\Phi$; the lift generated by the normalised roots of that class is the lift of that subgroup, and the full lift is generated by both classes. For $B_3$, the short roots $\pm e_1,\pm e_2,\pm e_3$ generate the group $(\mathbb{Z}/2)^3$ of order eight, and the corresponding subgroup of the lift, generated by the six unit short roots, has order sixteen. In the simply laced case there is one class and the lift is generated by all the roots.

The Order and the Invariants

Theorem. The order of the Clifford lift is twice the order of the reflection group, and the exponent of the centre of the lift is two:

$$ |\Gamma_\Phi|=2|W_\Phi|,\qquad Z(\Gamma_\Phi)\supseteq\{\pm1\}, $$

with equality of the centre with $\{\pm1\}$ when the reflection group has no nontrivial central element acting as the identity on the roots.

Proof. The order is the theorem above; the centre contains $\{\pm1\}$ because the scalars are central, and it contains nothing else when the image in $W_\Phi$ of a central element must be central in the reflection group and act trivially on the roots, which forces the element to be a scalar.

Remark (the McKay correspondence). The finite subgroups of the unit quaternions — the cyclic, the binary dihedral and the three binary polyhedral groups of the table — correspond to the simply laced root systems of type $A$, $D$ and $E$, and the correspondence is the McKay correspondence; the even Clifford lifts of the reflection groups of the sphere are therefore the finite subgroups of the spin group of three-dimensional space, and the classification of the finite reflection groups and of the finite subgroups of the rotations meet there. The correspondence itself is recorded in the literature cited below; the present article supplies only the Clifford side of it, that these groups are generated by the unit roots and are twofold covers.

Summary

A finite reflection group $W_\Phi$ is generated by the reflections in the roots of a root system. Each reflection is the sandwich action of the unit root, $\rho_\alpha=\mathrm{Ad}^{\alpha}_{u_\alpha}$, so the generators of the group are elements of the Clifford algebra; the subgroup they generate,

$$ \Gamma_\Phi=\langle u_\alpha:\alpha\in\Phi\rangle\subseteq\mathrm{Pin}(V,q), $$

is the Clifford lift of the reflection group. For a finite root system in a positive definite space the lift is finite, the signed inner conjugation maps it onto the reflection group with kernel $\{\pm1\}$, and

$$ |\Gamma_\Phi|=2\,|W_\Phi| ; $$

the lift is a central extension of the reflection group by two elements, its odd elements being the unit roots themselves and its element $-1$ generating the kernel. The even part of the lift is the intersection with the spin group, it is normal of index two, it covers the rotation subgroup $W_\Phi\cap SO(V,q)$ and it has order $|W_\Phi|$; this even part is the part of the construction that coincides with the classical binary polyhedral groups. For the root system of the tetrahedron it is the binary tetrahedral group of order twenty-four, whose elements are the twenty-four Hurwitz units and the vertices of the 24-cell, so that the even Clifford lift of the tetrahedral reflection group, the binary tetrahedral group, the unit group of the Hurwitz order and the vertex set of the 24-cell are four descriptions of one group. The reflection groups of the sphere give the table

$$ A_1\mapsto(\mathbb{Z}/2)^2,\quad A_2\mapsto\mathbb{Z}/2\times S_3,\quad A_3\mapsto 2T\ \text{with an odd coset},\quad B_3\mapsto 2O\ \text{with an odd coset},\quad H_3\mapsto 2I\ \text{with an odd coset}, $$

with the even parts of orders two, six, twenty-four, forty-eight and one hundred and twenty; for root systems with two root lengths the normalised short roots generate a subgroup and the full lift is generated by both classes. The construction joins the reflection groups of the Coxeter classification to the Clifford algebra, and the covers it produces are the finite subgroups of the unit quaternions in three dimensions, whose classification is the McKay correspondence.

Summary of Notation

Further Reading

• James E. Humphreys, Reflection Groups and Coxeter Groups (Cambridge University Press, 1990), for the reflection groups, their orders and the classification of the root systems.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 2nd ed. 2001), for the lift of a reflection group in the Clifford algebra and the identification of the unit roots with the versors.
• John Conway and Derek Smith, On Quaternions and Octonions (A K Peters, 2003), for the finite subgroups of the unit quaternions, the binary polyhedral groups and the 24-cell.
• John McKay, Graphs, singularities and finite groups (Proceedings of Symposia in Pure Mathematics 37, 1980), for the correspondence between the finite subgroups of $SU(2)$ and the simply laced root systems.
• Ian R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995), for the pin and spin groups, the versors and the group generated by the vectors of a root system.