Reflection Groups and the Pin Lift with Signed Hermitian Adjoint

Introduction

A finite reflection group is generated by the reflections in the roots of a finite root system, and the Clifford algebra lifts it to a finite subgroup of the pin group: to each root one attaches the unit vector on it, and the product of the unit roots over the reflections of a word is the lift of the corresponding group element. In the inverse formulation the lift is by the signed inner conjugation, $\rho_\alpha=\mathrm{Ad}^{\alpha}_{u_\alpha}$, and the lift is a central extension by two elements, of order twice that of the reflection group; this is the content of Reflection Groups and Clifford Algebras with Signed Inner Conjugation.

This article is the Hermitian reading. The lift attaches to a root its unit vector $u_\alpha$ as before, but the operator that realises the reflection is the signed Hermitian sandwich,

$$ \Theta^{\alpha}_u(v)=\alpha(u)\,v\,u^{\dagger}, $$

and the point of the Hermitian reading is the exact condition under which this operator is the reflection. On a vector $u$ of square $q(u)$ one has $u^{\dagger}=-\sigma(u)$ and hence

$$ \Theta^{\alpha}_u=-q(u)\,\rho_u,\qquad \Theta_u=q(u)\,\rho_u, $$

in agreement with the general vector proposition of Two-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint. So the signed Hermitian sandwich of a root is the reflection precisely when the root lies on the unitary slice, $q(u)=-1$, and it is the negative of the reflection when $q(u)=+1$; the unsigned member does the opposite. Since the corpus's definite algebras are $\mathrm{Cl}_{0,m}$ with $e_j^{2}=-1$, that is with $q$ negative definite, the unit roots of the standard root systems are scaled to square $-1$ and the signed Hermitian sandwich is the operator that lifts the reflection. The sign of the form, which is invisible in the inverse formulation, is therefore part of the statement here: the reflection is the signed Hermitian sandwich of a slice-normalised root, and the slice normalisation is a genuine normalisation of the roots.

Two things follow and are developed below. The first is that the Clifford lift is a subgroup of the unitary slice and of the Hermitian Clifford group, $\Gamma_\Phi\subseteq U\cap\Gamma_{\dagger}$, so it is a group to which the Hermitian sandwich restricts as an action by isometries, and the cover $\Gamma_\Phi\to W_\Phi$ is two-to-one with kernel $\{\pm1\}$. The second is that the even part is the lift of the rotation subgroup, of order $|W_\Phi|$, and the examples reproduce the classical binary polyhedral groups and the Hurwitz units.

The root systems, the reflection groups and the Clifford lift are Reflection Groups and Clifford Algebras with Signed Inner Conjugation; the pin group, the norm and the double cover are The Pin and Spin Groups with Signed Hermitian Adjoint; the operator and the vector formula are Two-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint; the slice is The Unitary Slice and the Compact Real Form with Hermitian Adjoint; the Hurwitz units and the 24-cell are Biquaternion Orders and Finite Groups of Units. Nothing owned by those entries is reproved.

Conventions. $V$ is a positive- or negative-definite space of dimension $n$ over $\mathbb{R}$, $q$ the form, $\rho_u(v)=v-2g(v,u)q(u)^{-1}u$, $\sigma=\mathrm{id}$ unless stated, $x^{\dagger}=\sigma(\alpha(x^{r}))$, $U=\{x:x^{\dagger}x=1\}$, $\varepsilon_x=(-1)^{|x|}$. The root system $\Phi$ is finite, its roots are normalised so that $q(u_\alpha)=-1$ when the form is negative definite, and $u_\alpha$ denotes the normalised root.

Reflection Groups and Their Roots

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 containing no multiple of a root except its negative. Its reflection group is

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

a finite Coxeter group generated by the reflections in the simple roots.

Definition. The normalised root on $\alpha$ is the vector $u_\alpha$ on the line $\mathbb{R}\alpha$ with $q(u_\alpha)=-1$ when the form is negative definite, equivalently the root scaled so that it lies on the unitary slice; for a positive definite form the same construction is made with $q(u_\alpha)=+1$, and the two conventions are compared below. The reflection is insensitive to the scaling, $\rho_{t\alpha}=\rho_\alpha$ for $t\neq0$, because it depends only on the line.

Proposition (the reflection as a Hermitian sandwich). For any nonzero $u\in V$,

$$ \Theta^{\alpha}_u=-q(u)\,\rho_u, \qquad \Theta_u=q(u)\,\rho_u . $$

Hence the signed Hermitian sandwich of a vector is the reflection exactly when $q(u)=-1$, and the negative of the reflection when $q(u)=+1$; on the unitary slice, $u^{\dagger}u=1$, the vector has $q(u)=-1$ and $\Theta^{\alpha}_u=\rho_u=\mathrm{Ad}^{\alpha}_u$.

Proof. For $u\in V$, $u^{\dagger}=-\sigma(u)$ and $\alpha(u)=-u$, so $\Theta^{\alpha}_u(v)=u\,v\,\sigma(u)=-q(u)\rho_u(v)$ and $\Theta_u(v)=-u\,v\,\sigma(u)=q(u)\rho_u(v)$ by the vector proposition of the companion article. On the slice, $u^{\dagger}u=-\sigma(u)u=-q(u)=1$, so $q(u)=-1$ and $\Theta^{\alpha}_u=\rho_u$; the equality with $\mathrm{Ad}^{\alpha}_u$ is the slice reduction.

Remark (the sign of the form is part of the statement). The inverse formulation has no such condition: $\mathrm{Ad}^{\alpha}_u=\rho_u$ for every non-isotropic vector, with no normalisation. The Hermitian formulation carries an extra sign because the dagger of an odd vector is $-\sigma(u)$, the minus coming from the grade involution inside the dagger. The reflection is therefore realised by the signed Hermitian sandwich on the slice, and by the unsigned Hermitian sandwich off it; the two members are exchanged by the sign of the form. In the corpus's definite algebras $\mathrm{Cl}_{0,m}$, where $e_j^{2}=-1$, the slice convention is the natural one and the signed member is the one that lifts the reflections.

The Clifford Lift

Definition. The Hermitian Clifford lift of $W_\Phi$ is the subgroup of the unit group generated by the normalised roots,

$$ \Gamma_\Phi=\langle u_\alpha:\alpha\in\Phi\rangle\subseteq U\cap\Gamma_{\dagger}, $$

where $U$ is the unitary slice and $\Gamma_{\dagger}$ the Hermitian Clifford group.

Proposition (the lift lies on the slice). Each normalised root $u_\alpha$ lies on the unitary slice and in $\Gamma_{\dagger}$; hence $\Gamma_\Phi\subseteq U\cap\Gamma_{\dagger}$ and the signed Hermitian sandwich restricts to $\Gamma_\Phi$ as a homomorphism into $O(V,q)$,

$$ \Theta^{\alpha}_{xy}=\Theta^{\alpha}_x\circ\Theta^{\alpha}_y,\qquad \Theta^{\alpha}_x\bigr|_{V}\in O(V,q),\quad x\in\Gamma_\Phi . $$

Proof. $q(u_\alpha)=-1$ gives $u_\alpha^{\dagger}u_\alpha=-q(u_\alpha)=1$, so $u_\alpha\in U$; and $\sigma(N(u_\alpha))^{2}=N(u_\alpha)^{2}=(-q(u_\alpha))^{2}=1$, so $u_\alpha\in\Gamma_{\dagger}$. The restriction statements are the multiplicativity and the isometry condition of the operator article.

Theorem (finiteness and the double cover). For a finite root system $\Phi$ in a definite space the lift is finite, the signed Hermitian sandwich restricts to a surjection

$$ \Theta^{\alpha}:\Gamma_\Phi\longrightarrow W_\Phi,\qquad \Theta^{\alpha}_{u_\alpha}=\rho_\alpha , $$

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

Proof. For a root $\beta$ one has $\Theta^{\alpha}_{u_\alpha}(u_\beta)=\rho_\alpha(u_\beta)=u_{\rho_\alpha(\beta)}$ up to a scalar, since the reflection permutes the roots and preserves the lines; so every element of $\Gamma_\Phi$ permutes the finite set of root lines, the image of $\Theta^{\alpha}$ lies in the finite group of those permutations, and it contains the reflections $\rho_\alpha$, hence equals $W_\Phi$. The kernel on the versors consists of the scalars, and a scalar in $\Gamma_\Phi$ is even with $N=1$, so the scalar is $\pm1$; hence $|\Gamma_\Phi|=2|W_\Phi|$. The identification $\Theta^{\alpha}_{u_\alpha}=\rho_\alpha$ is the vector proposition.

Remark (the extension class). Whether the extension splits depends on $\Phi$ and is not claimed here. It splits for $A_1$, where $\Gamma_{A_1}\cong(\mathbb{Z}/2)^{2}$, 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. The Hermitian formulation changes none of these group-theoretic facts, because on the slice the operator is the signed inner conjugation; what it changes is the presentation of the generators, which are now slice-normalised roots.

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

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

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

Proof. The parity of a versor is the determinant of the isometry it defines, and on the slice the signed Hermitian sandwich agrees with the signed inner conjugation, so the even elements of $\Gamma_\Phi$ are exactly those mapping into $SO$; the index and the order follow from the previous theorem.

Examples

One and Two Dimensions

Example ($A_1$). The roots are $\pm e_1$ in $\mathrm{Cl}_{0,1}$ with $e_1^{2}=-1$. The normalised root is $e_1$, on the slice, and

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

with $\Theta^{\alpha}_{e_1}=\rho_{e_1}$ the single reflection and $\Gamma^0_{A_1}=\{\pm1\}$.

Example ($A_2$). The roots of the hexagonal system in $\mathrm{Cl}_{0,2}\cong\mathbb{H}$, with $e_1^{2}=e_2^{2}=-1$. The normalised roots are the six unit vectors $\pm e_1,\pm e_2,\pm e_1e_2$ up to the algebra convention, all on the slice,

$$ \Gamma_{A_2}\cong\mathbb{Z}/2\times S_3,\qquad |\Gamma_{A_2}|=12, $$

and the even part is the lift of the rotation subgroup $C_3$ of order three, of order six, the binary cyclic group of the hexagonal lattice. The three signed Hermitian sandwiches $\Theta^{\alpha}_{u_\alpha}$ are the three reflections of the root system.

The Root System $A_3$ and the Hurwitz Units

Example ($A_3$). In $\mathrm{Cl}_{0,3}\cong\mathbb{H}\oplus\mathbb{H}$ with $e_j^{2}=-1$, the root system $A_3$ has twelve roots, and the normalised roots lie on the slice. The lift

$$ \Gamma_{A_3}=\langle u_\alpha\rangle\subseteq\mathrm{Pin} $$

has order $48$, the binary octahedral group, and its even part has order $24$, the binary tetrahedral group, the group of Hurwitz units up to sign. The signed Hermitian sandwich of the product of two normalised roots is the rotation by twice the angle between them, and the Cayley graph of the lift is the 24-cell, as recorded for the inverse formulation. The dictionary with the biquaternion algebra, where the same units appear in the physics corpus, is Biquaternion Orders and Finite Groups of Units.

The Other Polyhedral Cases

The icosahedral system in $\mathrm{Cl}_{0,3}$ gives the binary icosahedral group of order $120$ as the lift, with even part of order $60$, the binary icosahedral group; the dihedral systems in $\mathrm{Cl}_{0,2}$ give the binary dihedral groups; and the systems of rank one and two in the plane recover the cyclic and the dihedral cases. In each case the signed Hermitian sandwich of a normalised root is the reflection, the even part is the binary polyhedral group, and the order is twice that of the reflection group.

The Order and the Invariants

Proposition. For a finite root system in a definite space the orders are

$$ |\Gamma_\Phi|=2\,|W_\Phi|,\qquad |\Gamma^0_\Phi|=|W_\Phi|,\qquad |\ker\Theta^{\alpha}|=2, $$

and the map $\Theta^{\alpha}$ is the restriction to the lift of the covering of the orthogonal group.

Proof. The three statements are the two theorems above and the slice corollary of the group article.

Remark (invariants of the lift). The conjugacy classes, the characters and the presentation of $\Gamma_\Phi$ are invariants of the inverse formulation and are unchanged here, because on the slice the Hermitian sandwich is the signed inner conjugation and the lift is the same subgroup of the pin group. The Hermitian reading contributes the normalisation of the generators by the slice condition, the identification of the reflection with a Hermitian sandwich of a slice-normalised root, and the honest dependence on the sign of the form; it contributes no new finite group.

Summary

The Hermitian lift of a reflection group attaches to each root its slice-normalised vector $u_\alpha$, that is the vector on the root line with $q(u_\alpha)=-1$, and realises the reflection as a signed Hermitian sandwich. The realised sign is a genuine statement of the Hermitian formulation: on a vector one has $\Theta^{\alpha}_u=-q(u)\rho_u$ and $\Theta_u=q(u)\rho_u$, so the signed member is the reflection exactly on the unitary slice and the unsigned member exactly off it, and the two are exchanged by the sign of the form. In the corpus's definite algebras $\mathrm{Cl}_{0,m}$, with $e_j^{2}=-1$, the slice convention is the natural one. The Hermitian Clifford lift $\Gamma_\Phi=\langle u_\alpha\rangle$ lies in $U\cap\Gamma_{\dagger}$, the signed Hermitian sandwich restricts to it as $\Theta^{\alpha}_{u_\alpha}=\rho_\alpha$, and it is a finite central extension of the reflection group by two elements,

$$ \Theta^{\alpha}:\Gamma_\Phi\longrightarrow W_\Phi,\qquad \ker\Theta^{\alpha}=\{\pm1\},\qquad |\Gamma_\Phi|=2|W_\Phi|, $$

with even part $\Gamma_\Phi^0$ the lift of the rotation subgroup, $|\Gamma_\Phi^0|=|W_\Phi|$, and $\Theta^{\alpha}(\Gamma_\Phi^0)=W_\Phi\cap SO(V,q)$. The examples are those of the inverse formulation — the cyclic, dihedral, tetrahedral, octahedral and icosahedral lifts, the Hurwitz units and the 24-cell — because on the slice the two formulations coincide; the Hermitian reading adds the slice normalisation of the roots and the sign of the form, and no new finite group.

Summary of Notation

Symbol	Meaning
$\Phi$, $W_\Phi$	Root system, reflection group
$u_\alpha$, $q(u_\alpha)=-1$	Slice-normalised root
$\Theta^{\alpha}_u=-q(u)\rho_u$, $\Theta_u=q(u)\rho_u$	Reflection realised exactly on the slice
$\Gamma_\Phi=\langle u_\alpha\rangle\subseteq U\cap\Gamma_{\dagger}$	Hermitian Clifford lift
$\Theta^{\alpha}:\Gamma_\Phi\to W_\Phi$, kernel $\{\pm1\}$	Two-to-one cover
$\Gamma_\Phi^0$, $\Theta^{\alpha}(\Gamma_\Phi^0)=W_\Phi\cap SO$	Even part, lift of the rotations
$\|\Gamma_\Phi\|=2\|W_\Phi\|$, $\|\Gamma_\Phi^0\|=\|W_\Phi\|$	Orders
$A_1$: $(\mathbb{Z}/2)^2$; $A_3$: binary octahedral $48$	Examples, even part $24$ (Hurwitz units)