The Signed Left Multiplication on a Symmetry Group

Introduction

The left multiplication of a graded symmetry group by an element $a$ is the operator $x \mapsto ax$; the signed left multiplication is the same operator with the grade involution inserted,

$$ \Lambda^{\alpha}_a : G \longrightarrow G, \qquad \Lambda^{\alpha}_a(x) = a\,\alpha(x) = a\,\varepsilon(x)\,x, $$

the left multiplication twisted by the grade involution, exactly as the signed sandwich is the sandwich twisted by it. It is the one-sided member of the signed layer, and it behaves differently from the two-sided one in two precise ways: it is a bijection with an explicit inverse, and yet the set of signed left multiplications is not closed under composition — the composite of two of them is an ordinary left multiplication. Together with the ordinary left multiplications it forms the group $G \rtimes \langle\alpha\rangle$ generated by the left multiplications and the grade involution, and the signed ones are the nontrivial coset. Its fixed elements are determined exactly: the operator $\Lambda^{\alpha}_a$ has a fixed point only when $a$ is the unit or the sign element $z$, and then the fixed set is the whole even part or the whole odd part.

The article has four sections: the definition and the parity sign; the composition law and the coset structure; the inverse and the fixed elements; and the worked cases. It is the one-sided counterpart of The Signed Sandwich on a Symmetry Group and of Reflections as Signed Two-Sided Operators on a Symmetry Group, the two previous articles of this group. The graded symmetry group, the sign element $z$, the grade involution $\alpha$ and its parity sign are from those articles and are not re-defined; the Clifford and pin-group case, where $\alpha$ is the restriction of the algebra grade involution and $z = -1$, is The Clifford, Pin and Spin Groups with Signed Inner Conjugation. No adjoint is taken; the adjoint of the signed left multiplication is The Signed Adjoint of the Left Multiplication on a Symmetry Group, in the group - * Operator Theory.

Throughout, $G$ is a graded group with central involution $z$ and grading $\varepsilon : G \to \{e,z\}$, the grade involution is $\alpha(g) = \varepsilon(g)g$, and the left multiplication by $a$ is written $L_a(x) = ax$.

The Definition and the Parity Sign

Definition

Definition. The signed left multiplication by $a \in G$ is the operator

$$ \Lambda^{\alpha}_a : G \longrightarrow G, \qquad \Lambda^{\alpha}_a(x) = a\,\alpha(x), $$

and it is the composite of the left multiplication by $a$ with the grade involution, in either order:

$$ \Lambda^{\alpha}_a = L_a \circ \alpha = \alpha \circ L_{\alpha(a)} . $$

Proof. The first identity is the definition, with $\alpha$ applied to $x$; for the second, $\alpha\bigl(L_{\alpha(a)}(x)\bigr) = \alpha\bigl(\alpha(a)\,x\bigr) = \alpha^2(a)\,\alpha(x) = a\,\alpha(x)$, using that $\alpha$ is an automorphism of order two. The two orders agree because $\alpha L_{\alpha(a)} = L_a\alpha$ is the statement that $\alpha$ conjugates the left multiplication by $\alpha(a)$ into the left multiplication by $a$.

The Parity Sign

Proposition. For homogeneous $x \in G$ the signed left multiplication and the unsigned left multiplication are related by the grading,

$$ \Lambda^{\alpha}_a(x) = \varepsilon(x)\, L_a(x), $$

so that on the even part they agree and on the odd part they differ by the sign element:

$$ \Lambda^{\alpha}_a(x) = L_a(x) \ \ (x \in G_0), \qquad \Lambda^{\alpha}_a(x) = z\, L_a(x) \ \ (x \in G_1). $$

At the unit they agree, $\Lambda^{\alpha}_a(e) = a = L_a(e)$.

Proof. $\alpha(x) = \varepsilon(x)x$ with $\varepsilon(x) \in \{e,z\}$ central, so $a\alpha(x) = a\varepsilon(x)x = \varepsilon(x)ax = \varepsilon(x)L_a(x)$; the cases are $\varepsilon(x) = e$ and $z$.

Remark. The one-sided sign is the same as the two-sided one, and it is the whole difference from the unsigned operator. The signed left multiplication differs from the unsigned one on the odd coset alone, and there by the constant central sign $z$; a group with trivial grading, $G = G_0$, has $\alpha = \mathrm{id}$ and the two coincide.

The Composition Law and the Coset Structure

Composition

Proposition. The signed and the unsigned left multiplications compose by

$$ \Lambda^{\alpha}_a \circ \Lambda^{\alpha}_b = L_{a\,\alpha(b)}, \qquad \Lambda^{\alpha}_a \circ L_b = \Lambda^{\alpha}_{a\,b}, \qquad L_a \circ \Lambda^{\alpha}_b = \Lambda^{\alpha}_{a\,\alpha(b)} . $$

In particular the composite of two signed left multiplications is an unsigned one: the signed operators are not closed under composition, and every word in the signed and the unsigned left multiplications reduces to a single operator, signed or unsigned according to the number of signed factors modulo two.

Proof. For the first, $\Lambda^{\alpha}_a\bigl(\Lambda^{\alpha}_b(x)\bigr) = a\,\alpha\bigl(b\,\alpha(x)\bigr) = a\,\alpha(b)\,\alpha^2(x) = a\alpha(b)\,x = L_{a\alpha(b)}(x)$. The other two are the same computation with the middle factor even; the reduction of a word is the repeated application of the three rules.

Corollary (the generated group). The set $$ \langle L_a, \alpha : a \in G\rangle = \{L_a : a \in G\} \cup \{\Lambda^{\alpha}_a : a \in G\} $$ is a group, equal to the semidirect product $G \rtimes \langle\alpha\rangle$ in which the generator of the second factor acts on $G$ by $\alpha$; the unsigned left multiplications form the normal subgroup $\{L_a\} \cong G$ of index two when $\alpha \neq \mathrm{id}$, and the signed left multiplications $\Lambda^{\alpha}_a = L_a\alpha$ are the nontrivial coset.

Proof. Closure and inverses are the three composition rules and the inverse computed below; the identification with the semidirect product is $\alpha L_a \alpha^{-1} = L_{\alpha(a)}$, which is the defining action, and the coset statement is $\Lambda^{\alpha}_a = L_a\alpha$.

The One-Sidedness

Proposition. The signed left multiplication is a bijection of $G$ with inverse

$$ \bigl(\Lambda^{\alpha}_a\bigr)^{-1} = \Lambda^{\alpha}_{\alpha(a)^{-1}} = \alpha \circ L_{a^{-1}}, $$

and it is a left action up to the grade involution: $\Lambda^{\alpha}_a\Lambda^{\alpha}_b = L_{a\alpha(b)}$, which is a left multiplication and not a signed one, so the assignment $a \mapsto \Lambda^{\alpha}_a$ is not a homomorphism but a bijection of $G$ onto a coset of $G$ in $G \rtimes \langle\alpha\rangle$.

Proof. Compose $\Lambda^{\alpha}_a$ with $\Lambda^{\alpha}_{\alpha(a)^{-1}}$ and use the first composition rule: $L_{a\,\alpha(\alpha(a)^{-1})} = L_{a\,a^{-1}} = L_e = \mathrm{id}$, since $\alpha^2=\mathrm{id}$; the inverse is thus the stated signed operator. The non-homomorphism is immediate from the same rule: the parameter of the composite is $a\alpha(b)$, not $ab$.

Remark (the contrast with the sandwich). The signed sandwich $S^{\alpha}_{a,b}(x) = a\alpha(x)b$ is closed under composition, $S^{\alpha}_{a,b}S^{\alpha}_{c,d} = S^{\alpha}_{a\alpha(c),\alpha(d)b}$, because the right factor supplies the second half of the parameter; the signed left multiplication has no such factor, and its composite collapses to the unsigned layer. This is the operator form of the fact that a reflection needs two sides: the one-sided operator is not an involution of the space, and the two-sided one is.

The Inverse and the Fixed Elements

The Fixed Elements

Proposition. The signed left multiplication fixes the following elements and no others:

$$ \Lambda^{\alpha}_a(x) = x \quad\Longleftrightarrow\quad a = x\,\alpha(x)^{-1} = \begin{cases} e & x \in G_0 \\ z & x \in G_1 \end{cases}. $$

Hence $\Lambda^{\alpha}_a$ has a fixed point exactly when $a = e$ or $a = z$: the operator $\Lambda^{\alpha}_e = \alpha$ fixes the whole even part and inverts the sign on the odd part, the operator $\Lambda^{\alpha}_z$ fixes the whole odd part, and for every other $a$ the signed left multiplication is fixed-point-free.

Proof. The equation $a\alpha(x) = x$ is equivalent to $a = x\alpha(x)^{-1}$, and $\alpha(x)^{-1} = \alpha(x^{-1})$; for even $x$ this is $x x^{-1} = e$, and for odd $x$ with $\alpha(x) = zx$ it is $x(zx)^{-1} = x x^{-1}z^{-1} = z$, using $z^{-1} = z$ and the centrality of $z$. The last statements read off the two cases.

The Orbits

Proposition. For $a \notin \{e, z\}$ the signed left multiplication is fixed-point-free and its orbits are the pairs $\{x, a\alpha(x)\}$: it is an involution on the quotient by the relation generated by $x \sim a\alpha(x)$, so the action of the cyclic group generated by $\Lambda^{\alpha}_a$ has orbits of size two at most, and the orbit of $x$ has size one exactly when $x$ is fixed, which happens only for $a \in \{e,z\}$.

Proof. The square of the operator is $\Lambda^{\alpha}_a\Lambda^{\alpha}_a = L_{a\alpha(a)}$, and $\alpha(a) = a$ when $a$ is even and $a\alpha(a) = az$ when $a$ is odd; in neither case is the square the identity unless $a \in \{e,z\}$, so the orbit of $x$ has size two or four; the first return is the pair computed directly. For $a = e$ the operator is the grade involution, of order two, and its orbits are the pairs $\{x, \alpha(x)\}$ together with the fixed even points.

Remark. The fixed-point-free property is what distinguishes the signed left multiplication from the two-sided one: the signed inner conjugation always fixes the even part elementwise when the parameter is even, while the signed left multiplication with a nontrivial even parameter moves every point. The one-sided operator is a translation of the group layer, not an automorphism of it.

Worked Cases

Example (the Euclidean plane). Let $G$ be the pin group of $\mathbb{R}^2$ with the Euclidean form, and identify an even element $R(\theta)$ with the rotation through $\theta$ and an odd element $u = e^{i\varphi}$ with the unit vector at angle $\varphi$. The signed left multiplication by the odd element $u$ sends $x$ to $u\alpha(x)$, so it sends every rotation to a unit vector and every unit vector to a scalar of modulus one: it is the reflection-to-rotation shift of the plane, while the left multiplication by $u$ sends $x$ to $ux$, the same shift composed with the sign $z$ on the odd part. The fixed points are absent for $u \notin \{1, -1\} = \{e, z\}$.

Example (the dihedral group). Let $G = D_n$ with rotations even and reflections odd and sign element $z = r^{n/2}$ when $n$ is even (the central half-turn) and $z = e$ generically. For $a = s$ a reflection, $\Lambda^{\alpha}_s$ sends a rotation $r^k$ to $s r^k$ and a reflection $r^k s$ to $s\,z\, r^k s$; the composite of two signed left multiplications by reflections is the left multiplication by $s\alpha(s) = s z s$, a rotation, illustrating the collapse of the signed layer to the unsigned one.

Example (the odd-dimensional volume element). Let $V$ be odd-dimensional over $\mathbb{R}$ with the Euclidean form and let $\omega$ be the central odd volume element, $z = -1$. The signed left multiplication by $\omega$ is $\Lambda^{\alpha}_\omega(x) = \omega\alpha(x) = \varepsilon(x)\,\omega x$; since $\omega$ is central, this is $\varepsilon(x)\,x\,\omega$, so $\Lambda^{\alpha}_\omega = R_\omega \circ \alpha$ where $R_\omega$ is the right multiplication by $\omega$. The fixed points are the odd elements, on which $\Lambda^{\alpha}_\omega$ is the identity; on the even part it is multiplication by $-\omega$. The example shows that a central parameter turns the signed left multiplication into a right multiplication composed with the grade involution.

Summary

The signed left multiplication by $a$ is $\Lambda^{\alpha}_a(x) = a\alpha(x) = L_a\alpha = \alpha L_{\alpha(a)}$, the left multiplication twisted by the grade involution; it agrees with the unsigned left multiplication on the even part and differs by the central sign element $z$ on the odd part. Its composites are $\Lambda^{\alpha}_a\Lambda^{\alpha}_b = L_{a\alpha(b)}$, $\Lambda^{\alpha}_aL_b = \Lambda^{\alpha}_{ab}$ and $L_a\Lambda^{\alpha}_b = \Lambda^{\alpha}_{a\alpha(b)}$, so two signed factors give an unsigned operator and the signed left multiplications are not a group but the nontrivial coset of $G$ in the group $G\rtimes\langle\alpha\rangle$ that the left multiplications and the grade involution generate. The operator is a bijection with inverse $\Lambda^{\alpha}_{\alpha(a)^{-1}}$, it has a fixed point exactly when $a = e$ or $a = z$, and then the fixed set is the whole even part or the whole odd part; for every other parameter it is fixed-point-free with orbits of the pair $\{x, a\alpha(x)\}$. The central odd volume element of an odd-dimensional form makes the signed left multiplication a right multiplication composed with the grade involution.

The one-sidedness is the difference from the signed sandwich: without the right factor the signed layer collapses to the unsigned one on composition, which is the operator form of the fact that a reflection requires two sides. No adjoint is taken; the adjoint of the signed left multiplication is the group - * Operator Theory.

Summary of Notation

Symbol	Meaning
$\Lambda^{\alpha}_a$	the signed left multiplication, $\Lambda^{\alpha}_a(x) = a\alpha(x)$
$L_a$	the left multiplication, $L_a(x) = ax$
$\alpha = \Lambda^{\alpha}_e$	the grade involution
$z$	the central sign element of the grading
$G \rtimes \langle\alpha\rangle$	the group generated by the left multiplications and $\alpha$
$\Lambda^{\alpha}_a\Lambda^{\alpha}_b = L_{a\alpha(b)}$	the collapse of two signed factors to an unsigned one
$\Lambda^{\alpha}_{\alpha(a)^{-1}}$	the inverse of $\Lambda^{\alpha}_a$