nLab Pin(2)

Idea

The Pin group in dimension 2.

Properties

Let $\mathbb{H}$ be the quaternions generated from unit imaginaries $\mathbf{i}$, $\mathbf{j}$ with $\mathbf{k} \coloneqq \mathbf{i} \, \mathbf{j}$, with a copy of the complex numbers identified as

$$\begin{array}{ccc} \mathbb{C} &\xhookrightarrow{\phantom{---}}& \mathbb{H} \\ x + \mathrm{i} y &\mapsto& x + \mathbf{i} y \mathrlap{\,.} \end{array}$$

Identify the circle group with the unit circle in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way

$$\begin{array}{ccccc} SO(2) &\simeq& S\big( \mathbb{C}\big) &\xhookrightarrow{\phantom{--}}& \mathbb{H} \\ && x + \mathrm{i} y &\mapsto& x + \mathbf{i} y \end{array} \phantom{-------} \begin{array}{l} x,y \in \mathbb{R} \\ x^2 + y^2 = 1 \mathrlap{\,,} \end{array}$$

with group structure now thought of as given by multiplication of quaternions. Then the Pin group $Pin_-(2)$ is isomorphic to the subgroup of the group of units $\mathbb{H}^\times$ of the quaternions which consists of this copy of SO(2), together with the multiples of the imaginary quaternion $\mathbf{j}$ with this copy:

(1)$$Pin_-(2) \;\simeq\; S\big( \mathbb{C}\big) \;\cup\; \mathbf{j} \cdot S\big( \mathbb{C}\big) \;\subset\; \mathbb{H}^\times \,.$$

This is in fact inside the unit sphere $S(\mathbb{H}) \simeq$ Spin(3), whence we have a canonical inclusion

(2)$$Pin_-(2) \xhookrightarrow{\phantom{--}} Spin(3) \,.$$

Properties

Relation to orthogonal and binary dihedral groups

By the classification of the finite subgroups of Spin(3) this means that the restriction of $Pin_-(2)$ to the inclusion of the dihedral group $D_{2n}$ into the orthogonal group $O(2)$ is the binary dihedral group $2 D_{2n}$:

(3)$$\array{ 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow \\ D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }$$

To make this fully explicit:

With the presentation (1) we are to identify the plane with the linear span of the generators $\mathbf{j}$ and $\mathbf{k}$

$$\mathbb{R}^2 \;\simeq\; \langle \mathbf{j}, \mathbf{k}\rangle \;\subset\; \mathbb{H}_{\mathrm{im}} \mathrlap{\,,}$$

where we may think of

• $\langle \mathbf{j} \rangle$ as the x-axis,

• $\langle \mathbf{k} \rangle$ as the y-axis.

The action of $Pin_-(2)$ on $\mathbb{R}^2$ is by conjugation in $\mathbb{H}$. Hence we have plane rotation operators

$$R_{\alpha} \;\coloneqq\; Ad_{ \cos(\alpha/2) + \sin(\alpha/2) \mathbf{i} } \;\in\; S(\mathbb{C}) \subset S(\mathbb{H})$$

acting on $\mathbb{R}^2$ as

$$\begin{array}{rcl} R_\alpha(\mathbf{j}) &\equiv& \big( \cos(\alpha/2) + \sin(\alpha/2)\mathbf{i} \big) \, \mathbf{j} \, \big( \cos(\alpha/2) - \sin(\alpha/2)\mathbf{i} \big) \\ &=& \big( \cos(\alpha/2)^2 - \sin(\alpha/2)^2 \big) \, \mathbf{j} + \big( 2 \sin(\alpha/2) \cos(\alpha/2) \big) \, \mathbf{k} \\ &=& \cos(\alpha) \, \mathbf{j} + \sin(\alpha) \, \mathbf{k} \end{array}$$

(in the last line we used the sum-of-angles trigonometric identities)

and in addition the reflection operator $I_y \coloneqq Ad_{\mathbf{j}}$ acting as

$$I_y \mathbf{j} \;=\; \mathbf{j} \, \mathbf{j} \, (- \mathbf{j}) \;=\; \mathbf{j}$$

$$I_y \mathbf{k} \;=\; \mathbf{j} \, \mathbf{k} \, (- \mathbf{j}) \;=\; - \mathbf{k}$$

(hence inverting the y-axis).

Together these clearly generate the group O(2). Since the conjugation action assignment

$$\begin{array}{ccc} Pin_-(2) &\longrightarrow& O(2) \\ g &\mapsto& Ad_g \end{array}$$

has kernel $\{\pm 1\} \in Pin_-(2)$, this exhibits $Pin_-(2)$ as the double cover of O(2).

To note here that under the inclusion $Pin_-(2) \hookrightarrow Spin(3)$ (2), the above inversion of the $y$ axis in 2D corresponds to rotation around the x-axis, in 3D!

Appearance in Seiberg-Witten theory and Floer homology

On a spin 4-manifold, the $U(1)$-symmetry of the Seiberg-Witten equations enhances to a $Pin(2)$-symmetry (Furuta 01). Because of this, $Pin(2)$-equivariance appears in Seiberg-Witten theory and Floer homology. See the references below.

Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

$Pin(2)$-equivariant homotopy theory/equivariant cohomology theory

application to Seiberg-Witten theory and Floer homology:

• Mikio Furuta, Monopole equation and the $11/8$-conjecture Mathematical Research Letters, Volume 8, Number 3, (2001).

• Ciprian Manolescu, $Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture (arXiv:1303.2354)

• Matthew Stoffregen, $Pin(2)$-equivariant Seiberg-Witten Floer homology of Seifert fibrations (arXiv:1505.03234)

• Matthew Stoffregen, A Remark on $Pin(2)$-Equivariant Floer Homology, Michigan Math. J. Volume 66, Issue 4 (2017), 867-884 (euclid:1508810818)

$Pin(2)$-equivariant KO-theory:

• Jianfeng Lin, Pin(2)-equivariant KO-theory and intersection forms of spin four-manifolds, Algebr. Geom. Topol. 15 (2015) 863-902 (arXiv:1401.3264)