# nLab Pin(5)

Contents

group theory

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

# Contents

## Idea

The pin group in dimension 5.

## Properties

### Exceptional isomorphisms

###### Proposition

The exceptional isomorphism Spin(5) $\simeq$ Sp(2) (this Prop.) generalizes to

$Pin^\pm(5) \;\simeq\; Sp(2) \sqcup \omega Sp(2) \phantom{AAA} \omega^2 = \pm e$

where $\omega \in Z\big( Pin^+(5)\big)$ is an element in the center which, for $Pin^+(5)$, squares to the the neutral element $e$ (corresponding to the Clifford algebra element $+1$) or, for $Pin^-(5)$, to $-e$ (the Clifford algebra element $-1$).

(e.g. Varlamov 99, Theorem 5)

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)