nLab SemiSpin(16)

Idea

The semi-spin group in dimension 16.

Properties

As a subgroup of $E_8$

The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the semi-spin group in that dimension

$$SemiSpin(16) \;\subset\; E_8$$

On the other hand, the special orthogonal group $SO(16)$ is not a subgroup of $E_8$ (e.g. McInnes 99a, p. 11).

In heterotic string theory

In heterotic string theory with gauge group the direct product group $E_8 \times E_8$ it is usually only this subgroup $Semispin(16) \times SemiSpin(16)$ which is considered (but typically denoted $Spin(16)/\mathbb{Z}_2$, see also Distler-Sharpe 10, Sec. 1).

Related entries

• spin group

• Spin(n1).Spin(n2)

• heterotic string theory, Green-Schwarz anomaly cancellation

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

• Brett McInnes, The Semispin Groups in String Theory, J. Math. Phys. 40:4699-4712, 1999 (arXiv:hep-th/9906059)

• Brett McInnes, Gauge Spinors and String Duality, Nucl. Phys. B577:439-460, 2000 (arXiv:hep-th/9910100)