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Chevalley action
- How are normed division algebras related to the Chevalley action?
John Baez's favorite numbers 8 and 24
He describes the action of a vector multiplying a spinor to give a spinor. The dimension of the spinor space is given by the dimension of the irreducible representation of the exterior algebra {$\wedge^\bullet W$} of the subspace of the vector space {$V=W\oplus W'\oplus U$}, as follows:
| {$n$} | vectors | spinors | normed division algebra |
| {$1$} | {$\mathbb{R}^1$} | {$\mathbb{R}^1$} | real numbers |
| {$2$} | {$\mathbb{R}^2$} | {$\mathbb{R}^2$} | complex numbers |
| {$3$} | {$\mathbb{R}^3$} | {$\mathbb{R}^4$} | |
| {$4$} | {$\mathbb{R}^4$} | {$\mathbb{R}^4$} | quaternions |
| {$5$} | {$\mathbb{R}^5$} | {$\mathbb{R}^8$} | |
| {$6$} | {$\mathbb{R}^6$} | {$\mathbb{R}^8$} | |
| {$7$} | {$\mathbb{R}^7$} | {$\mathbb{R}^8$} | |
| {$8$} | {$\mathbb{R}^8$} | {$\mathbb{R}^8$} | octonions |
When the dimension of the vector space equals the dimension of the spinor space, then the dimension supports a normed division algebra.
Bott periodicity implies that adding {$n=8$} multiplies the spinor dimension by {$16$}. So these are the only normed division algebras.