occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Friday, November 16, 2007

### M Theory Lesson 125

The $D_4$ triality in Garrett's paper (slide 16) leaves invariant the $W^{3}$. Deleting this component of the $4 \times 4$ rotation matrix reduces it to the matrix

010
001
100

which is the $(231)$ Fourier basis $3 \times 3$ circulant familiar to M theorists. Observe that this matrix permutes the $\frac{1}{2} \omega_{R}^{3}$, $\frac{1}{2} \omega_{L}^{3}$ and $B_{1}^{3}$ gravity fields.