### M Theory Lesson 126

As Ebeling explains, by taking the seven rows of the Hamming circulant and adding a check bit to each row we can write down the seven vectors

$v_1 = \frac{1}{\sqrt{2}} (0,1,1,0,1,0,0,1)$

$v_2 = \frac{1}{\sqrt{2}} (0,0,1,1,0,1,0,1)$

$v_3 = \frac{1}{\sqrt{2}} (0,0,0,1,1,0,1,1)$

$v_4 = \frac{1}{\sqrt{2}} (1,0,0,0,1,1,0,1)$

$v_5 = \frac{1}{\sqrt{2}} (0,1,0,0,0,1,1,1)$

$v_6 = \frac{1}{\sqrt{2}} (1,0,1,0,0,0,1,1)$

$v_7 = \frac{1}{\sqrt{2}} (1,1,0,1,0,0,0,1)$

in $\mathbb{R}^{8}$. These satisfy the rule for a root lattice, $v^{2} = 2$. With the change of variables $e_1 = v_1$, $e_2 = v_2 - v_1$, $e_3 = v_3 - v_2$, $e_4 = v_4 - v_3$, $e_5 = v_5 - v_4$, $e_6 = v_6 - v_5$ and $e_7 = v_7 - v_6$, and the addition of the vector

$e_8 = \frac{1}{\sqrt{2}} (-1,-1,0,0,1,0,-1,0)$

we have a basis for the $E8$ lattice. Since $u^{2} = 2$ for such vectors, it follows that $u \cdot v \in \{ 0, \pm 1, \pm 2 \}$. Hopefully these numbers are familiar.

$v_1 = \frac{1}{\sqrt{2}} (0,1,1,0,1,0,0,1)$

$v_2 = \frac{1}{\sqrt{2}} (0,0,1,1,0,1,0,1)$

$v_3 = \frac{1}{\sqrt{2}} (0,0,0,1,1,0,1,1)$

$v_4 = \frac{1}{\sqrt{2}} (1,0,0,0,1,1,0,1)$

$v_5 = \frac{1}{\sqrt{2}} (0,1,0,0,0,1,1,1)$

$v_6 = \frac{1}{\sqrt{2}} (1,0,1,0,0,0,1,1)$

$v_7 = \frac{1}{\sqrt{2}} (1,1,0,1,0,0,0,1)$

in $\mathbb{R}^{8}$. These satisfy the rule for a root lattice, $v^{2} = 2$. With the change of variables $e_1 = v_1$, $e_2 = v_2 - v_1$, $e_3 = v_3 - v_2$, $e_4 = v_4 - v_3$, $e_5 = v_5 - v_4$, $e_6 = v_6 - v_5$ and $e_7 = v_7 - v_6$, and the addition of the vector

$e_8 = \frac{1}{\sqrt{2}} (-1,-1,0,0,1,0,-1,0)$

we have a basis for the $E8$ lattice. Since $u^{2} = 2$ for such vectors, it follows that $u \cdot v \in \{ 0, \pm 1, \pm 2 \}$. Hopefully these numbers are familiar.

## 5 Comments:

Marni, how do I get the LaTeX to render in Explorer?

Smells like ternary logic to me. ;)

Sorry, but explorer isn't the way to view MathML. Kea.

Smells? Or

reeks, perhaps?So I discovered... I'm now using mathml fonts and viewing with Firefox and all is well. Thanks.

By the way, now that you've been doctored what are your plans? There have recently been a few positions advertised in Australia and New Zealand. Sydney Uni advertised two positions in pure maths. Otago and Waikato also had positions.

If you're open to working on things other than theoretical physics, there appear to be quite a few postdocs in statistics, applied maths, financial modelling and so on.

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