occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Monday, July 21, 2008

### M Theory Lesson 209

Recall that the $6 \times 6$ operator $2K$, with basic permutations as eigenvalues, is of the form for circulants $(1)$ and $(2)$. What is the eigenvector? Let $K$ act on an object $(X,Y)$. Then one can solve the eigenvalue equation for $\lambda = (231)$ to obtain provided we do arithmetic mod 7. Try it yourself. The cyclic nature of the linear equations forces the eigenvector to live in such a ring. Choosing $K$ instead, rather than $2K$, we find that the same vector is an eigenvector for the other 1-circulant, $(312)$.