### M Theory Lesson 275

Last time we saw how a symmetric magic matrix could be decomposed into one three dimensional and one two dimensional piece. Sadly, AF has neglected to mention some lovely properties of Combescure matrices, acting via conjugation. For example, on the two dimensional circulant piece we have so $R_3$ simply permutes the indices. And recall that $R_3$, being a $1$-circulant, fixes any $1$-circulant. On three dimensional $2$-circulants it acts like the basic permutation operator $(231)$. That is, $R_3$ naturally encodes several permutation actions.

## 2 Comments:

Yup, this is how triality arises for cubic Jordan algebras. If you were working over the octonions, you'd be mapping between the three equivalent ways of embedding SO(9) into the exceptional group F4.

SO(9) in string/M-theory is the light-cone little group, which classifies the massless degrees of freedom of D=11 supergravity, with a triplet of representations.

SO(9) can arise in a quantum information context in a novel way by extending Duff and Ferarra's extremal black hole/qutrit correspondence, where the three equivalent ways of embedding SO(9) correspond to the three ways of rotating the black hole qutrit computational basis in J(3,O), a twenty-seven dimensional real Hilbert space.

Excellent, kneemo. And as far as the underlying category theory is concerned, octonions are just as nice as complex numbers. I'm just too stupid to be adept at manipulating non-associative operators.

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