D_n: Θ_{D_n} = ½(θ(z)_3^n + θ(z)_4^n) = sum_{m=0}^∞r_n(2m)q^{2m}

Or equal to

Θ_{D_n} = Π_{m=1}^∞(1 – q^{2m})^n((1 + q^{2m-1})^2n + (1 – q^{2m-1})^{2n}).

Cheers LC

]]>Cheers LC

]]>F4, [3,4,3] — (1152)

B4, [4,3,3] — (384)

D4, [3^{1,1,1}] — (192)

And the 1152 is the number of Hurwitz quaternions. With F_4 there is a quotient system with B_4 ~ SO(9), and the B_4 representation with 16 roots plus 8 long roots. The 52 dimensional space of F_4 contains the 36-dim of SO(9), and the remaining 16 dimensions is the 16-dimensional Cayley plane. I am less certain about what is meant by F_4/D_4. The quotient with the 28-dimensional SO(8) leaves a 24-dimensional subspace. The short exact sequence

F_4/B_4:1 – -> so(9) – ->F_{52/16} – -> OP^2 – -> 1,

Where F_{52/16} restricts from 52 dimension of 16 spinors plus the 36 of SO(9). OP^2 is the projective Cayley plane in 16 dimensions, where F_4 is the automorphism of the Jordan matrix algebra.

This relationship between D_4 ~ SO(4,4) and B_4 ~ SO(5,4) seems to be suggest a structure for 5-qubit entanglements with holographic content. The BFSS M-theory is on the infinite momentum frame an SO(9) theory. This suggests that the 5-qubit system might be a holographic projection from a 4-qubit system.

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