Any elementary matrix, denoted $E_{ij}$, can be expressed as a Fourier transform of the
democratic matrix.
In the $3 \times 3$ case, the nine choices for $F$ give the $9$ elementary matrices. Thus any matrix at all may be expressed as a combination of such Fourier transforms. For example, the circulant permutation $(231)$ uses the three Fourier matrices that sum to
A general $1$-circulant is therefore a combination of three transforms, each of this form. An alternative choice of phases for $(231)$ would have given us
which is a bit more reminiscent of
CKM symmetries. Since the
tribimaximal mixing matrix may be expressed as a product $F_3 F_2$, one hopes that the CKM matrix is also easily written in terms of a natural transform.
In the $3 \times 3$ case, the nine choices for $F$ give the $9$ elementary matrices. Thus any matrix at all may be expressed as a combination of such Fourier transforms. For example, the circulant permutation $(231)$ uses the three Fourier matrices that sum to 
A general $1$-circulant is therefore a combination of three transforms, each of this form. An alternative choice of phases for $(231)$ would have given us 
which is a bit more reminiscent of CKM symmetries. Since the tribimaximal mixing matrix may be expressed as a product $F_3 F_2$, one hopes that the CKM matrix is also easily written in terms of a natural transform.
Aloha! I am always happy to read your M-theory posts. Working on recommendation as soon as I get to my Houston desk.
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