To recap the previous post we began by combining the SU(2) spin-1/2 and U(1) operators into 2×2 matrices. We then showed that the leptons were solutions of the idempotency equation UU = U for 2×2 matrices subject to the additional requirement that the solutions be eigenstates of electric charge Q. For pure density matrix formalism, individual particle states are represented by primitive idempotents (with trace = 1), so we then converted these idempotents into primtive form by embedding them into 4×4 matrices. In doing this, we found that the idempotents given by the 2×2 matrices were composite, each being composed of two sub particles.

**1-Circulant and 2-circulant matrices**

In this post, we add the quarks to the picture. To do this, we need to use the 1-circulant and 2-circulant 3×3 matrices Kea talks about. We will write the general 1-circulant and 2-circulant matrices as follows:

Where I, J, K, R, G, and B are complex numbers. Note that there are only 6 complex degrees of freedom in the 1-circulant and 2-circulant matrices, one cannot create an arbitraray 3×3 matrix, with 9 complex degrees of freedom, from 1-circulant and 2-circulant matrices. In addition, setting R=G=B=1 gives a matrix of 1s, the same as setting I=J=K=1. Consequently, the 1-circulant and 2-circulant matrices together, have only 5 complex degrees of freedom, about half that of the 3×3 matrices in general. Writing a 3×3 matrix as a sum of a 1-circulant and a 2-circulant matrix is very restrictive; to write it as just a 1-circulant is even more so.

One obtains the basis for these matrices by setting one of the elements to 1 and the rest to zero. For example, putting I=1, J=0, K=0 gives the unit matrix. These 3×3 basis matrices correspond to permutations on three elements. We will think of the three elements being permuted as red, green, and blue, hence the labels R, G, and B for the 2-circulant matrices (i.e. R labels the permutation that leaves red unchanged and swaps green and blue, etc.). Similarly, “I” labels the permutation that leaves nothing changed, while J and K are the non trivial even permutations.

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