Unitary Circulant Matrices and Idempotentcy

Let be a complete set of N annihilating primitive idempotents. That is, , and when j is not k, and for all k. Also suppose that each is Hermitian.

Let be a set of N real numbers. Then:
is unitary.
I’m guessing that the reader will find the proof immediately. If not, ask in the comments and I’ll give the short proof.

I’ve been working on the CKM matrix recently, which is a 3×3 unitary matrix. For 3×3 matrices, the simplest complete set of annihilating primitive idempotents is the diagonal primitive idempotents, that is, the matrices that are zero except for a single one somewhere on the diagonal:

The unitary matrices generated by this set are simply the diagonal matrices with complex phases down the diagonal.

The 1-Circulant Primitive Idempotents

Around here, our favorite complete set of mutually annihilating primitive idempotents for the 3×3 matrices are the 1-circulant ones:

Label the above three 3×3 matrices as .

Interpreted as density matrix states, these are generated from the bra / ket states as shown in the right side of the above. Just as all other pure density matrix states created from state vectors, they are Hermitian. In addition, they are all magic. The rows and columns of sum to unity, while the rows and columns of the and sum to zero.

So these are just what we need to write down an elegant parameterization of the 1-circulant unitary matrices:

This is a unitary matrix whose rows and columns sum to .

The previous post showed a parameterization of the magic unitary 3×3 matrices. These included 2-circulant parts as well as 1-circulant. In the above, the primitive idempotent is both 1-circulant and 2-circulant, while the other two matrices are 2-circulant. Can we get an elegant parameterization of the magic matrices this way?