Spectral matrix $$S_6 = S_6^{(0)}=\left[\begin{array}{llllll} 1& 1& 1& 1& 1& 1\\ 1& 1& w& w& w^2& w^2\\ 1& w& 1& w^2& w^2& w\\ 1& w& w^2& 1& w& w^2\\ 1& w^2& w^2& w& 1& w\\ 1& w^2& w& w^2& w& 1\end{array}\right]:\quad w=\exp(2\pi i/3)$$ was first introduced in the work of A. T. Butson in an equivalent form [51]. Later it was rediscovered independently by G. Moorhouse [36] and by T. Tao [89].
It is symmetric, isolated and belongs to the Butson class $BH(6, 3)$, so its entries depend on the third root of unity.