Construction of eigenstates for the cyclotomic polygons.

To repeat our earlier discussion, it is not difficult te see that, for domains with the symmetry group of the regular n-gons whose vertices are determined by the nth root of unity, $z^n=1$ and translation invariant differential operator with this symmetry group, the capacitance matrix will have a block circulant structure. Using the notation of Davis [16],

$$C_n={\rm bcirc}(A_1,A_2,\ldots,A_n).$$

Each block $A_j$ is associated with the jth root of the nth degree cyclotomic equation, $z^n=1$. Since $C_n$ is symmetric and block circulant we find that for $j=1,2,\ldots,n$

1. For real $z_j$, $A_j=A_j^t.$
2. For complex $z_j$, $A_j=A_{n+2-j}^t.$

By a result of Davis ([16], p 180) a block circulant matrix can be block diagonalized by

$$C_n=(F_n\otimes I_m)^* {\rm diag}(P_1,P_2,\ldots,P_n)(F_n\otimes I_m),$$

where $\otimes$ is the direct product and $F_n$ is the Fourier matrix $F^*_n=\frac{1}{\sqrt n}(w^{(i-1)(j-1)}$ with $w=e^{\imath 2\pi/n}$.

Again, with $z_j=w^{(j-1)}$,

1. For real $z_j$, $P_j=P_j^t.$
2. For complex $z_j$, $P^*_j=P_j=\overline{P}_{n+2-j}.$

The analysis of the square eigenstates can be easily generalized. The principal point is the existence of the operators

$$R_j= \left( \begin{array}{ll} z_j & \\ & J \end{array}\right)$$

so that

1. For real $z_j$, $P_j=R_jP_j.R_j$
2. For complex $z_j$, $\overline{P}_j=R_jP_jR_j^*.$

With these operators a complete description of the capacitance matrix eigenspaces can be found for all of the cyclotomic polygons. Although the equilateral triangle, the square and hexagon are the only regular polygons that tile the plane the general cyclotomic polygon might be useful in more general domain decompositions.