& -1 & 2 & -1 \\
-1 & & & -1 &2
\end{array} \right)
.$$ The mass eigenvalues will then be given by those of the $\Omega$ matrix with a multiplicity given by $M^2-1$, the dimension of the gauge index space. The diagonalization of $\Omega$ follows by writing $\Omega$ as $2-C-C^{\dagger}$, where $C$ is the matrix of cyclic permutations $$C=\left( \begin{array}{rrrr}
 & 1 &   & \\
 &   & \ddots &\\
 &   &   & 1\\
1 &&& \\
\end{array} \right),$$ whose eigenvectors are given by $(1,\omega_k,\omega_k^2, \ldots ,\omega_k^{N-1})$, and eigenvalues by $\omega_k$, where $\omega_k=e^{i\, 2\pi k/N}$, $k=0,\dots,(N-1)$, are the $N^{\rm th}$ roots of unity. From this the mass eigenvalues are [@ACG; @Fermi1; @Fermi2]: $$m_k= 2 \sqrt{2} g v \sin \frac{k \pi}{N}, \qquad 0\leq k\leq N-1.
\label{eq:mkomega}$$ corresponding to the normalized eigenvectors: $$\label{eq:eigenbosons}
\tilde{A}^a_{k\, \mu}
=\frac{1}{\sqrt{N}}\, \sum_{j=1}^{N} \omega_k^{j-1} A^a_{j\, \mu},
\qquad 0\leq k\leq N-1$$ This mode decomposition is just the discrete analogue of the usual continuous Fourier expansion (see Fig. \[fig:lat\](a)). Each mass level is $M^2-1$ degenerate, forming an adjoint representation of the unbroken diagonal $SU(M)$ gauge group (on top of this gauge degeneracy, there is also an accidental lattice degeneracy since $m_k=m_{N-k}$). For small enough $k$ the spectrum approximates the 