Example. Real Fourier sectors of a cycle [fdip-004W]
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  • October 31, 2026

Let \(n\ge 3\), \(V=E=\mathbb Z/n\mathbb Z\), with edge \(e_v\) from \(v\) to \(v+1\), uniform weight \(w>0\) and mass \(m>0\). The generator \(r\) of \(C_n\) sends \(v\) to \(v+1\) and \(e_v\) to \(e_{v+1}\), with every edge sign positive. Its action on fields is \((\rho (r)f)(v)=f(v-1)\). The cyclic characters in [serre1977linear, Section 5.1, p. 35] and the real projectors yield the following direct calculation.

Set \(\theta _k=2\pi k/n\). The normalized constant vector \(u_0(v)=1/\sqrt n\) spans the trivial sector. For \(1\le k<n/2\), the sector \(H_k\) has orthonormal basis \[c_k(v)=\sqrt {2/n}\cos (\theta _kv),\qquad s_k(v)=\sqrt {2/n}\sin (\theta _kv).\] When \(n\) is even there is also \(u_{n/2}(v)=(-1)^v/\sqrt n\). These give all sectors indexed by \(0\le k\le \lfloor n/2\rfloor \). In the ordered basis \((c_k,s_k)\), the rotation is \[\rho (r)|_{H_k}=\begin {pmatrix} \cos \theta _k&-\sin \theta _k\\ \sin \theta _k&\cos \theta _k \end {pmatrix}.\] For \(0<\theta _k<\pi \) it has no real eigenline, so \(H_k\) is real irreducible. Its complex constituents have characters \(r\mapsto e^{i\theta _k}\) and \(r\mapsto e^{-i\theta _k}\).

Proof. 

  • October 31, 2026

The finite geometric sum \(\sum _{v=0}^{n-1}e^{2\pi i jv/n}\) equals \(n\) when \(n\) divides \(j\) and zero otherwise. Taking real and imaginary parts proves all stated norms and orthogonality. There are \(n\) basis vectors, so they span \(H\). The addition formulas give the displayed rotation. A real rotation through \(0<\theta _k<\pi \) has eigenvalues \(e^{\pm i\theta _k}\), neither real, proving the irreducibility claim.

The precision and each sector eigenvalue are \[\begin {aligned} (Qf)(v)&=(m^2+2w)f(v)-wf(v-1)-wf(v+1),\\ Q|_{H_k}&=q_kI,\\ q_k&=m^2+2w(1-\cos \theta _k). \end {aligned}\] Substitution of each sine, cosine and exceptional one-dimensional mode proves the formula. The orthogonal projector kernels are \[\begin {aligned} P_0(v,u)&=1/n,\\ P_k(v,u)&=\frac 2n\cos (\theta _k(v-u))\quad (0<k<n/2),\\ P_{n/2}(v,u)&=(-1)^{v-u}/n\quad (n\text { even}). \end {aligned}\] They follow by summing the outer products of the displayed orthonormal basis vectors, so their sum is the identity. For \(n=3\), this reads \(P_0=\mathbf 1\mathbf 1^T/3\) and \(P_1=I-P_0\); the eigenvalues are \(m^2\) and \(m^2+3w\), the latter twice. The later circle comparison rescales these same finite modes; no continuum limit is used in this calculation.