1.4 Rational character-order projectors
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It is helpful to distinguish the size of an ambient signal from the order of a character. Fix a positive ambient length \(N\). For each divisor \(d\) of \(N\), define
The set \(S_d\) contains the labels of characters of exact order \(d\). Put \(D_d=\operatorname{diag}(\mathbf1_{a\in S_d})\) and
The superscript records the ambient dimension. In particular, the kernel normalization below is \(1/N\), even when \(d<N\).
Theorem QA06: Rational kernels and complete divisor decomposition
For every \(d\mid N\),
The matrices have rational entries and satisfy
Proof
A label in \(S_d\) has the unique form \(a=(N/d)b\), where \(0\le b<d\) and \(\gcd(b,d)=1\). Indeed,
Direct multiplication therefore gives
QA04 proves rationality. Each \(D_d\) is a self-adjoint diagonal projector. The sets \(S_d\) partition \(I_N\), because every character has exactly one order dividing \(N\). Hence \(D_dD_e=\delta_{de}D_d\) and \(\sum_{d\mid N}D_d=\operatorname{Id}_N\). Conjugating these identities by the unitary \(F_N\) proves the projector identities. The rank is unchanged under multiplication by invertible matrices, so it equals the number of selected diagonal entries, \(|S_d|=\varphi(d)\). The diagonal kernel entries are all \(c_d(0)/N=\varphi(d)/N\), whose sum is \(\varphi(d)\). \(\square\)
This orthogonal divisor-shell decomposition has established antecedents in Ramanujan subspaces and filter banks [2], [3], [4]. The general theorem above is proved independently in this text. The formal companion certifies the concrete \(N=d=12\) projector and its transport identity, rather than an arbitrary-divisor Lean theorem.
1.4.1 A shift-operator realization and its sign
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Let \(A\) be the cyclic shift \((Ax)(n)=x(n-1)\). A Fourier column \(f_a(n)=N^{-1/2}\zeta_N^{na}\) satisfies \(Af_a=\zeta_N^{-a}f_a\). Therefore
acts as the identity on \(f_a\) and as zero on every other column, by QA02. Summing \(E_a\) over \(S_d\) gives
This formula also follows entrywise because \((A^\ell)_{nm}\) is one exactly when \(n-m=\ell\) modulo \(N\). A group-algebra convention with the opposite shift sign writes individual idempotents with the opposite phase sign. The divisor shells are unchanged under \(a\mapsto -a\), and \(c_d(-\ell)=c_d(\ell)\), so the rational shell formula remains the same.
Proposition 21.11 of Stratified Universality for Resolved Representations, gives the rational cyclic order-projector formula in the group algebra. Boundary 21.12 distinguishes it from formal-series coefficient-residue selection [5]. The present shift calculation makes the representation and the sign explicit.
1.4.2 Unions of order classes
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If \(\mathcal D\) is any set of divisors of \(N\), then
is a rational orthogonal projector of rank \(\sum_{d\in\mathcal D}\varphi(d)\). Its complement is the sum over the remaining divisors. Individual split Fourier projectors can have nonrational entries. Summing complete order classes is a sufficient mechanism for rationality. No converse classification of every rational selector is needed for the present bridge.