dedekind::numbers::Character< N > Struct Template Reference

The de Moivre CHARACTER \(\chi_m : \mathbb{Z}/N \to \mu_N\), \(k \mapsto \zeta_N^{\,mk}\) — the plane wave "as such", carried SYMBOLICALLY in UnitRoot<N> (consumes the μ_N layer; resolve() collapses to exact ℂ at N=8). More...

Public Types
using	Domain = dedekind::morphologies::Modular<N>
using	Codomain = UnitRoot<N>

Public Member Functions
constexpr UnitRoot< N >	operator() (dedekind::morphologies::Modular< N > k) const

Public Attributes
dedekind::morphologies::Modular< N >	m {}

Detailed Description

template<unsigned N>
struct dedekind::numbers::Character< N >

The de Moivre CHARACTER \(\chi_m : \mathbb{Z}/N \to \mu_N\), \(k \mapsto \zeta_N^{\,mk}\) — the plane wave "as such", carried SYMBOLICALLY in UnitRoot<N> (consumes the μ_N layer; resolve() collapses to exact ℂ at N=8).

An IsArrow (position k → harmonic), so it feeds the function-space precompose / reflect combinators directly. Its codomain UnitRoot<N> is a group, not a semiring, so for bra-ket contraction (inner_product, which needs IsSemiring) it is first resolve()d into ℂ — see the ChiKet / ChiBra wrappers in plane_wave_spike_test. root8 is its eagerly-ℂ-collapsed m=1 instance. A group homomorphism in k ( \(\chi_m(k)\,\chi_m(k')=\chi_m(k+k')\)) AND in the dual m (it is the self-dual pairing \(\langle m,k\rangle=\zeta^{mk}\), the DFT kernel). The conjugate bra is just the negated frequency: \(\overline{\chi_m}=\chi_{-m}\) ( \(\overline{\zeta^{mk}}=\zeta^{-mk}\)) — so no separate conjugation is needed, an equality/negation suffices.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/numbers/complex.cppm