Manuscript Oai069-A@1 · Oai069-A.tex · proposition · a4913aab

Proposition: Fourier identities

An explicit abelian local-system category › Constructive Fourier realization · lines 533–554 · proof included or linked

Oai069-A@1#prop:fourier, Oai069-A@1#eq:finite-weyl

The printed matrix $F_{d}$ is unitary, and

F_{d} v_{k} = ∣ k ⟩, S_{a} v_{k} = ω^{ak} v_{k},

F_{d} S_{a} F_{d}^{†} = Z_{a}, F_{d} Z_{b} F_{d}^{†} = S_{- b} .

Thus the regular-representation local system is explicitly identified with its character decomposition.

Proof. The inner products of distinct rows of $F_{d}$ vanish by the finite sum in Lemma lem:projectors; each row has norm one. The remaining equations follow by substituting the finite sums and changing the summation index modulo $d$ .

Displayed formulas in this passage

Oai069-A@1#eq:finite-weyl · line 540
$F_{d} S_{a} F_{d}^{†} = Z_{a}, F_{d} Z_{b} F_{d}^{†} = S_{- b} .$
unlabeled · line 536
$F_{d} v_{k} = ∣ k ⟩, S_{a} v_{k} = ω^{ak} v_{k},$

Proposition: Fourier identities

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