Reading the figure
Translation and multiplication operators feed a common representation space. Their noncommutation is the central scalar relation; together they generate the full matrix algebra in the supplied finite model.
Mathematical labels
\(X\) translations
\(T_x f(u)=f(u+x)\)
\(X^*\) multipliers
\(M_\xi f(u)=\psi(\xi(u))f(u)\)
\(\mathcal{H}=\mathbf{C}^X\)
\(T_xM_\xi=\psi(\xi(x))M_\xi T_x\)
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X translations Txf(u) = f(u + x) X * multipliers Mξf(u) = ψ(ξ(u))f(u) = C X TxMξ = ψ(ξ(x))MξTx Noncommuting operators generate the full matrix algebra.
Original figure caption
The commutator is central and nontrivial when the pairing is nonzero.
For odd \(p\) let \(V=X\oplus X^*\) and
The group \(H(V)=V\times{\mathbb F}_p\) has product
On \(\mathcal H={\mathbb C}^X\) the Schrödinger operator is
Translations and character multipliers generate every matrix unit by finite Fourier inversion. Thus the representation is irreducible and has dimension \(|X|=\sqrt{|V|}\).