From the Applicable to the Abstruse: An Example in Representation Theory
The operations of time shift (ƒ(t) → ƒ(t+1)) and frequency shift (ƒ(t) → e2πiωtƒ(t)) are fundamental ingredients of applied Fourier analysis, and the group of operators on L2(ℝ) that they generate gives a unitary representation of the so-called discrete Heisenberg group. How does this representation de- compose into irreducible representations? The answer provides illustrations of (i) some useful tools of modern harmonic analysis, when ω is rational, and (ii) some pathological phenomena from the dark side of representation theory, when ω is irrational. We shall discuss these results after providing a bit of background on unitary representation theory.