Editing Projective representation (section)

==First example: discrete Fourier transform==
{{see also|Oscillator representation|Theta representation}}
Consider the field <math>\mathbb Z/p</math> of integers mod <math>p</math>, where <math>p</math> is prime, and let <math>V</math> be the <math>p</math>-dimensional space of functions on <math>\mathbb Z/p</math> with values in <math>\mathbb C</math>. For each <math>a</math> in <math>\mathbb Z/p</math>, define two operators, <math>T_a</math> and <math>S_a</math> on <math>V</math> as follows:
:<math>\begin{align}
  (T_a f)(b) &= f(b - a) \\
  (S_a f)(b) &= e^{2\pi iab/p}f(b).
\end{align}</math>

We write the formula for <math>S_a</math> as if <math>a</math> and <math>b</math> were integers, but it is easily seen that the result only depends on the value of <math>a</math> and <math>b</math> mod <math>p</math>. The operator <math>T_a</math> is a translation, while <math>S_a</math> is a shift in frequency space (that is, it has the effect of translating the [[discrete Fourier transform]] of <math>f</math>).

One may easily verify that for any <math>a</math> and <math>b</math> in <math>\mathbb Z/p</math>, the operators <math>T_a</math> and <math>S_b</math> commute up to multiplication by a constant:
:<math>T_a S_b = e^{-2\pi iab/p}S_b T_a</math>.

We may therefore define a projective representation <math>\rho</math> of <math>\mathbb Z/p\times \mathbb Z/p</math> as follows:
:<math>\rho(a, b) = [T_a S_b]</math>,
where <math>[A]</math> denotes the image of an operator <math>A\in\mathrm{GL}(V)</math> in the quotient group <math>\mathrm{PGL}(V)</math>. Since <math>T_a</math> and <math>S_b</math> commute up to a constant, <math>\rho</math> is easily seen to be a projective representation. On the other hand, since <math>T_a</math> and <math>S_b</math> do not actually commute—and no nonzero multiples of them will commute—<math>\rho</math> cannot be lifted to an ordinary (linear) representation of <math>\mathbb Z/p\times \mathbb Z/p</math>.

Since the projective representation <math>\rho</math> is faithful, the central extension <math>H</math> of <math>\mathbb Z/p\times \mathbb Z/p</math> obtained by the construction in the previous section is just the preimage in <math>\mathrm{GL}(V)</math> of the image of <math>\rho</math>. Explicitly, this means that <math>H</math> is the group of all operators of the form
:<math>e^{2\pi ic/p}T_a S_b</math>

for <math>a,b,c\in\mathbb Z/p</math>. This group is a discrete version of the [[Heisenberg group]] and is isomorphic to the group of matrices of the form
::<math>\begin{pmatrix}
 1 & a & c\\
 0 & 1 & b\\
 0 & 0 & 1\\
\end{pmatrix}</math>

with <math>a, b, c\in\mathbb Z/p</math>.