Editing Stone–von Neumann theorem (section)

== Representations of finite Heisenberg groups ==
The Heisenberg group {{math|''H<sub>n</sub>''(''K'')}} is defined for any commutative ring {{mvar|K}}.  In this section let us specialize to the field {{math|''K'' {{=}} '''Z'''/''p'''''Z'''}} for {{mvar|p}} a prime.  This field has the property that there is an embedding {{mvar|ω}} of {{mvar|K}} as an [[abelian group|additive group]] into the circle group {{math|'''T'''}}.  Note that {{math|''H<sub>n</sub>''(''K'')}} is finite with [[cardinality]] {{math|{{!}}''K''{{!}}<sup>2''n'' + 1</sup>}}.  For finite Heisenberg group {{math|''H<sub>n</sub>''(''K'')}} one can give a simple proof of the Stone–von Neumann theorem using simple properties of [[Character theory|character function]]s of representations.  These properties follow from the [[orthogonality relations]] for characters of representations of finite groups.

For any non-zero {{mvar|h}} in {{mvar|K}} define the representation {{math|''U<sub>h</sub>''}} on the finite-dimensional [[inner product space]] {{math|{{ell}}<sup>2</sup>(''K''<sup>''n''</sup>)}} by
<math display="block">\left[U_h \mathrm{M}(a, b, c) \psi\right](x) = \omega(b \cdot x + h c) \psi(x + ha). </math>

{{math theorem | For a fixed non-zero {{mvar|h}}, the character function {{mvar|χ}} of {{math|''U<sub>h</sub>''}} is given by:
<math display="block">\chi (\mathrm{M}(a, b, c)) = \begin{cases}
  |K|^n\, \omega(hc) & \text{if } a = b = 0 \\
                   0 & \text{otherwise}
\end{cases}</math>}}

It follows that
<math display="block"> \frac{1}{\left|H_n(\mathbf{K})\right|} \sum_{g \in H_n(K)} |\chi(g)|^2 = \frac{1}{|K|^{2n+1}} |K|^{2n} |K| = 1. </math>

By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups {{math|''H<sub>n</sub>''('''Z'''/''p'''''Z''')}}, particularly:
* Irreducibility of {{math|''U<sub>h</sub>''}}
* Pairwise inequivalence of all the representations {{math|''U<sub>h</sub>''}}.
Actually, all irreducible representations of {{math|''H<sub>n</sub>''(''K'')}} on which the center acts nontrivially arise in this way.{{r|Hall 2013|p=Chapter 14, Exercise 5}}