The unique representation of Heisenberg Group. How can we construct the unique-up-to-isomorphism irreducible representation of Heisenberg Group.

There is a result of David Mumford (see also the related result of Marshall Stone and John von Neumann) stating that there
exists a unique irreducible representation of the algebraic Heisenberg group such that its center acts normally.
Here algebraic Heisenberg groups are affine group schemes $G$ over an algebraically closed field $k$ lying in a central extension
$$
1\rightarrow G_m \rightarrow G \rightarrow K \rightarrow 0,
$$
where $K$ is a finite abelian affine group scheme and $G_m = Spec ( k[t, t^{−1}])$ is the center of $G$. A proof with references can found in the thesis of Matt Collins.
On the other hand, the real Heisenberg group has many linear representations, and in particular a faithful linear one with upper-trinagular matrices of size $3$ (see below resp. above).
