  Weyl's unitary matrices, which were introduced in Weyl's 1927 paper on group
theory and quantum mechanics, are $p\times p$ unitary matrices given by the
diagonal matrix whose entries are the $p$-th roots of unity and the cyclic
shift matrix. Weyl's unitaries, which we denote by $\mathfrak u$ and $\mathfrak
v$, satisfy $\mathfrak u^p=\mathfrak v^p=1_p$ (the $p\times p$ identity matrix)
and the commutation relation $\mathfrak u\mathfrak v=\zeta \mathfrak v\mathfrak
u$, where $\zeta$ is a primitive $p$-th root of unity. We prove that Weyl's
unitary matrices are universal in the following sense: if $u$ and $v$ are any
$d\times d$ unitary matrices such that $u^p= v^p=1_d$ and $ u v=\zeta vu$, then
there exists a unital completely positive linear map $\phi:\mathcal M_p(\mathbb
C)\rightarrow\mathcal M_d(\mathbb C)$ such that $\phi(\mathfrak u)= u$ and
$\phi(\mathfrak v)=v$. We also show, moreover, that any two pairs of $p$-th
order unitary matrices that satisfy the Weyl commutation relation are
completely order equivalent.
  When $p=2$, the Weyl matrices are two of the three Pauli matrices from
quantum mechanics. It was recently shown that $g$-tuples of Pauli-Weyl-Brauer
unitaries are universal for all $g$-tuples of anticommuting selfadjoint unitary
matrices; however, we show here that the analogous result fails for positive
integers $p>2$.
  Finally, we show that the Weyl matrices are extremal in their matrix range,
using recent ideas from noncommutative convexity theory.
