Universality of Weyl unitaries (Q2052177)

Let \(p,d\) be positive integers and let \(\omega\) be a primitive \(p\)-th root of the unity. With the help of trace and determinant it is shown that any two unitary matrices \(u,v\in M_d(\mathbb{C})\) which satisfy Weyl's relations \[u^p=v^p=1\quad\hbox { and }\quad uv=\omega vu\] have the spectrum equal to \(\{1,\omega,\dots,\omega^{p-1}\}\) with all eigenvalues having the same multiplicity. As a consequence, it is further shown that the matrix size \(d\) is divisible by \(p\) and every such pair of unitary matrices \((u,v)\) is simultaneously unitarily similar to a standard pair \[ \mathbf{u}=\mathrm{diag}(1,\omega,\dots,\omega^{p-1})\otimes I\quad\hbox{ and }\quad \mathbf{v}=c\otimes I\tag{\(\ast\)}, \] where \(c\in M_p(\mathbb{C})\) is a cyclic permutation matrix. More precisely, the authors first prove that \((u,v)\) is simultaneously unitarily similar to \[\mathrm{diag}(1,\omega,\dots,\omega^{p-1})\otimes I\] and a block-matrix \[(v_2^\ast v_3^\ast\dots v_p^\ast)E_{p1}+\sum v_{i} E_{(i+1)i}\] for some unitary matrices \(v_2,\dots,v_{p}\). After that, they apply the theory of completely positive maps. However, a concrete simultaneous unitary similarity to the standard pair \((\ast)\) is given by a block-diagonal unitary matrix \(\mathrm{diag}(I,v_2, v_3 v_2,v_4v_3v_2,\dots)\). The authors further provide a recursive construction which, for each integer \(k\), produces an ordered tuple \((u_1,\dots,u_{2k+1})\) of unitary matrices in \(\bigotimes_1^k M_{p}(\mathbb{C})\) that satisfy \(u_i^p=1\) and \(u_iu_j=\omega u_j u_i\), \(1\le i<j\le 2k+1\). It is shown, among other results, that each such tuple is irreducible (i.e., it has no proper nontrivial invariant subspace in common) and is a matrix extreme point as well as an absolute extreme point of its matrix range.