Editing Unitary group (section)

== Topology ==
The unitary group U(''n'') is endowed with the [[relative topology]] as a subset of {{nowrap|M(''n'', '''C''')}}, the set of all {{nowrap|''n'' × ''n''}} complex matrices, which is itself homeomorphic to a 2''n''<sup>2</sup>-dimensional [[Euclidean space]].

As a topological space, U(''n'') is both [[compact space|compact]] and [[connected space|connected]]. To show that U(''n'') is connected, recall that any unitary matrix ''A'' can be [[diagonalized]] by another unitary matrix ''S''. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the [[main diagonal]]. We can therefore write
: <math>A = S\,\operatorname{diag}\left(e^{i\theta_1}, \dots, e^{i\theta_n}\right)\,S^{-1}.</math>

A [[path (topology)|path]] in U(''n'') from the identity to ''A'' is then given by
: <math>t \mapsto S \, \operatorname{diag}\left(e^{it\theta_1}, \dots, e^{it\theta_n}\right)\,S^{-1} .</math>

The unitary group is not [[simply connected]]; the fundamental group of U(''n'') is infinite cyclic for all ''n'':<ref>{{harvnb|Hall|2015}} Proposition 13.11</ref>
: <math>\pi_1(\operatorname{U}(n)) \cong \mathbf{Z} .</math>

To see this, note that the above splitting of U(''n'') as a [[semidirect product]] of SU(''n'') and U(1) induces a topological product structure on U(''n''), so that
: <math>\pi_1(\operatorname{U}(n)) \cong \pi_1(\operatorname{SU}(n)) \times \pi_1(\operatorname{U}(1)).</math>

Now the first unitary group U(1) is topologically a [[circle]], which is well known to have a [[fundamental group]] isomorphic to '''Z''', whereas SU(''n'') is simply connected.<ref>{{harvnb|Hall|2015}} Proposition 13.11</ref>

The determinant map {{nowrap|det: U(''n'') → U(1)}} induces an isomorphism of fundamental groups, with the splitting {{nowrap|U(1) → U(''n'')}} inducing the inverse.

The [[Weyl group]] of U(''n'') is the [[symmetric group]] S<sub>''n''</sub>, acting on the diagonal torus  by permuting the entries:
: <math>\operatorname{diag}\left(e^{i\theta_1}, \dots, e^{i\theta_n}\right) \mapsto \operatorname{diag}\left(e^{i\theta_{\sigma(1)}}, \dots, e^{i\theta_{\sigma(n)}}\right)</math>