Systems of subspaces of a unitary space (Q1938712)

For a finite poset \(\mathcal P=\{p_{1},\dots ,p_{t}\}\), the authors study systems \((U_{1},\dots ,U_{t})_{U}\) of subspaces of a unitary space \(U\) such that \(U_{i}\subseteq U_{j}\) if \(p_{i}\prec p_{j}\). Two systems \((U_{1},\dots ,U_{t})_{U}\) and \((V_{1},\dots ,V_{t})_{V}\) are said to be isometric if there exists an isometry \(\varphi :U\to V\) such that \(\varphi (U_{i})=V_{i}\). Such systems are classified up to isometry if \(\mathcal P\) is a semichain. The authors prove that the problem of their classification is unitarily wild if \(\mathcal P\) is not a semichain. A classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space, which is hopeless in a certain sense.