Automorphisms and some geodesic properties of ortho-Grassmann graphs (Q2121741)

Summary: Let \(H\) be a complex Hilbert space. Consider the ortho-Grassmann graph \(\Gamma^{\perp}_k(H)\) whose vertices are \(k\)-dimensional subspaces of \(H\) (projections of rank \(k)\) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-\(k\) projections commute and their difference is an operator of rank \(2)\). Our main result is the following: if \(\dim H\ne 2k\), then every automorphism of \(\Gamma^{\perp}_k(H)\) is induced by a unitary or anti-unitary operator; if \(\dim H=2k\geqslant 6\), then every automorphism of \(\Gamma^{\perp}_k(H)\) is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when \(\dim H=2k=4\) the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.