Automorphisms and some geodesic properties of ortho-Grassmann graphs

TL;DR

This paper characterizes automorphisms of ortho-Grassmann graphs in complex Hilbert spaces, showing they are mostly induced by unitary or anti-unitary operators, with specific exceptions when the space dimension equals twice the subspace dimension.

Contribution

It provides a complete description of automorphisms of ortho-Grassmann graphs, linking them to unitary or anti-unitary operators and analyzing geodesic properties and compatibility.

Findings

Automorphisms are induced by unitary or anti-unitary operators when dim H ≠ 2k.

In the case dim H=2k≥6, automorphisms include compositions with orthocomplementary maps.

The characterization extends to generalized ortho-Grassmann graphs related to finite-rank operators.

Abstract

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 $\dim H=2k\ge 6$, 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.