Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators

TL;DR

This paper generalizes Grassmann graphs to conjugacy classes of finite-rank self-adjoint operators, characterizing their automorphisms as compositions of unitary or anti-unitary transformations and eigenspace permutations.

Contribution

It extends the adjacency concept to broader conjugacy classes and classifies automorphisms, linking them to unitary, anti-unitary, and semilinear transformations.

Findings

Automorphisms are composed of unitary/anti-unitary operators and eigenspace permutations.

Classical Chow's theorem applies to two-eigenvalue cases, involving semilinear automorphisms.

Generalized graphs reveal structure of conjugacy classes beyond classical Grassmann graphs.

Abstract

Two distinct projections of finite rank $m$ are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is $(m-1)$-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let ${\mathcal C}$ be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than $1$. Under the assumption that operators from ${\mathcal C}$ have at least three eigenvalues we prove that every automorphism of the corresponding generalized Grassmann graph is the composition of an automorphism induced by a unitary or anti-unitary operator and the automorphism obtained from a permutation of eigenspaces with the same dimensions. The case when the operators from ${\mathcal C}$ have two eigenvalues only is covered by classical Chow's theorem which says that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality.