2020/06/28 by Pankov, Mark, Petelczyc, Krzysztof, Zynel, Mariusz
#Combinatorics (math.CO) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2006.15581
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.