2021/11/04 by Mark Pankov, Pankov, Mark, Krzysztof Petelczyc +3
Materials Science · Mathematics · #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Lanthanide and Transition Metal Complexes #Magnetism in coordination complexes #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2111.02837
openalex publication_date 2021/11/04 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
We consider the graph whose vertex set is a conjugacy class \mathcal C consisting of finite-rank self-adjoint operators on a complex Hilbert space H. The dimension of H is assumed to be not less than 3. In the case when operators from \mathcal C have two eigenvalues only, we obtain the Grassmann graph formed by k-dimensional subspaces of H, where k is the smallest dimension of eigenspaces. Classical Chow's theorem describes automorphisms of this graph for k>1. Under the assumption that operators from \mathcal C have more than two eigenvalues we show that every automorphism of the graph is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces with the same dimensions. In contrast to this result, Chow's theorem states that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality if \mathcal C is formed by operators with precisely two eigenvalues.