Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators
| Autor: | Krzysztof Petelczyc, Mark Pankov, Mariusz Żynel |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory 010102 general mathematics Dimension (graph theory) Mathematics - Operator Algebras 010103 numerical & computational mathematics Automorphism 01 natural sciences Combinatorics Permutation Mathematics::Group Theory Operator (computer programming) Conjugacy class FOS: Mathematics Discrete Mathematics and Combinatorics Rank (graph theory) Mathematics - Combinatorics Combinatorics (math.CO) Geometry and Topology 0101 mathematics Operator Algebras (math.OA) Self-adjoint operator Eigenvalues and eigenvectors Mathematics |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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