A note on adjacency preservers on Hermitian matrices over finite fields
Let \(F\) be a finite field with nonidentical involution denoted by \(\;\bar{ }:F\to F\), and let \(n\geq 2\). A matrix \(A=\bigl(a_{ij}\bigr)_{ij}\in M_n(F)\) is hermitian if it equals its conjugated transpose \(A^\ast:=\bar{A}^{tr}=\bigl(\bar{a}_{ij}\bigr)_{ij}^{tr}\). The set of all Hermitian matrices, \(H_n(F)\) forms an undirected graph, called a graph of Hermitian matrices, whose vertex set equals \(H_n(F)\), and where vertices, corresponding to matrices \(A,B\) are connected if rank\((A-B)=1\). The author proves that this is a core graph, that is, every homomorphism is actually an isomorphism. Using the well-known results on geometry of Hermitian matrices it then follows that every homomorphism of the graph of Hermitian matrices takes the form \[ A\mapsto PA^\sigma P^\ast+B \] for some invertible \(P\) and some field isomorphism \(\sigma\) commuting with involution. Actually, a finite field \(F\) possesses at most one nonidentical involution, which forces \(\sigma\) to be given by \(x\mapsto x^{\alpha}\), \(\alpha=\text{char}(F)^j\) for some \(j\). This gives a negative answer to a natural question posed by \textit{W.-L. Huang} [Aequationes Math. 75, No.~1--2, 51--64 (2008; Zbl 1146.15001)] if there exists an adjacency preserving map on \(H_n(F)\) whose image contains only adjacent matrices. It is a folklore that in infinite fields such maps can exist. Say, when \(F=C\) is the infinite field of complex numbers, the map \(A\mapsto \text{Trace}(A)E_{11}\) preserves adjacency.
scientific article; zbMATH DE number 5603984
scientific article