Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution
Given a division ring \(D\) with an involution \(a\to \bar{a}\), let \( \mathcal{H}_{n}(D)\) be the space of \(n\times n\) Hermitian matrices, \(n\geq 2\), with respect to the involution. For \(H_{1},H_{2}\in \mathcal{H}_{n}(D)\) the arithmetic distance ad\((H_{1},H_{2})\) is defined to be the rank of \(H_{1}-H_{2},\) and two matrices are said to be adjacent if their arithmetic distance is 1. Let \(\varphi :\mathcal{H}_{n}(D) \to \mathcal{H}_{n}(D)\) be an adjacency-preserving map which is a bijection. An important theorem of the geometry of Hermitian matrices with involution says that, under certain conditions, \(\varphi (X)=\alpha ^{t} \bar{P}X^{\sigma }P+H_{0}\) for all \(X\in \mathcal{H}_{n}(D)\) where \(\alpha \) is a nonzero element of \(D\) such that \(\bar{\alpha}=\alpha ,\) \(P\in \text{GL}_{n}(D),\;\sigma :D\to D\) an automorphism commuting with the involution, and \(H_{0}\) a fixed element of \( \mathcal{H}_{n}(D).\) One of these ``certain conditions'' is that \(\varphi ^{-1}\) also preserve the adjacency. Let \(P\in \text{GL}_{n}(D).\) In this paper, it is shown that any adjacency-preserving bijective map \(\varphi \) on \(\mathcal{H}_{n}(D)\) such that \(\varphi (0)=0\) then there is a \(Q\in \text{GL}_{n}(D)\) such that \(\varphi (^{t}\bar{P} \mathcal{M}_{i}P)=\,^{t}\bar{Q}\mathcal{M}_{i}Q,\) where \(\mathcal{M} _{i}\) is the subset of \(\mathcal{H}_{n}(D)\) generated over \(F\) by the matrix consisting of a one in the \(i,i\)-position and zeros everywhere else. Additionally, the author investigates whether the condition on \(\varphi ^{-1} \) can be removed. It is proved that any adjacency-preserving bijection \( \varphi \) has the property that \(\varphi ^{-1}\) also preserves the adjacency. This answers an open question by Z. X. Wan. Furthermore, if \( \mathcal{H}_{n}(D)\neq \mathcal{S}_{3}(\mathbb{F}_{2})\) then \(\varphi \) in fact preserves arithmetic distance, where \( \mathcal{S}_{3}(\mathbb{F}_{2})\) is the set of \(3\times 3\) symmetric matrices over \(\mathbb{F}_{2}.\)
scientific article; zbMATH DE number 5080003
scientific article