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Geometry of symmetric matrices and its applications. I. (English) Zbl 0806.15010

Let \(F\) and \(F'\) be fields of characteristic \(\neq 2\), \(n\) and \(n'\) integers \(\geq 2\) and \({\mathcal S}_ n (F)\) the set of \(n \times n\) symmetric matrices over \(F\). Denote by \(\Gamma ({\mathcal S}_ n(F))\) the graph whose vertices are the elements of \({\mathcal S}_ n(F)\) and in which an edge joins vertices \(S_ 1\) and \(S_ 2\) if and only if \(\text{rank} (S_ 1 - S_ 2) = 1\). The author determines all graph isomorphisms between \(\Gamma ({\mathcal S}_ n (F))\) and \(\Gamma ({\mathcal S}_{n'} (F'))\): they exist only when \(n = n'\) and \(F\) is isomorphic to \(F'\); and then a mapping \({\mathcal A} : {\mathcal S}_ n (F) \to {\mathcal S}_{n'} (F')\) provides a graph isomorphism if and only if it has the form \({\mathcal A} (X) = a P^ T X^ \sigma P + S_ 0\), where \(a \in F'{}^*\), \(P \in GL_ n (F')\), \(\sigma\) is an isomorphism from \(F\) to \(F'\) and \(S_ 0 \in {\mathcal S}_ n(F')\). The special case where \(n=n'\) and \(F=F'\) is due to L. K. Hua [Ann. Math., II. Ser. 50, 8-31 (1949, Zbl 0034.157)]. The (quite complicated) proof is modelled on Hua’s proof of a cognate result for rectangular matrices. Both Hua’s and the author’s results are closely related to those of W. L. Chow [Ann. Math., II. Ser. 50, 32-67 (1949; Zbl 0040.229)]: cf. Ch. III. §3, in J. Dieudonné [La Géométrie des Groupes Classiques, 2nd ed. (1963; Zbl 0111.031)].
The paper contains as (not quite immediate) corollaries two further isomorphism theorems of a similar kind. The first describes the isomorphisms between the modified graphs \(\Gamma^* ({\mathcal S}_ n(F))\) and \(\Gamma^* ({\mathcal S}_{n'} (F'))\) in which an edge joins vertices \(S_ 1\) and \(S_ 2\) if and only if \(\text{rank} (S_ 1 - S_ 2) = 1\) or 2. The second determines the isomorphisms between \({\mathcal S}_ n (F)\) and \({\mathcal S}_{n'} (F')\) as Jordan rings.
Reviewer: G.E.Wall (Sydney)

MSC:

15A30 Algebraic systems of matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices
51D20 Combinatorial geometries and geometric closure systems
05E30 Association schemes, strongly regular graphs
17C30 Associated groups, automorphisms of Jordan algebras
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