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Spaces of similarities. I: The Hurwitz problem. (English) Zbl 0358.15024
Let \(\sigma\) and \(q\) be nonsingular quadratic forms over a field. The statement \(\sigma < Sim(q)\) means that \(q\) ”admits composition” with \(\sigma\); that is, there exists a formula \(\sigma(X)q(Y) = q(Z)\), where \(X\) and \(Y\) are vectors of indeterminates and each entry of \(Z\) is a bilinear form in \(X\) and \(Y\). In this situation, \(\sigma\) is called a subspace of similarities of \(q\), because each element of the quadratic space \((S,\sigma)\) acts as a similarity on the space \((V,q)\). It was proved 50 years ago by Radon and Hurwitz, that when \(\sigma\) and \(q\) are sum-of-squares forms over the reals or complexes, and \(\dim q=n\), then: \(\sigma < Sim(q)\) if and only if \(\dim \sigma \leq \rho(n)\). Here, the Hurwitz-Radon function \(\rho\) is defined by: \(\rho(n) =8a+ 2^b\), for \(n=2^m n_0\), where \(n_0\) is odd and \(m=4a +b\), \(0\leq b\leq 3\). In this paper the following more general theorem is proved. Theorem. Let \(\sigma\) and \(q\) be quadratic forms over \(F\) with \(\dim q= n\). If \(\sigma<Sim(q)\), then \(\dim \sigma \leq \rho(n)\). Conversely, if \(q \cong \varphi \otimes \omega\), where \(\varphi\) is a Pfister form and \(\dim \omega\) is odd, then there exists a form \(\sigma\) with \(\sigma < Sim(q)\) and \(\dim \sigma = \rho(n)\). The proof involves the study of representations of the Clifford algebra associated to \(\sigma\). Some generalizations and other types of Hurwitz functions are also considered.

MSC:
15A63 Quadratic and bilinear forms, inner products
15A66 Clifford algebras, spinors
11E16 General binary quadratic forms
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