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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.

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