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