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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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##### References:
 [1] Adams, J.F; Lax, P.D; Phillips, R.S; Adams, J.F; Lax, P.D; Phillips, R.S, On matrices whose real linear combinations are nonsingular, (), 945-947 · Zbl 0168.02404 [2] Curtis, C, The four and eight square problem and division algebras, () · Zbl 0196.30806 [3] Fröhlich, A; McEvett, A.W, Forms over rings with involution, J. algebra, 12, 79-104, (1969) · Zbl 0256.15017 [4] Gabel, M.R, Generic orthogonal stably free projectives, J. algebra, 29, 477-488, (1974) · Zbl 0297.16014 [5] Geramita, A.V; Pullman, N.J, A theorem of Hurwitz and Radon and orthogonal projective modules, (), 51-56 · Zbl 0279.13007 [6] Hurwitz, A, Über die komposition der qudratischen formen, Math. ann., 88, 1-25, (1923) [7] Jacobson, N, Composition algebras and their automorphisms, Rend. circ. mat. Palermo, 7, 2, 55-80, (1958) · Zbl 0083.02702 [8] Lam, T.Y, The algebraic theory of quadratic farms, (1973), Benjamin New York · Zbl 0259.10019 [9] McEvett, A.W, Forms over semisimple algebras with involution, J. algebra, 12, 105-113, (1969) · Zbl 0256.15018 [10] Radon, J, Lineare scharen orthogonaler matrizen, Abh. math. sem. univ. Hamburg, 1, 1-14, (1922) · JFM 48.0092.06 [11] Shapiro, D.B, Cancellation of semisimple Hermitian pairings, J. algebra, 41, 212-223, (1976) · Zbl 0341.16005 [12] Shapiro, D.B, Spaces of similarities. II. Pfister factors, J. algebra, 46, 171-181, (1977) · Zbl 0358.15025 [13] Wolf, J.A, Geodesic spheres in Grassmann manifolds, Illinois J. math., 7, 425-446, (1963) · Zbl 0114.37002 [14] Wolfe, W, Amicable orthogonal designs—existence, Canad. J. math., 28, 1006-1020, (1976) · Zbl 0326.05021
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