On the geometric invariant theory of infinite groups generated by skew reflections

Ignatenko, V. F.

doi:10.1007/bf02684079

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R0 = R~((1 -)Ty~+t) and = Aortx~ (8)1

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“…By (2). (13) When u > Ul > 0 (u_, = u3 = 0) and h + u < A. we select some w-plane H" = fl"OH v' (w = u -u~) and place new coordinate axes in it.…”

Section: R0 = R~((1 -)Ty~+t) and = Aortx~ (8)mentioning

confidence: 99%

On algebraic surfaces with an infinite set of planes of skew symmetry

Ignatenko¹,

Diban²

1998

J Math Sci

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We consider (m-1)-dimensional noncylindrical algebraic surfaces in a real space R m with infinite symmetry groups G generated by skew reflections with respect to planes, along with the relative locations of linear hulls of four G-orbits of the directions of symmetry.Let F. be an (m-1)-dimensional noncylindrical surface of order n in a real space Era: assume that F is invariant with respect to an infinite group G that is generated by skew reflections relative to planes and does not admit extension. In proving the theorem (Sections l~ ~ we obtain equations for the surfaces F,, that are invariant under the corresponding groups G. t ~ We set dj = 1 and cr = v (there is no loss of generality). In the space E m we specify a Cartesian coordinate system Oyl ...y4zl ...z,zl ...zr (m = r + r + 4). We define a surface Fn (n > 2) with complete symmetry group G by means of the equation

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“…By (2). (13) When u > Ul > 0 (u_, = u3 = 0) and h + u < A. we select some w-plane H" = fl"OH v' (w = u -u~) and place new coordinate axes in it.…”

Section: R0 = R~((1 -)Ty~+t) and = Aortx~ (8)mentioning

confidence: 99%