Log24

Tuesday, October 19, 2010

Savage Logic continued…

Filed under: General,Geometry — m759 @ 9:36 AM

CHAPTER V

THE KALEIDOSCOPE

"This is an account of the discrete groups generated by reflections…."

Regular Polytopes , by H.S.M. Coxeter (unabridged and corrected 1973 Dover reprint of the 1963 Macmillan second edition)

"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields…. Let V  be an n-dimensional vector space over a field F, and let G ≤ Gln (F) be a finite group…. An element of finite order in Gl(V ) is a reflection if its fixed point space in V  is a hyperplane, called the reflecting hyperplane. There are two types of reflections: the diagonalizable reflections in Gl(V ) have a single nonidentity eigenvalue which is a root of unity; the nondiagonalizable reflections in Gl(V ) are called transvections and have determinant 1 (note that they can only occur if the characteristic of F is positive)…. A reflection group is a finite group G  generated by reflections."

— Julia Hartmann and Anne V. Shepler, "Reflection Groups and Differential Forms," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955-971

"… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field. The reflections in Gl(V ) not only include diagonalizable reflections (with a single nonidentity eigenvalue), but also transvections, reflections with determinant 1 which can not be diagonalized. The transvections in Gl(V ) prevent one from developing a theory of reflection groups mirroring that for Coxeter groups or complex reflection groups."

— Julia Hartmann and Anne V. Shepler, "Jacobians of Reflection Groups," Transactions of the American Mathematical Society , Vol. 360, No. 1 (2008), pp. 123-133 (Pdf available at CiteSeer.)

See also A Simple Reflection Group of Order 168 and this morning's Savage Logic.

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