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Reflection Groups in Finite Geometry

by Steven H. Cullinane, Dec. 7, 2007

A survey paper in the current Bulletin of the American Mathematical Society (Vol. 45, No. 1, January 2008, pp. 1-60) is titled "Reflection Groups in Algebraic Geometry."  That paper deals with groups defined over fields of characteristic zero; this note is to point out some references for reflection groups over fields of positive characteristic.

Recall that a reflection group may be defined as a group of linear transformations of a vector space over a (possibly finite) field that is generated by reflections-- transformations that fix a hyperplane pointwise.

Characteristic Two:

For characteristic two, there exist easily visualized reflection groups acting on the 2x2 square, the 2x2x2 cube, the 4x4 square, and the 4x4x4 cube.  For details, see Binary Coordinate Systems and Finite Geometry of the Square and Cube.

Characteristic Greater than Two:

See, for instance, the following--

1. Linear Groups Generated by Reflection Tori (pdf), by Arjeh M. Cohen, Hans Cuypers, and Hans Sterk (Canad. J. Math. Vol. 51 (6), 1999, pp. 1149-1174).

2. From a page for Mathematics of the USSR-- Izvestiya:

by A. E. Zalesskii and V. N. Serezhkin

A complete classification is given of all finite irreducible linear groups generated by reflections over an arbitrary field of characteristic not 2.

Bibliography: 10 titles.

DOI 10.1070/IM1981v017n03ABEH001369
Citation  A E Zalesskii, V N Serezhkin, "FINITE LINEAR GROUPS GENERATED BY REFLECTIONS", MATH USSR IZV, 1981, 17 (3), 477-503.
Classification  AMS MSC: Primary: 20H15, 51F15
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