Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1992. Reflection transformation is one of the four types of transformations in geometry. When describing a reflection, you need to state the line which the shape has been reflected in. Goodman, The Mathematics of Mirrors and Kaleidoscopes, American Mathematical Monthly. A reflection is like placing a mirror on the page. In fact, if you have done MA453 Lie Algebras then you are already familiar with classification of semisimple Lie algebras, which is essentially the classification of crystallographic reflection groups.īesides classifications, we will concentrate on examples and polynomial invariants.
The question appeared in 1920s in the works of Cartan and Weyl as the Weyl group is a finite crystallographic reflection group. Students make and identify patterns that involve translation, reflection. The main goal of the module is to classify finite groups (of linear transformations) generated by reflections. In this unit students explore patterns involving transformations of shapes. The dihedral group can be generated by a pair of reflections. Synergies: The following modules go well together with Reflection Groups:Ĭontent: A reflection is a linear transformation that fixes a hyperplane and multiplies a complementary vector by -1. Useful background: Interest in Group Theory: MA251 Algebra I: Advanced Linear Algebra: Assumed knowledge: The groups theory and some geometric ideas from the second year Maths core: A reflection is an isometry, which means the original and image are congruent, that can be described as a flip.
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