By Omer Cabrera
Desk of Contents
Chapter 1 - Symmetry
Chapter 2 - staff (Mathematics)
Chapter three - staff Action
Chapter four - commonplace Polytope
Chapter five - Lie aspect Symmetry
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Additional info for Applications of Symmetry in Mathematics, Physics & Chemistry
Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
The set of all orbits of X under the action of G is written as X /G (or, less frequently: G \X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.
The group is acting on a vector space, such as the three-dimensional Euclidean space R3. A representation of G on an n-dimensional real vector space is simply a group homomorphism ρ: G → GL(n, R) from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. Given a group action, this gives further means to study the object being acted on On the other hand, it also yields information about the group.