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wdlang
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how to prove that the symmetry group of a regular polygon has only 1 and 2 dim irreducible representations?
morphism said:The group in question, i.e. the dihedral group, has an abelian subgroup of index 2 (the one generated by the reflection). Thus any irreducible representation is at most 2 dimensional. I'll let you fill in the details. Post back if you need more help!
Yup - sorry! (The reflection subgroup has order 2!)wdlang said:maybe you mean the rotation subgroup is of index 2?
morphism said:The group in question, i.e. the dihedral group, has an abelian subgroup of index 2 (the one generated by the reflection). Thus any irreducible representation is at most 2 dimensional. I'll let you fill in the details. Post back if you need more help!
A symmetry group of a regular polygon is a set of all the transformations that preserve the shape and size of the polygon. These transformations include rotations, reflections, and translations.
Proving that a symmetry group of a regular polygon has 1 and 2 dimensional irreducible representations helps us understand the structure and properties of the polygon. It also has implications in fields such as crystallography, where the symmetry of regular polygons is used to describe the arrangement of atoms in a crystal lattice.
To prove the existence of 1 and 2 dimensional irreducible representations in a symmetry group of a regular polygon, we use group theory and representation theory. We first identify the generators and relations of the symmetry group, and then use techniques such as character tables and group characters to determine the number and dimension of irreducible representations.
Yes, a symmetry group of a regular polygon can have more than 2 dimensional irreducible representations. In fact, the number and dimension of irreducible representations can vary depending on the order and type of symmetry of the polygon.
Knowing the 1 and 2 dimensional irreducible representations of a symmetry group of a regular polygon allows us to predict and determine various properties of the polygon, such as its rotational and reflectional symmetry axes, its point group, and its crystallographic space group. It also helps in understanding the relationships between different symmetry groups and their representations.