A crash course on group theory
This document provides an overview of group theory concepts including: 1) The definition of a group which requires closure, associativity, identity, and inverse. An example is the D3 point group. 2) Group representations can be reducible or irreducible. Reducible representations can be broken into irreducible representations, while irreducible representations cannot. 3) Conjugacy classes are sets of elements that are conjugate to each other. The character of a class is the trace of the representation's matrix for elements in that class. Character tables display the characters of each class for different representations. 4) Irreducible group representations satisfy orthogonal relations where their characters form an orthogonal vector space. This allows application to problems using orthogonal relations
A crash course on group theory
- 1. A Crash Course on Group Theory 12 May, 2014 Dongwoook Go
- 2. References • George B. Arfken, Hans J. Weber, Frank E. Harris, Mathematical Methods for Physicists, 7th ed, 2013 Elsevier • Wikipedia • ChemWiki http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/ Group_Theory%3A_Theory • L. E. Laverman’s lecture note – Introduction to the Chemical Applications of Group Theory
- 3. Definition of Group (1) Closure, (2) Associativity, (3) Identity, (4) Inverse * Example : D3 point group
- 4. Group Representation Faithful representation ; group isomorphism
- 5. Reducible and Irreducible Representation Reducible : there exists a similarity transform which transforms the matrix into block-diagonal form. Thus, the representation is a direct sum of irreducible representations. Irreducible : there’s no such transform.
- 6. Example : D3 point group
- 7. Classes : set of elements conjugate to each other : A and B are conjugate to each other Why important ? – traces are all the same for the elements in the same class. Example : C3v point group
- 9. Character of the Class = Trace Example : D3 point group c.f. E representation Character table
- 10. Orthogonal Relations Dimensionality Theorem Caution : Both holds for irreducible group representations ! Characters form a vector space being orthogonal to each other.
- 11. Application of Orthogonal Relation