Crash Course onGroup Theory

12 May, 2014

Dongwoook Go

References

• George B. Arfken, Hans J. Weber, Frank E. Harris, Mathematical Methods for Physicists, 7th ed, 2013 Elsevier

• Wikipedia

• ChemWikihttp://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Group_Theory%3A_Theory

• L. E. Laverman’s lecture note – Introduction to the Chemical Applications of Group Theory

Definition of Group

(1) Closure, (2) Associativity, (3) Identity, (4) Inverse

* Example : D3 point group

Group Representation

Faithful representation ; group isomorphism

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.

Example : D3 point group

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

Character of the Class = Trace

Example : D3 point group

c.f. E representation

Character table

Orthogonal Relations

Dimensionality Theorem

Caution : Both holds for irreducible group representations !

Characters form a vector space being orthogonal to each other.

Application of Orthogonal Relation