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A crash course on group theory
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Transcript of A crash course on group theory
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Crash Course onGroup Theory
12 May, 2014
Dongwoook Go
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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
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Definition of Group
(1) Closure, (2) Associativity, (3) Identity, (4) Inverse
* Example : D3 point group
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Group Representation
Faithful representation ; group isomorphism
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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.
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Example : D3 point group
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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
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Character of the Class = Trace
Example : D3 point group
c.f. E representation
Character table
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Orthogonal Relations
Dimensionality Theorem
Caution : Both holds for irreducible group representations !
Characters form a vector space being orthogonal to each other.
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Application of Orthogonal Relation
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