Matrices and representations

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Matrices and representations
MT 4.3.1, Ogden 2.1-2.3 Office hours are changed
to MWF, 2pm.
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Matrix notations for geometric transformations

• Any point in space column matrix
• The identity
• Inversion

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Reflections

• If a plane of reflection is chosen to coincide
  with a principal Cartesian plane (i.e. an xy, xz,
  or yz plane), reflection of a general point has
  the effect of changing the sign of the coordinate
  measured - to the plane while leaving unchanged
  the two coordinates whose axes define the plane.

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Proper rotations
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Improper rotations
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Multiple operations
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Reducible and irreducible representations

• MT 4.3.2 - 4.3.3, Ogden 2.4 2.6, 3

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Outline

• Representations
• Irreducible
• Reducible
• Character tables
• Reduction of reducible representations
• Inspection method
• Systematic method

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The concept of a representation
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Definition of a representation

• A representation of a group may be defined as a
  set of matrices, each corresponding to a single
  operation in the group, that can be combined
  among themselves similarly to how the group
  elements (symmetry operations) combine.
• The dimension of a representation is the order of
  the square matrices that constitute it.

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Representations in C2v
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