12.2 Characters of Representations

Characters of representations are powerful tools in group theory. They simplify complex matrix calculations by encoding key information about representations into single numbers. By studying characters, we can analyze group structures and decompose representations without dealing with full matrices.

Character tables organize this information compactly, displaying characters for all irreducible representations. They reveal crucial properties of representations and enable efficient calculations. Understanding characters is essential for applying representation theory to various fields, from physics to chemistry.

Representation Character

Defining Character and Its Importance

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• Character of a representation assigns to each group element the trace of its corresponding matrix in the representation
• Provides concise encoding of representation information invariant under similarity transformations
• Plays crucial role in determining decomposition of reducible representations into irreducible components
• Simplifies many representation theory problems without needing full matrices
• Functions as class function remaining constant on conjugacy classes of the group
• Summarized in character table displaying characters of all irreducible representations

Character Calculations and Properties

• Calculated as trace of matrix ρ(g) for each group element g in representation ρ: G → GL(V)
• Always 1 for all group elements in trivial representation
• Equals |G| (group order) for identity element and 0 for others in regular representation
• Sum of individual representation characters for direct sum of representations
• Product of corresponding character values for tensor product of representations
• Complex conjugate of original representation's character for conjugate representation
• Calculated using character formula for induced representations involving sum over coset representatives

Character Values for Representations

Basic Representation Characters

• Trivial representation character always 1 for all group elements
• Regular representation character |G| for identity, 0 for others
• Character of direct sum equals sum of individual representation characters
• Tensor product representation character obtained by multiplying factor representation characters

Advanced Representation Characters

• Conjugate representation character complex conjugate of original
• Induced representation character calculated using formula with coset representative sum
• Character values for symmetric and alternating tensor powers derived from original representation character
• Adjoint representation character related to structure constants of associated Lie algebra

Character Tables for Analysis

Structure and Properties of Character Tables

• Display characters of all irreducible representations organized by conjugacy classes and irreducible representations
• First row corresponds to trivial representation, first column to identity element
• Number of irreducible representations (rows) equals number of conjugacy classes (columns)
• Dimensions of irreducible representations read directly as character value for identity element
• Verify orthogonality relations of characters

Applications of Character Tables

• Decompose reducible representations into irreducible components using inner product of characters
• Identify real, complex, and quaternionic representations based on table entries
• Determine character of tensor product representations by multiplying corresponding entries
• Calculate dimensions of fixed point subspaces under group action
• Analyze symmetry properties of molecular orbitals in chemical applications

Irreducible Representations and Orthogonality

Determining Irreducibility

• Representation irreducible if and only if its character satisfies orthogonality relation ⟨χ,χ⟩ = 1
• Inner product of characters defined as ⟨χ,ψ⟩ = (1/|G|) Σg∈G χ(g)ψ(g)* (|G| group order, * complex conjugation)
• Number of times irreducible representation appears in reducible representation determined by inner product of characters
• Sum of squared dimensions of all irreducible representations equals group order (|G| = Σi dim(ρi)²)

Orthogonality Relations and Applications

• Characters of irreducible representations ρ and σ satisfy ⟨χρ,χσ⟩ = δρσ (Kronecker delta)
• Reducible representation ρ decomposed as ρ = ⊕i miρi, (mi = ⟨χρ,χρi⟩, ρi irreducible representations)
• Second orthogonality relation: Σi χi(g)χi(h)* = |CG(g)|δgh (CG(g) centralizer of g in G)
• Used to verify character tables and determine conjugacy classes
• Helps in constructing projection operators onto irreducible subspaces