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 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 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
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)