2.2 Matrix representations and group algebras

Matrix representations and group algebras are powerful tools in representation theory. They allow us to study abstract group structures using concrete linear algebra. These concepts bridge the gap between group theory and linear transformations.

Group algebras extend the idea of matrix representations. They provide a unified framework for studying all representations of a group simultaneously. This algebraic approach reveals deep connections between group structure and representation properties.

Matrix Representations

Matrix notation for linear representations

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• Linear representations of groups map group elements to invertible linear transformations (homomorphism from G to GL(V))
• Matrix representation chooses basis for vector space V, represents group elements as matrices
• Properties preserve group structure: matrix multiplication corresponds to group operation, identity element becomes identity matrix
• Dimension of representation determined by size of matrices (2x2, 3x3)
• Examples: rotation matrices for cyclic groups, permutation matrices for symmetric groups, reflection matrices for dihedral groups

Group algebra construction

• Group algebra combines vector space over field F with group structure
• Basis elements correspond to group elements, allowing formal linear combinations $\sum_{g \in G} a_g g$ with $a_g \in F$
• Dimension equals order of group (number of elements)
• Multiplication defined by group operation, distributive over addition
• Examples: group algebra of cyclic group $C_3$ has basis $\{e, g, g^2\}$, symmetric group $S_3$ has 6-dimensional algebra

Group Algebras and Matrix Representations

Group algebra vs matrix representations

• Regular representation of group algebra acts on itself via left multiplication
• Matrix representations correspond to algebra homomorphisms from group algebra
• Group algebra elements expressed as linear combinations of representation matrices
• Maschke's theorem decomposes group algebra into simple modules (irreducible representations)
• Character theory connects traces of matrix representations to group algebra elements

Products in group algebra

• Multiplication rules based on group operation, linear in both arguments
• Distributive property: $(a + b)c = ac + bc$ and $a(b + c) = ab + ac$
• Associativity inherited from group operation $(ab)c = a(bc)$
• Computation steps:
  1. Expand linear combinations
  2. Apply group multiplication
  3. Collect like terms
• Examples: $(1 + g)(1 - g) = 1 - g + g - g^2$ in cyclic group algebra, $(s + r)(s - r) = s^2 - r^2$ in dihedral group algebra