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 ∑ g ∈ G a g g \sum_{g \in G} a_g g ∑ g ∈ G a g g with a g ∈ F a_g \in F a g ∈ 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 C_3 C 3 has basis { e , g , g 2 } \{e, g, g^2\} { e , g , g 2 } , symmetric group S 3 S_3 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 = a c + b c (a + b)c = ac + bc ( a + b ) c = a c + b c and a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c
Associativity inherited from group operation ( a b ) c = a ( b c ) (ab)c = a(bc) ( ab ) c = a ( b c )
Computation steps:
Expand linear combinations
Apply group multiplication
Collect like terms
Examples: ( 1 + g ) ( 1 − g ) = 1 − g + g − g 2 (1 + g)(1 - g) = 1 - g + g - g^2 ( 1 + g ) ( 1 − g ) = 1 − g + g − g 2 in cyclic group algebra, ( s + r ) ( s − r ) = s 2 − r 2 (s + r)(s - r) = s^2 - r^2 ( s + r ) ( s − r ) = s 2 − r 2 in dihedral group algebra