2.1 Definition and examples of linear representations

Linear representations map group elements to linear transformations, preserving group structure. This powerful tool allows us to study abstract groups using familiar linear algebra techniques, bridging the gap between group theory and linear algebra.

In practice, we verify if a function is a linear representation by checking key properties. These include the homomorphism property, linearity of transformations, preservation of identity, and invertibility. These checks ensure the representation accurately captures the group's structure.

Fundamentals of Linear Representations

Linear representations of groups

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• Linear representation of a group maps group elements to linear transformations preserves group structure (homomorphism) (rotation groups, symmetry groups)
• Homomorphism from group G to general linear group GL(V) of vector space V preserves algebraic structure
• Each group element corresponds to linear transformation on V maintains group properties

Group elements and linear transformations

• Group elements correspond to linear maps $\rho(g): V \to V$ preserves vector space structure
• Identity element maps to identity transformation $\rho(e) = I_V$ leaves vectors unchanged
• Group operation corresponds to composition of linear maps $\rho(gh) = \rho(g) \circ \rho(h)$ maintains group structure
• Matrix representation expresses linear transformations as matrices in GL(n, F) for n-dimensional vector space over field F

Vector spaces for group representations

• Representation space V acts as domain for group action via linear transformations
• Characteristics include dimension (finite or infinite) and underlying field (real, complex, quaternions)
• Examples: $\mathbb{R}^n$ or $\mathbb{C}^n$ for finite-dimensional representations, function spaces for infinite-dimensional representations
• Basis choice affects matrix representation of group elements influences computations

Functions as linear group representations

• Trivial representation maps all group elements to identity transformation preserves structure trivially
• Regular representation acts on group algebra by left multiplication captures group structure faithfully
• Permutation representation describes group elements as permutations on a set (symmetric groups)
• Sign representation of symmetric group maps even permutations to 1, odd to -1 captures parity information

Verifying Linear Representations

Determine if a given function is a linear representation of a group

• Check homomorphism property $\rho(gh) = \rho(g)\rho(h)$ for all g, h in G preserves group operation
• Confirm linearity of transformations $\rho(g)(av + bw) = a\rho(g)(v) + b\rho(g)(w)$ for all g in G, v, w in V, and scalars a, b
• Verify preservation of identity $\rho(e) = I_V$ where e is the identity element of G
• Check invertibility of $\rho(g)$ for all g in G ensures bijective mapping
• Ensure dimension consistency all $\rho(g)$ map V to itself preserving vector space structure