2.3 Equivalence and reducibility of representations
2 min read•july 25, 2024
Representation equivalence and reducibility are key concepts in understanding how groups act on vector spaces. These ideas help us break down complex representations into simpler parts, making it easier to analyze group actions.
show the same group structure in different vector spaces. Reducible representations can be split into smaller pieces, while irreducible ones can't. These concepts are crucial for simplifying and studying group representations.
Representation Equivalence and Reducibility
Define equivalent representations and provide criteria for equivalence
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Adjoint Representation [The Physics Travel Guide] View original
Equivalent representations describe two representations of a group related by preserving group structure and action on vector spaces
Criteria for equivalence requires existence of between representation spaces preserving under the linear map
Formal definition states representations ρ1:G→GL(V1) and ρ2:G→GL(V2) are equivalent if invertible linear map T:V1→V2 exists such that T∘ρ1(g)=ρ2(g)∘T for all g∈G
Ensures group elements act consistently across both representations (rotation matrices, permutation matrices)
Explain the concept of reducible and irreducible representations
Reducible representations contain proper non-trivial invariant subspaces allowing decomposition into simpler representations (direct sum of smaller representations)
Irreducible representations lack proper non-trivial invariant subspaces preventing further decomposition (fundamental building blocks)
Importance in representation theory stems from irreducible representations serving as building blocks for understanding complex representations and fundamental objects of study
Analogous to prime numbers in number theory or simple groups in group theory
Describe methods for determining if a representation is reducible
method identifies non-trivial proper subspaces preserved by group action (eigenspaces, fixed point sets)
Character theory analyzes comparing with known characters (trace of matrices)