2.3 Equivalence and reducibility of representations

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.

Equivalent representations 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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• Equivalent representations describe two representations of a group related by similarity transformation preserving group structure and action on vector spaces
• Criteria for equivalence requires existence of invertible linear map between representation spaces preserving group action under the linear map
• Formal definition states representations $\rho_1: G \to GL(V_1)$ and $\rho_2: G \to GL(V_2)$ are equivalent if invertible linear map $T: V_1 \to V_2$ exists such that $T \circ \rho_1(g) = \rho_2(g) \circ T$ for all $g \in G$
  • 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

• Invariant subspace method identifies non-trivial proper subspaces preserved by group action (eigenspaces, fixed point sets)
• Character theory analyzes representation character comparing with known irreducible representation characters (trace of matrices)
• Schur's Lemma application tests for irreducibility by examining intertwining operators
• Projection operators construct and analyze projections onto invariant subspaces revealing reducibility structure (orthogonal projections)

Explain the process of decomposing a reducible representation into irreducible components

• Direct sum decomposition expresses representation space as direct sum of invariant subspaces (vector space decomposition)
• Isotypic decomposition groups isomorphic irreducible representations together (symmetry-based grouping)
• Methods for finding irreducible components include:
  1. Character projections
  2. Iterative subspace decomposition
• Uniqueness of decomposition determined by multiplicity of each irreducible component and isomorphism classes of irreducible representations (representation stability)

Discuss the significance of Maschke's Theorem in the context of reducibility

• Maschke's Theorem states every representation of finite group over field of characteristic zero is completely reducible (semisimplicity guarantee)
• Implications include guaranteed existence of decomposition into irreducible representations simplifying study of finite group representations
• Conditions for applicability limited to finite groups and fields of characteristic zero (complex numbers, real numbers)
• Proof outline involves averaging method and construction of complementary invariant subspaces
  • Relies on group finiteness and field characteristic to ensure well-defined averages