Reducible Representations

Definition

Reducible representations are those representations of a group that can be expressed as a direct sum of two or more non-trivial invariant subspaces. This means that the representation can be decomposed into simpler components, making them easier to analyze. Understanding reducible representations is crucial in studying more complex structures like irreducible representations, which cannot be broken down further.

5 Must Know Facts For Your Next Test

1. Reducible representations can often be identified through the existence of invariant subspaces under the action of the group.
2. The decomposition of a reducible representation helps in understanding its structure and simplifying computations related to it.
3. The process of determining whether a representation is reducible involves analyzing the action of group elements on vectors in the vector space.
4. Not all representations are reducible; irreducible representations represent the simplest form of group action on a vector space.
5. In the context of the GNS construction, understanding reducible representations aids in constructing Hilbert spaces that reflect the symmetry properties of quantum states.

Review Questions

• How does the concept of invariant subspaces relate to reducible representations?
  • Invariant subspaces are key to understanding reducible representations because these are the subspaces that remain unchanged under the group action. A representation is deemed reducible if there exists at least one non-trivial invariant subspace. Identifying these invariant subspaces allows us to decompose the representation into simpler parts, which ultimately aids in analyzing its structure.
• Discuss how the decomposition of reducible representations can impact the study of more complex systems in functional analysis.
  • The decomposition of reducible representations into simpler components allows mathematicians and physicists to better understand and manipulate complex systems. By breaking down a reducible representation into its irreducible parts, one can study the behavior and interactions within those components individually. This process enhances the ability to solve problems related to symmetry and linear transformations in functional analysis, making it easier to apply these concepts to various fields such as quantum mechanics.
• Evaluate the role of reducible representations within the GNS construction framework and their implications for quantum mechanics.
  • In the GNS construction framework, reducible representations play a significant role as they help in generating Hilbert spaces that reflect underlying symmetries associated with quantum states. The ability to identify and decompose reducible representations into irreducible components means that one can construct spaces where quantum observables can be effectively studied. This decomposition leads to a clearer understanding of how different quantum states relate to one another and provides insights into the physical implications of symmetries in quantum mechanics.

Related terms

Irreducible Representation: An irreducible representation is a representation that has no proper non-trivial invariant subspaces, meaning it cannot be decomposed into simpler parts.

Group Homomorphism: A group homomorphism is a structure-preserving map between two groups that respects the operation of the groups, often used in the study of representations.

Invariant Subspace: An invariant subspace is a subspace that remains unchanged when a linear transformation (from a representation) is applied to it.