Character of a representation

Definition

The character of a representation is a function that assigns to each group element the trace of its corresponding linear transformation in a given representation. It provides an important way to study the properties of representations of groups, especially in the context of Lie groups and their associated Lie algebras, allowing one to relate different representations and their symmetries.

5 Must Know Facts For Your Next Test

1. The character of a representation is invariant under equivalence, meaning if two representations are equivalent, they will have the same character.
2. Characters are particularly useful in analyzing finite-dimensional representations since they can simplify the study of irreducibility and decompositions.
3. For compact Lie groups, characters can be used to classify irreducible representations and understand their structure.
4. The orthogonality relations for characters allow one to extract information about the structure of the group and its representations.
5. The character can often provide insights into physical systems, particularly in quantum mechanics where symmetry and group representations play crucial roles.

Review Questions

• How does the character of a representation help in understanding different representations of Lie groups?
  • The character of a representation allows mathematicians to compare different representations of a Lie group through the traces of their corresponding linear transformations. By analyzing these characters, one can determine whether two representations are equivalent or distinct, as equivalent representations will share the same character. This comparison simplifies the process of studying irreducibility and decomposition within representations.
• What role do characters play in the classification of irreducible representations of compact Lie groups?
  • Characters play a central role in classifying irreducible representations of compact Lie groups by providing critical information about their structure. The orthogonality relations for characters indicate how these irreducible representations interact with each other, allowing for an organized framework to categorize them. Essentially, characters serve as powerful tools to understand symmetries within compact Lie groups and their representations.
• Evaluate how understanding characters can impact fields like physics, particularly in quantum mechanics.
  • Understanding characters has significant implications in physics, especially in quantum mechanics where symmetry plays a vital role. Characters allow physicists to analyze how quantum systems behave under symmetry transformations, which can lead to conservation laws and selection rules. The ability to classify particles and interactions using characters contributes to a deeper understanding of fundamental physical principles, showcasing how mathematics intersects with theoretical physics.

Related terms

Lie Group: A Lie group is a group that is also a differentiable manifold, where the group operations are smooth. This structure allows for the application of calculus and algebraic techniques.

Lie Algebra: The Lie algebra is an algebraic structure associated with a Lie group, consisting of elements that can be thought of as infinitesimal transformations of the group. It captures the local behavior of the group near the identity.

Representation: A representation is a homomorphism from a group to the general linear group of a vector space, effectively turning group elements into linear transformations on that space.