Character

Definition

In the context of linear representations, the character of a representation is a function that associates each group element with the trace of its representing matrix. This concept is crucial in understanding the properties of representations, as it captures how the representation behaves under group actions, and it ties into key results such as Schur's lemma and the orthogonality relations, which further elucidate the structure and decomposition of representations.

5 Must Know Facts For Your Next Test

1. Characters are class functions, meaning they are constant on conjugacy classes of the group, reflecting the symmetry in group actions.
2. The character of a representation can be used to distinguish non-isomorphic representations; if two representations have different characters, they cannot be equivalent.
3. Orthogonality relations among characters state that characters of distinct irreducible representations are orthogonal with respect to a specific inner product on the space of functions defined on the group.
4. The character table summarizes important information about the group's irreducible representations, including degrees and orthogonality relations.
5. Characters play a key role in determining whether a representation is reducible or irreducible, providing insights into the structure of the representation.

Review Questions

• How do characters serve as a tool for distinguishing between different representations?
  • Characters act as powerful tools for distinguishing between representations because they provide a unique numerical signature for each representation. If two representations have different characters for even one group element, it indicates that these representations cannot be equivalent. This characteristic makes characters essential in classifying representations and understanding their relationships.
• Discuss how Schur's Lemma relates to the concept of characters in representation theory.
  • Schur's Lemma is closely tied to characters because it provides insights into the nature of homomorphisms between representations. It states that if there is a non-zero homomorphism between two irreducible representations, then it must be an isomorphism. Characters help identify when these homomorphisms exist by revealing information about the structure of the representations involved, emphasizing their distinct behaviors through their associated characters.
• Evaluate how orthogonality relations among characters enhance our understanding of group representations and their decompositions.
  • Orthogonality relations among characters enhance our understanding by establishing criteria for decomposing representations into irreducible components. These relations imply that distinct irreducible representations correspond to orthogonal characters, allowing us to employ inner product techniques to analyze and break down complex representations. This framework not only aids in classifying representations but also provides tools for computing decomposition coefficients, making it easier to understand the underlying structure of group actions.

Related terms

Representation: A homomorphism from a group to the general linear group of some vector space, illustrating how group elements can act as linear transformations.

Trace: The sum of the diagonal entries of a square matrix, which is significant in capturing certain properties of linear transformations in relation to characters.

Schur's Lemma: A fundamental result in representation theory that describes conditions under which certain linear maps between representations are either zero or isomorphisms.