Character of a Representation

Definition

The character of a representation is a homomorphism from a group (or algebra) to the complex numbers, capturing essential features of how a group acts on a vector space. This function helps in understanding the representation's structure and allows for the classification of representations, especially when considering irreducible representations and their relationships through concepts like Schur's Lemma and characters of finite-dimensional representations.

5 Must Know Facts For Your Next Test

1. The character of a representation provides crucial information about the representation's structure and can be used to distinguish between different representations.
2. Characters are additive in nature, meaning that the character of a direct sum of representations is the sum of the characters of each representation.
3. For finite-dimensional representations, characters can be calculated using traces of the corresponding linear transformations.
4. Characters are constant on conjugacy classes, which simplifies computations in group theory since you only need to calculate them for representative elements.
5. The character table of a group summarizes all the characters of its irreducible representations and is a powerful tool in the study of group representations.

Review Questions

• How does the character of a representation relate to Schur's Lemma and why is this connection important?
  • The character of a representation plays a significant role in understanding irreducible representations, which is where Schur's Lemma comes into play. Schur's Lemma states that any homomorphism between irreducible representations is either zero or an isomorphism. This means that characters can help identify when two representations are equivalent or distinct by examining their values, making it easier to classify and understand their relationships.
• In what ways do characters facilitate the study of finite-dimensional representations?
  • Characters simplify the study of finite-dimensional representations by allowing for easy computation through traces, which capture essential information about linear transformations. Since characters are constant on conjugacy classes, researchers only need to compute values for representatives of these classes, significantly reducing computational complexity. This enables scholars to create character tables that summarize critical features of the group's representations and aid in distinguishing different representations.
• Evaluate how the concept of characters impacts the classification and understanding of various representations in algebra.
  • The concept of characters greatly enhances the classification and understanding of various representations by providing a systematic way to analyze their properties. Characters allow mathematicians to determine when two representations are equivalent and facilitate comparisons among them through their character tables. Furthermore, since characters reveal symmetries and structural relationships within the algebraic systems being studied, they serve as essential tools for deeper insights into representation theory, leading to advancements in both pure mathematics and applications in physics.

Related terms

Representation:

A representation is a way to express elements of an algebraic structure (like a group or Lie algebra) as linear transformations on a vector space, preserving the algebraic operations.

Irreducible Representation:

An irreducible representation is one that has no proper invariant subspaces, meaning it cannot be decomposed into smaller representations.

Schur's Lemma:

Schur's Lemma states that if two irreducible representations of a group are equivalent, then any intertwining operator between them is either zero or an isomorphism.