Character inner product

Definition

The character inner product is a mathematical tool used in the representation theory of compact Lie groups, which measures the 'overlap' between two characters of representations. Specifically, it is defined as the integral over the group of the product of two characters, normalized by the volume of the group. This concept reveals important relationships between different representations and plays a crucial role in understanding the structure of these groups and their representations.

5 Must Know Facts For Your Next Test

1. The character inner product is typically denoted as \( \langle \chi_1, \chi_2 \rangle \) for characters \( \chi_1 \) and \( \chi_2 \).
2. It can be computed using the formula: \( \langle \chi_1, \chi_2 \rangle = \frac{1}{|G|} \int_G \chi_1(g) \overline{\chi_2(g)} \, dg \), where \( |G| \) is the volume of the group.
3. The character inner product is particularly useful in determining orthogonality among characters, which helps identify irreducible representations.
4. For compact Lie groups, characters provide critical information about the representations, such as their dimensions and decompositions.
5. If two characters are from irreducible representations of a compact Lie group, their inner product will be zero unless they correspond to the same representation.

Review Questions

• How does the character inner product provide insight into the relationships between different representations of a compact Lie group?
  • The character inner product reveals how different representations relate to each other through their characters. When two characters have a non-zero inner product, it indicates that their corresponding representations are not entirely distinct; they share some common structure. Conversely, an inner product of zero implies that the representations are orthogonal, meaning they do not overlap in terms of their action on vector spaces. This understanding helps classify and analyze the representation theory of compact Lie groups.
• Discuss how the normalization factor in the character inner product influences its properties in representation theory.
  • The normalization factor in the character inner product ensures that it behaves consistently across different representations. By dividing by the volume of the group, this normalization allows for meaningful comparisons between characters from potentially different dimensions or groups. It helps establish orthogonality relations more cleanly and maintains homogeneity across various cases, which is crucial when analyzing irreducible representations and their interactions. This property is essential in building a cohesive framework for understanding representation theory.
• Evaluate how the character inner product contributes to the classification of irreducible representations of compact Lie groups.
  • The character inner product plays a pivotal role in classifying irreducible representations by establishing clear criteria for orthogonality among them. Since irreducible representations are fundamental components in building more complex representations, understanding their relationships through inner products aids in identifying which representations can be combined or decomposed into others. The property that if two characters have a non-zero inner product, they must correspond to the same representation provides a powerful tool for this classification process. Thus, it contributes significantly to understanding how representations fit together within the larger structure of compact Lie groups.

Related terms

Character: A character is a homomorphism from a group to the multiplicative group of complex numbers, representing the action of a group on a vector space.

Compact Lie Group: A compact Lie group is a group that is both a manifold and compact, meaning it is closed and bounded, with smooth structure allowing for the application of differential geometry.

Representation: A representation of a group is a way to express the group's elements as linear transformations on a vector space, providing insights into the structure and actions of the group.