<.,.>

Definition

<.,.> denotes the inner product or inner pairing of characters in the context of representation theory. This mathematical notation is used to define the orthogonality relations for characters, indicating how distinct irreducible representations relate to each other through their characters. The significance of this term is crucial for understanding how characters function as tools for analyzing representations and their properties, particularly in the context of group theory.

5 Must Know Facts For Your Next Test

1. <.,.> is utilized to calculate the inner product between characters of irreducible representations, leading to important results about their relationships.
2. The orthogonality relations state that if $ ho$ and $ au$ are distinct irreducible representations, then <$ ho$, $ au$> = 0.
3. For a given irreducible representation $ ho$, <$ ho$, $ ho$> equals the order of the group divided by the dimension of the representation.
4. <.,.> allows us to understand how different representations can be distinguished based on their characters, which simplifies many calculations in representation theory.
5. The notation <.,.> is not only essential in finite group representation theory but also finds applications in harmonic analysis and quantum mechanics.

Review Questions

• How does the notation <.,.> help in determining the relationships between different characters of irreducible representations?
  • <.,.> serves as a mathematical tool to compute the inner product of characters from different irreducible representations. By calculating these inner products, we can identify whether two representations are distinct or equivalent. If <$ ho$, $ au$> = 0 for two different irreducible representations $ ho$ and $ au$, it confirms that they are orthogonal, meaning they do not share common features in their structure.
• Discuss the implications of the orthogonality relations expressed through <.,.> for the study of finite groups.
  • The orthogonality relations illustrated by <.,.> indicate that the characters of distinct irreducible representations form an orthogonal basis in the space of class functions on the group. This means that we can decompose functions into components associated with different representations. As a result, this property greatly simplifies many computations and provides insights into how groups can be represented using their character tables, leading to deeper understanding in group theory.
• Evaluate how <.,.> contributes to broader applications in mathematics and physics beyond just representation theory.
  • <.,.> not only facilitates calculations within representation theory but also extends its reach into areas like harmonic analysis and quantum mechanics. The concept of orthogonality plays a critical role in simplifying problems in these fields, such as Fourier analysis, where functions can be expressed as sums of orthogonal basis functions. Moreover, in quantum mechanics, <.,.> allows for determining states' probabilities and overlaps, showcasing its fundamental importance across various mathematical and physical contexts.

Related terms

Characters:

Functions that assign a complex number to each group element, providing a way to analyze representations of a group.

Orthogonality Relations:

Mathematical statements that describe the conditions under which characters of different irreducible representations are orthogonal to each other.

Irreducible Representation:

A representation that cannot be decomposed into smaller representations, providing fundamental building blocks for understanding group actions.