Irreducible Representation

Definition

An irreducible representation is a representation of a group that cannot be decomposed into smaller representations. This concept is vital in understanding how groups can act on vector spaces, and it plays a key role in character theory, which studies the properties of these representations through characters—functions that summarize the action of group elements.

5 Must Know Facts For Your Next Test

1. Irreducible representations can be used to classify representations of finite groups and help in understanding their structure.
2. The number of distinct irreducible representations of a finite group is equal to the number of conjugacy classes in that group.
3. Every finite-dimensional representation can be expressed as a direct sum of irreducible representations, known as Maschke's theorem for finite groups.
4. Characters of irreducible representations are orthogonal with respect to the inner product defined on the space of functions from the group to complex numbers.
5. In the context of compact Lie groups, irreducible representations are closely linked to the topology and geometry of the underlying manifold.

Review Questions

• How does an irreducible representation differ from reducible representations, and why is this distinction important?
  • An irreducible representation is one that cannot be decomposed into simpler representations, while reducible representations can be broken down into a direct sum of irreducible ones. This distinction is crucial because it allows us to classify and understand the structure of representations more effectively. Knowing whether a representation is irreducible helps in simplifying problems and finding solutions in representation theory.
• Discuss the significance of characters in relation to irreducible representations and how they contribute to character theory.
  • Characters serve as vital tools in understanding irreducible representations by summarizing their properties in a single function. The character of an irreducible representation captures essential information about its action on vector spaces, including how group elements behave under this action. In character theory, characters from different irreducible representations are used to establish orthogonality relations, which provide deep insights into the group's structure and its representations.
• Evaluate the implications of Maschke's theorem for finite groups concerning irreducible representations and representation rings.
  • Maschke's theorem asserts that every finite-dimensional representation of a finite group can be decomposed into irreducible components, highlighting the importance of these components in representation theory. This theorem has significant implications for representation rings because it ensures that these rings can be built using irreducible representations. This foundational result not only aids in classifying representations but also helps us understand how they interact within the algebraic framework provided by representation rings, ultimately enriching our understanding of group actions.

Related terms

Character: A character is a complex-valued function that assigns to each group element the trace of its representation matrix, capturing important information about the representation's structure.

Representation Ring: The representation ring is an algebraic structure that encodes information about all finite-dimensional representations of a group, allowing for the study of their direct sums and irreducible components.

Group Homomorphism: A group homomorphism is a function between two groups that preserves the group operation, helping to understand how groups can relate to one another through their representations.