Schur's Lemma

Definition

Schur's Lemma is a fundamental result in representation theory that characterizes the homomorphisms between irreducible representations of a group or algebra. It states that if two irreducible representations are equivalent, then any intertwining operator between them is either an isomorphism or zero, providing crucial insights into the structure of representations and their relationships.

5 Must Know Facts For Your Next Test

1. Schur's Lemma applies to both finite groups and Lie algebras, emphasizing its versatility in representation theory.
2. The lemma states that if a linear map commutes with the action of an irreducible representation, then it must be a scalar multiple of the identity operator.
3. In cases where the representations are not equivalent, Schur's Lemma asserts that any intertwining operator must be zero.
4. The applications of Schur's Lemma extend to proving the uniqueness of irreducible representations and character theory.
5. When analyzing tensor products of representations, Schur's Lemma aids in understanding how these products can be decomposed into irreducible components.

Review Questions

• How does Schur's Lemma relate to the concept of irreducibility in representations?
  • Schur's Lemma highlights the nature of irreducible representations by indicating that any linear map between two equivalent irreducible representations is either a non-zero scalar multiple of the identity operator or zero if they are not equivalent. This property reinforces the idea that irreducible representations cannot have non-trivial invariant subspaces, as any homomorphism must respect this structure. Thus, Schur's Lemma serves as a key tool in distinguishing between different types of representations based on their irreducibility.
• Discuss how Schur's Lemma contributes to our understanding of intertwining operators between representations.
  • Schur's Lemma provides a clear characterization of intertwining operators by stating that they can only exist as non-zero maps between equivalent irreducible representations. If the representations are not equivalent, the only intertwining operator possible is the zero map. This insight helps us understand how different representations interact with each other and allows for more profound implications in representation theory, especially when considering the structure and decomposition of various representations.
• Evaluate the implications of Schur's Lemma in relation to character theory and the classification of irreducible representations.
  • Schur's Lemma has significant implications for character theory as it helps establish relationships between characters of different irreducible representations. Since characters are associated with equivalence classes of representations, understanding how these characters relate through intertwining operators leads to powerful results in classifying and decomposing representations. Additionally, using Schur's Lemma allows for determining multiplicities in decompositions and aids in deriving orthogonality relations among characters, which are crucial for classifying all irreducible representations within a given context.

Related terms

Irreducible Representation: A representation that has no non-trivial invariant subspaces, meaning it cannot be decomposed into smaller representations.

Intertwining Operator: A linear transformation that maps one representation to another while respecting the group action, often used to analyze the relationship between different representations.

Character: A function that assigns a complex number to each group element, representing the trace of the corresponding group element's matrix in a given representation.