Decomposition

Definition

Decomposition refers to the process of breaking down a complex mathematical object into simpler, more manageable components. In the context of representation theory, this often involves expressing a representation as a direct sum of irreducible representations, which helps in understanding the structure and behavior of the representation in a clearer way.

5 Must Know Facts For Your Next Test

1. Decomposition is essential for understanding representations, as it reveals the building blocks of more complex structures.
2. In representation theory, every finite-dimensional representation over a field can be decomposed into irreducible components, making the study of these irreducible representations crucial.
3. The characters of irreducible representations play a key role in determining how a representation can be decomposed and analyzed.
4. Decomposition often utilizes tools like Schur's lemma, which provides conditions under which certain maps between representations vanish.
5. The process of decomposition can lead to important results like the complete reducibility theorem, which states that under certain conditions, every representation can be decomposed into a direct sum of irreducible representations.

Review Questions

• How does decomposition assist in understanding the structure of representations in representation theory?
  • Decomposition allows us to break down complex representations into simpler, irreducible components, which makes it easier to analyze their properties. By studying these irreducible representations, we can gain insights into how they combine to form larger structures. This simplification is critical because it reduces the complexity and helps reveal underlying patterns within the representation.
• Discuss how characters are used in the decomposition of representations and their significance in representation theory.
  • Characters serve as an important tool in the decomposition of representations by providing a way to classify and distinguish between different irreducible representations. Since the character is invariant under conjugation, it acts as a powerful means of analysis. In many cases, the orthogonality relations of characters can be used to determine how a given representation decomposes into irreducible parts, highlighting their role as essential functions in representation theory.
• Evaluate the implications of Schur's lemma on the process of decomposition in representation theory and provide examples.
  • Schur's lemma has significant implications for decomposition as it establishes criteria under which morphisms between representations can vanish. This means that if you have an irreducible representation and any inter-representation map is non-zero, it must be an isomorphism. This insight helps identify when decompositions yield unique components or when additional analysis is needed. For example, using Schur's lemma, one can conclude that if a non-zero endomorphism exists between two distinct irreducible representations, they must actually be equivalent, simplifying the overall structure in which these representations exist.

Related terms

Irreducible Representation:

A representation that cannot be expressed as a direct sum of two non-trivial representations.

Character:

A function that assigns to each group element a complex number reflecting the trace of its corresponding matrix in a representation.

Direct Sum:

A construction that combines multiple mathematical objects (like vector spaces or representations) into a new object that retains the structure of each component.