Linear representation

Definition

A linear representation is a homomorphism from a group to the general linear group of a vector space, which allows elements of the group to be expressed as linear transformations on the vector space. This concept connects algebraic structures with geometric representations, showing how groups can act on vector spaces through matrices. Understanding linear representations helps bridge the gap between abstract algebra and geometry, enabling deeper insights into both fields.

5 Must Know Facts For Your Next Test

1. Linear representations provide a way to study groups by analyzing their actions on vector spaces, translating abstract group elements into concrete matrix operations.
2. Every finite-dimensional representation can be expressed as a direct sum of irreducible representations, helping to break down complex structures into simpler components.
3. The kernel of a linear representation is the set of group elements that act trivially on the vector space, which can provide insights into the structure of the group.
4. Schur's lemma is a key result in representation theory that characterizes when two irreducible representations are equivalent or when a linear operator commutes with all operators in an irreducible representation.
5. The character of a representation is a function that assigns to each group element the trace of its corresponding matrix, which helps to study and classify representations through their traces.

Review Questions

• How does the concept of linear representation connect groups with vector spaces, and what role do homomorphisms play in this relationship?
  • Linear representations serve as a bridge between groups and vector spaces by associating group elements with linear transformations. A homomorphism establishes this connection by mapping each element of the group to an invertible matrix in the general linear group, showing how the group's structure can influence geometric actions in vector spaces. This mapping allows for the analysis of group behavior through matrix properties, enhancing our understanding of both algebra and geometry.
• What is Schur's lemma, and why is it significant in the study of linear representations?
  • Schur's lemma states that if two irreducible representations are equivalent, any linear operator that commutes with all operators from these representations must be a scalar multiple of the identity operator. This result is crucial because it provides insight into the structure and relationships between different representations. It simplifies the study of representations by indicating how irreducibility constrains the types of transformations that can exist within those representations.
• Analyze the importance of characters in understanding linear representations and their classification within representation theory.
  • Characters play a vital role in representation theory by providing a way to classify and analyze linear representations through their traces. By associating each group element with its character, we gain valuable information about the representation's structure and can identify equivalences between different representations. The use of characters simplifies many aspects of representation theory, allowing for powerful tools such as orthogonality relations and character tables that facilitate deeper exploration into the nature of groups and their actions.

Related terms

homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups or rings, that respects their operations.

general linear group: The general linear group is the group of invertible matrices over a field, which represents all possible linear transformations of a finite-dimensional vector space.

vector space: A vector space is a collection of vectors, which can be added together and multiplied by scalars, adhering to specific rules that allow for linear combinations.