Regular Representation

Definition

The regular representation is a way of representing a group as linear transformations on a vector space, specifically associated with the group acting on itself by left multiplication. This representation captures the structure of the group and is particularly significant in the study of compact Lie groups, as it provides a concrete way to understand how these groups can act on vector spaces and relate to their representations in terms of matrices.

5 Must Know Facts For Your Next Test

1. The regular representation is always finite-dimensional if the group is finite, and its dimension equals the order of the group.
2. In the case of compact Lie groups, the regular representation decomposes into irreducible representations, allowing for deeper insights into their structure.
3. The regular representation can be constructed using the group's elements acting on functions defined on the group itself, leading to an action on L² spaces.
4. For compact groups, the regular representation is unitary, meaning it preserves inner products and provides a geometric interpretation of the group's structure.
5. The concept of regular representation plays a vital role in harmonic analysis on groups, linking representation theory with analysis.

Review Questions

• How does the regular representation illustrate the relationship between group actions and linear transformations?
  • The regular representation showcases this relationship by demonstrating how each element of a group can be represented as a linear transformation acting on a vector space. Specifically, in this context, the group acts on itself through left multiplication, which translates into matrix representations. This framework allows us to study various properties of groups and their representations in a more tangible manner, particularly emphasizing the algebraic structure inherent within the group's actions.
• Discuss the significance of irreducibility in the context of the regular representation for compact Lie groups.
  • Irreducibility plays an essential role when examining the regular representation of compact Lie groups because it helps determine how these representations can be decomposed into simpler components. The regular representation itself can be expressed as a direct sum of irreducible representations. Understanding this decomposition not only reveals valuable insights into the structure of the compact Lie group but also shows how its elements interact within different dimensions, which is critical for applications in both mathematics and physics.
• Evaluate how the regular representation contributes to harmonic analysis on compact Lie groups and its broader implications.
  • The regular representation significantly contributes to harmonic analysis on compact Lie groups by establishing a framework for understanding functions on these groups through their representations. This connection enables techniques such as Fourier analysis to be applied, resulting in powerful tools for decomposing functions into series involving characters of irreducible representations. The broader implications include applications in theoretical physics, such as quantum mechanics and symmetry analysis, where understanding these representations can lead to breakthroughs in modeling physical systems and phenomena.

Related terms

Compact Lie Group:

A compact Lie group is a group that is both a Lie group and compact as a topological space, meaning it is closed and bounded.

Irreducible Representation:

An irreducible representation is a representation of a group that has no non-trivial invariant subspaces under the action of the group.

Character of a Representation:

The character of a representation is a function that assigns to each group element the trace of its corresponding matrix in the representation, providing important information about the representation's properties.