Understanding Induced Representations in Group Theory and Representation Theory

When a group G is a subgroup of another group H, the concept of an induced representation emerges as a powerful tool in representation theory. Induced representations allow us to transform representations of G into representations of H, facilitating a deeper understanding of how larger groups act on vector spaces. This article provides a detailed exploration of the definition and technicalities of induced representations, along with their relevance to Lie groups and practical examples.

Basics of Representation Theory and Group Theory

Before delving into induced representations, it's essential to familiarize oneself with the fundamental concepts of representation theory and group theory. A group is a set of elements together with an operation that combines any two of its elements to form a third element in such a way that the operation is associative, and that each element has an inverse within the set. The group operation also must have an identity element.

A representation of a group G is a homomorphism from G to the group of invertible linear transformations of a vector space. In simpler terms, it is a way to describe group elements as matrices acting on vector spaces. Understanding this requires a basic knowledge of Linear Algebra, including vector spaces and linear transformations.

Construction of Induced Representations

Given a subgroup G of a group H, an induced representation is an operation that takes a representation of G and extends it to a representation of H. The construction involves creating a larger vector space for H to act on, which is built from the vector space where G acts.

To construct an induced representation, consider a vector space V over which a group G acts. For each element g in G, there is a corresponding linear transformation T_g of V. If we want to extend this to a representation of H, we need to consider how H acts on a larger space U that contains V. The space U can be thought of as the space of all functions from H to V, with the action of H on U determined by its action on H.

Relevance to Lie Groups

Lie groups, named after Sophus Lie, are continuous groups that are also smooth manifolds. They are particularly important in physics, where rotational symmetries of space are often modeled using Lie groups. For instance, the group of rotations in three-dimensional space, denoted as SU(2), is a Lie group and is fundamental in quantum mechanics and particle physics.

The induced representation construction is especially relevant to Lie groups because it allows us to extend discrete representations (coming from smaller subgroups) to continuous symmetries (characterizing larger groups). This is crucial in theoretical physics, where symmetry principles play a central role.

Practical Examples

To better understand the induced representation, consider an exercise involving finite groups. Let G be the Symmetric Group on Two Elements, which is a non-trivial one-dimensional representation. Inducing this representation to the Symmetric Group on Three Elements can be a useful exercise in grasping the definition and mechanism of induced representations.

Another example involves rotations in three-dimensional space. The SU(2) group, which is a Lie group, can be used to construct induced representations that extend from discrete rotations to continuous rotations. This is particularly important in the context of spin in quantum mechanics, where the SU(2) group is used to describe the spin of fermions.

Induction vs. Restriction

In representation theory, there is a duality between induction and restriction. Restriction is the process of turning a representation of H (the larger group) into a representation of G (the subgroup) by ignoring the elements of H that are not in G. This process is more straightforward and can be performed explicitly.

Induction, however, is more abstract and involves constructing a larger vector space and defining an action of H on this space. Despite its complexity, induction is a powerful tool that allows us to relate representations of different groups, making it an essential concept in the study of Lie groups and representation theory.

Conclusion

The concept of induced representations is a cornerstone of representation theory, providing a bridge between representations of smaller and larger groups. Understanding induced representations requires a solid foundation in group theory and linear algebra. By exploring the technicalities and practical applications, one can gain insight into the rich structure of Lie groups and their relevance in various fields, including physics and mathematics.