Double cosets

Definition

Double cosets are subsets formed in the context of group theory, where a group is partitioned by two subgroups. Specifically, given a group G and two subgroups H and K, the double coset of an element g in G with respect to H and K is the set of all elements of the form h*g*k for h in H and k in K. This concept plays a vital role in understanding the interactions between different groups through induction and restriction functors, as well as in the formulation and applications of Mackey's theorem.

5 Must Know Facts For Your Next Test

1. Double cosets can be denoted as H ackslash G / K, representing the structure where H and K act on G from different sides.
2. They provide a way to study how elements of a group can relate to two different subgroups simultaneously, facilitating deeper analysis in representation theory.
3. The number of distinct double cosets formed by G, H, and K can be finite or infinite, depending on the properties of the groups involved.
4. In Mackey's theorem, double cosets help characterize how irreducible representations can be constructed when dealing with restrictions and inductions between different groups.
5. Double cosets also play a crucial role in determining the decompositions of representations, especially when analyzing how they behave under subgroup actions.

Review Questions

• How do double cosets relate to the concepts of induction and restriction in representation theory?
  • Double cosets serve as an essential bridge between induction and restriction processes by allowing us to analyze how representations transform under the action of two different subgroups. When inducing a representation from a subgroup H to a group G while considering another subgroup K, double cosets provide insight into how elements interact across both subgroups. This relationship helps us understand which components of the representation remain invariant or change when viewed through the lenses of H and K.
• Discuss the implications of Mackey's theorem concerning double cosets and irreducible representations.
  • Mackey's theorem states that irreducible representations of a group can be decomposed into representations from its subgroups using double cosets. This theorem highlights that when analyzing how a representation behaves under the actions of subgroups, one can use double cosets to obtain information about how irreducible components are formed. It also emphasizes that the structure provided by double cosets is crucial for establishing correspondences between representations of different groups.
• Evaluate the importance of double cosets in the study of group actions and their applications in modern mathematics.
  • Double cosets are fundamentally important in understanding complex interactions within group actions, allowing mathematicians to dissect intricate relationships between groups and their substructures. They offer insights into symmetry operations, which are applicable across various fields such as physics, chemistry, and computer science. Moreover, their role in representation theory enhances our understanding of how groups represent transformations, contributing significantly to advancements in modern mathematical research and applications.

Related terms

Coset: A coset is a subset formed by multiplying all elements of a subgroup by a fixed element from the group, either on the left or right.

Induction:

A method in representation theory that constructs representations of a larger group from representations of a smaller subgroup.

Restriction:

The process of taking a representation of a group and limiting it to a subgroup, essentially 'restricting' the domain of the representation.