Tensor product of representations

Definition

The tensor product of representations is an operation that combines two representations of a group into a new representation, allowing for the construction of more complex representations from simpler ones. This concept plays a crucial role in representation theory as it helps to understand how representations can interact and generate new structures, and it provides a method to analyze and classify representations of groups.

5 Must Know Facts For Your Next Test

1. The tensor product of two representations is denoted as $V \otimes W$, where $V$ and $W$ are vector spaces associated with each representation.
2. For finite-dimensional representations, the dimension of the tensor product is given by the product of the dimensions of the two individual representations: $\text{dim}(V \otimes W) = \text{dim}(V) \times \text{dim}(W)$.
3. The tensor product preserves certain properties; for example, if $V$ and $W$ are both irreducible representations, then their tensor product may not be irreducible.
4. The action of a group on the tensor product is defined in such a way that for any group element $g$, the representation acts on $V \otimes W$ as $(g \cdot (v \otimes w)) = (g \cdot v) \otimes (g \cdot w)$ for $v \in V$ and $w \in W$.
5. Tensor products can be used to construct representations of larger groups from smaller ones, making them essential for understanding the structure and classification of representations.

Review Questions

• How does the tensor product of two representations interact with their underlying vector spaces?
  • The tensor product combines two representations by taking the Cartesian product of their underlying vector spaces, resulting in a new vector space that embodies both original representations. Specifically, if you have representations $V$ and $W$, their tensor product $V \otimes W$ creates vectors that are linear combinations of pairs $(v,w)$ with $v \in V$ and $w \in W$. This operation captures interactions between the two representations, leading to richer structure and new elements that reflect their combined actions.
• What are some key properties preserved under the tensor product operation for representations?
  • One key property preserved under the tensor product operation is dimensionality; specifically, if $V$ has dimension $m$ and $W$ has dimension $n$, then the dimension of their tensor product $V \otimes W$ will be $m \cdot n$. Additionally, while the irreducibility may not always be preserved—meaning that even if both $V$ and $W$ are irreducible, $V \otimes W$ might not be—certain other structural features like the action of the group on the resultant space remain intact, maintaining consistency in how these representations behave under group actions.
• Evaluate how the tensor product of representations contributes to the overall understanding and classification of group representations.
  • The tensor product of representations significantly contributes to understanding and classifying group representations by enabling mathematicians to construct new representations from known ones. By exploring how simpler, irreducible representations combine through this operation, one can derive more complex structures that reveal insights into symmetries and invariants within group actions. This process not only enriches the field but also aids in organizing all possible representations into an overarching framework that categorizes them based on their interactions and derived properties.

Related terms

representation: A representation is a homomorphism from a group to the general linear group of a vector space, translating group elements into linear transformations.

direct sum of representations: The direct sum of representations combines multiple representations into a single representation where each original representation acts independently.

module:

A module is a generalization of vector spaces where scalars come from a ring instead of a field, allowing for the study of representations in a broader algebraic context.