Dimension of a representation

Definition

The dimension of a representation refers to the size of the vector space associated with a representation of a group. It indicates the number of basis elements needed to describe the representation, and is crucial in understanding how the group acts on the space. This dimension connects to matrix representations, where the dimension corresponds to the number of rows or columns in the matrices, and plays a significant role in character theory by influencing the characters’ values.

5 Must Know Facts For Your Next Test

1. The dimension of a representation can be determined by counting the number of independent vectors in the vector space associated with that representation.
2. For finite groups, the sum of the squares of the dimensions of all irreducible representations equals the order of the group.
3. The dimension directly affects the structure and properties of characters; higher-dimensional representations may yield more complex character tables.
4. In practical applications, the dimension often reflects physical systems, where each dimension corresponds to a different state or degree of freedom.
5. Understanding dimensions helps in classifying representations and identifying equivalences between different representations of the same group.

Review Questions

• How does the dimension of a representation influence its structure and properties?
  • The dimension of a representation greatly influences its structure as it determines how many independent states or vectors are available in the vector space. A higher dimension allows for richer structures, potentially leading to more complex characters and interactions. Additionally, dimensions play a key role in decomposing representations into irreducible components, which helps in classifying them and understanding their relationships with other representations.
• What is the relationship between the dimensions of irreducible representations and the order of a group?
  • There is an important relationship between the dimensions of irreducible representations and the order of a group, expressed by the formula that states that the sum of squares of the dimensions of all irreducible representations equals the order of that group. This means if you know the dimensions of all irreducible representations, you can compute crucial properties about the group itself. This relationship underscores how closely linked representation theory is to group structure.
• Evaluate how understanding the dimension of representations can facilitate applications in fields like physics or chemistry.
  • Understanding the dimension of representations can greatly facilitate applications in fields such as physics and chemistry by providing insights into system behaviors based on their state spaces. For example, in quantum mechanics, each state may correspond to a different dimension representing various energy levels or particle states. By grasping how these dimensions interact through group actions, scientists can predict outcomes and behaviors of physical systems, making this knowledge invaluable for research and development in these areas.

Related terms

Representation:

A representation is a homomorphism from a group to the group of invertible matrices, illustrating how groups can act on vector spaces.

Character:

A character is a function that assigns a complex number to each group element in a representation, reflecting the trace of the associated matrix.

Irreducible representation: An irreducible representation is one that has no proper invariant subspaces, meaning it cannot be decomposed into smaller representations.