Representation dimensions

Definition

Representation dimensions refer to the size of the matrices that represent the linear transformations associated with a particular group or algebra in representation theory. These dimensions provide insight into how many independent ways a group can act on a vector space, highlighting the complexity of the group's structure and the nature of its representations.

5 Must Know Facts For Your Next Test

1. The representation dimension of a finite group is equal to the number of distinct irreducible representations it has over the complex numbers.
2. In cases where groups have complex representations, the dimensions can be interpreted as providing insight into how those groups can be represented in higher-dimensional spaces.
3. Understanding representation dimensions helps in classifying representations, leading to results such as Schur's lemma and the complete reducibility of representations for semisimple groups.
4. The orthogonality relations among characters are connected to representation dimensions, as they imply that the sum of squares of the dimensions of all irreducible representations equals the order of the group.
5. Representation dimensions also play a crucial role in applications to physics and chemistry, where they help in analyzing symmetries and transformations in various systems.

Review Questions

• How do representation dimensions relate to irreducible representations and their significance within representation theory?
  • Representation dimensions give an important measure of how many distinct irreducible representations exist for a group. Each irreducible representation corresponds to a unique way in which the group can act on a vector space. By examining these dimensions, we can understand not only how the group is structured but also how it can be decomposed into simpler components, which is essential for further analysis in representation theory.
• Discuss how orthogonality relations among characters impact the calculation of representation dimensions for a given group.
  • Orthogonality relations among characters provide crucial tools for calculating representation dimensions by linking them directly to the group's order. Specifically, these relations state that when summed over all characters, the squares of their dimensions equal the order of the group. This relationship implies that understanding one part of this equation can lead to insights about others, allowing us to derive dimensions based on known character values.
• Evaluate the implications of representation dimensions on practical applications in fields like physics and chemistry, particularly concerning symmetry analysis.
  • Representation dimensions have profound implications in fields such as physics and chemistry because they help quantify how systems exhibit symmetry through their transformations. For instance, knowing the representation dimensions allows scientists to predict behaviors under various symmetry operations, leading to insights into molecular structures or particle interactions. As such, understanding these dimensions not only advances theoretical knowledge but also enhances practical applications in technology and materials science.

Related terms

character: A character is a function that assigns to each group element the trace of its representing matrix in a given representation, offering a way to study representations through their traces.

irreducible representation: An irreducible representation is a representation that has no proper invariant subspaces, meaning it cannot be decomposed into smaller representations, indicating the simplest form of representation for a group.

dimension formula: The dimension formula relates the representation dimensions of a group to its subgroup structure, providing a way to calculate or estimate the dimensions based on the properties of subgroups.