Matrix coefficients

Definition

Matrix coefficients are complex-valued functions that arise in the study of representations of groups, specifically in the context of unitary representations. They serve as the building blocks that relate the abstract group elements to linear transformations in a vector space, allowing for a deeper understanding of how groups act on these spaces through linear algebra.

5 Must Know Facts For Your Next Test

1. Matrix coefficients are often denoted as $ ho(g)_{ij}$, where $g$ is an element of the group and $i,j$ index the basis elements of the representation.
2. They provide essential information about the structure of representations, including their irreducibility and decomposition properties.
3. Matrix coefficients can be used to define positive definite functions on groups, which play a crucial role in harmonic analysis.
4. The Peter-Weyl theorem states that matrix coefficients can be decomposed into finite sums involving characters of the group, linking representation theory with harmonic analysis.
5. In quantum mechanics, matrix coefficients are significant as they relate to transition amplitudes between quantum states in unitary representations.

Review Questions

• How do matrix coefficients relate to unitary representations and their significance in understanding group actions?
  • Matrix coefficients connect unitary representations with how groups act on vector spaces by defining linear transformations corresponding to each group element. They capture how basis elements transform under the action of the group, providing insight into the structure and behavior of the representation. Understanding these coefficients helps analyze symmetries and facilitates connections with other areas such as harmonic analysis.
• Discuss the role of matrix coefficients in the context of the Peter-Weyl theorem and their implications for harmonic analysis.
  • Matrix coefficients play a central role in the Peter-Weyl theorem, which states that every continuous unitary representation can be decomposed into finite-dimensional irreducible representations. The matrix coefficients associated with these representations allow us to express functions on compact groups as series involving these irreducible components. This decomposition not only clarifies the structure of representations but also connects them to harmonic analysis through the concept of orthogonal function systems on groups.
• Evaluate how matrix coefficients contribute to understanding transition amplitudes in quantum mechanics within the framework of representation theory.
  • Matrix coefficients are crucial in quantum mechanics as they correspond to transition amplitudes between different quantum states. Within representation theory, these coefficients reflect how physical systems evolve under symmetry operations represented by unitary operators. This relationship allows physicists to utilize mathematical tools from representation theory to analyze and predict outcomes in quantum systems, highlighting the interplay between abstract mathematics and physical reality.

Related terms

Unitary Representation: A representation of a group where each group element is associated with a unitary operator on a Hilbert space, preserving inner products.

Group Algebra: An algebraic structure that combines the elements of a group with coefficients from a field, often used to study representations of groups.

Irreducible Representation: A representation that has no non-trivial invariant subspaces under the action of the group, meaning it cannot be decomposed into smaller representations.