The is a game-changer for understanding compact groups. It shows that irreducible representations form a basis for square-integrable functions, connecting group structure to function spaces. This links abstract algebra to analysis in a beautiful way.
This theorem is crucial for harmonic analysis on groups. It generalizes to compact groups, allowing us to decompose functions into simpler components. This powerful tool opens up new ways to study group actions and symmetries.
Compact Groups and Representations
Compact Groups and Their Properties
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Compact groups are topological groups that are compact as a topological space
implies several important properties such as being Hausdorff, second-countable, and locally compact
Examples of compact groups include the circle group S1, the torus Tn, and the special unitary group SU(n)
Compact groups have a unique (up to scaling) which allows for integration on the group
Unitary Representations of Compact Groups
A of a G is a continuous homomorphism ρ:G→U(H), where U(H) is the group of unitary operators on a H
Unitary representations preserve the inner product structure of the Hilbert space
The dimension of the Hilbert space H is called the dimension of the representation
Examples of unitary representations include the trivial representation, the regular representation, and the irreducible representations
Irreducible Representations and the Group Algebra
An is a unitary representation that has no non-trivial invariant subspaces
Irreducible representations are the building blocks of all unitary representations via the decomposition
The L2(G) is the space of square-integrable functions on the group G with the convolution product
The group algebra can be decomposed into a direct sum of irreducible representations, each appearing with a multiplicity equal to its dimension
Peter-Weyl Theorem and Fourier Series
The Peter-Weyl Theorem
The Peter-Weyl theorem states that the of the irreducible unitary representations form an for the space L2(G)
This theorem provides a natural generalization of the classical Fourier series to compact groups
The matrix coefficients are given by dπ⟨π(g)ei,ej⟩, where π is an irreducible representation, dπ is its dimension, and {ei} is an orthonormal basis for the representation space
Fourier Series on Compact Groups
The Fourier series of a function f∈L2(G) is given by f(g)=∑πdπ∑i,jf^(π)ijdπ⟨π(g)ei,ej⟩
The Fourier coefficients f^(π)ij are given by the inner product ⟨f,dπ⟨π(⋅)ei,ej⟩⟩L2(G)
The Fourier series converges to f in the L2 norm, and under additional regularity assumptions, it converges pointwise
Character Theory and Its Applications
The character of a representation π is the function χπ(g)=Tr(π(g)), where Tr denotes the trace
Characters are conjugation invariant and satisfy the relations ⟨χπ,χσ⟩L2(G)=δπσ
The of a compact group encodes important information about its representations and can be used to decompose the group algebra and compute multiplicities
Orthogonality and Irreducibility
Orthogonality Relations for Matrix Coefficients
The matrix coefficients of irreducible unitary representations satisfy the orthogonality relations ∫G⟨π(g)ei,ej⟩⟨σ(g)fk,fl⟩dg=dπ1δπσδikδjl
These relations express the orthogonality between different irreducible representations and between different matrix coefficients within the same representation
The orthogonality relations are a consequence of and the Peter-Weyl theorem
Schur's Orthogonality Relations for Characters
Schur's orthogonality relations state that ∫Gχπ(g)χσ(g)dg=δπσ
These relations express the orthogonality between the characters of different irreducible representations
Schur's orthogonality relations can be derived from the orthogonality relations for matrix coefficients by taking the trace
Decomposition of Unitary Representations
Every unitary representation of a compact group can be decomposed into a direct sum of irreducible representations
The multiplicity of an irreducible representation π in a unitary representation ρ is given by the inner product ⟨χρ,χπ⟩L2(G)
The decomposition of a unitary representation into irreducibles is unique up to isomorphism
The orthogonality relations and the decomposition theorem provide powerful tools for studying the representation theory of compact groups