14.4 Peter-Weyl theorem and representation theory

The Peter-Weyl theorem 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 Fourier series 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
• Compactness implies several important properties such as being Hausdorff, second-countable, and locally compact
• Examples of compact groups include the circle group $S^1$, the torus $T^n$, and the special unitary group $SU(n)$
• Compact groups have a unique Haar measure (up to scaling) which allows for integration on the group

Unitary Representations of Compact Groups

• A unitary representation of a compact group $G$ is a continuous homomorphism $\rho: G \to U(\mathcal{H})$, where $U(\mathcal{H})$ is the group of unitary operators on a Hilbert space $\mathcal{H}$
• Unitary representations preserve the inner product structure of the Hilbert space
• The dimension of the Hilbert space $\mathcal{H}$ 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 irreducible representation is a unitary representation that has no non-trivial invariant subspaces
• Irreducible representations are the building blocks of all unitary representations via the direct sum decomposition
• The group algebra $L^2(G)$ 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 matrix coefficients of the irreducible unitary representations form an orthonormal basis for the space $L^2(G)$
• This theorem provides a natural generalization of the classical Fourier series to compact groups
• The matrix coefficients are given by $\sqrt{d_\pi} \langle \pi(g) e_i, e_j \rangle$, where $\pi$ is an irreducible representation, $d_\pi$ is its dimension, and $\{e_i\}$ is an orthonormal basis for the representation space

Fourier Series on Compact Groups

• The Fourier series of a function $f \in L^2(G)$ is given by $f(g) = \sum_\pi d_\pi \sum_{i,j} \hat{f}(\pi)_{ij} \sqrt{d_\pi} \langle \pi(g) e_i, e_j \rangle$
• The Fourier coefficients $\hat{f}(\pi)_{ij}$ are given by the inner product $\langle f, \sqrt{d_\pi} \langle \pi(\cdot) e_i, e_j \rangle \rangle_{L^2(G)}$
• The Fourier series converges to $f$ in the $L^2$ norm, and under additional regularity assumptions, it converges pointwise

Character Theory and Its Applications

• The character of a representation $\pi$ is the function $\chi_\pi(g) = \text{Tr}(\pi(g))$, where $\text{Tr}$ denotes the trace
• Characters are conjugation invariant and satisfy the orthogonality relations $\langle \chi_\pi, \chi_\sigma \rangle_{L^2(G)} = \delta_{\pi\sigma}$
• The character table 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 $\int_G \langle \pi(g) e_i, e_j \rangle \overline{\langle \sigma(g) f_k, f_l \rangle} dg = \frac{1}{d_\pi} \delta_{\pi\sigma} \delta_{ik} \delta_{jl}$
• 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 Schur's lemma and the Peter-Weyl theorem

Schur's Orthogonality Relations for Characters

• Schur's orthogonality relations state that $\int_G \chi_\pi(g) \overline{\chi_\sigma(g)} dg = \delta_{\pi\sigma}$
• 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 $\pi$ in a unitary representation $\rho$ is given by the inner product $\langle \chi_\rho, \chi_\pi \rangle_{L^2(G)}$
• 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