Unitary GroupEdit
The unitary group, usually denoted U(n), is a cornerstone of both pure mathematics and theoretical physics. It consists of all n×n complex matrices that preserve the standard Hermitian inner product on the complex vector space C^n. In symbols, U(n) = { U ∈ M_n(C) : U† U = I }, where U† denotes the conjugate transpose and I is the identity matrix. This condition implies that each element of U(n) preserves norms and angles, so unitary matrices are precisely the linear operators that conserve the usual notion of length in quantum and signal-processing contexts.
Beyond its definition as a collection of matrices, U(n) is a group under matrix multiplication. It is a compact Lie group of real dimension n^2, and its natural action on C^n makes it a group of unitary transformations. The determinant map det: U(n) → U(1) is a continuous surjective homomorphism with kernel equal to the special unitary group SU(n); hence there is a short exact sequence 1 → SU(n) → U(n) → U(1) → 1. The center Z(U(n)) consists of the scalar matrices e^{iθ} I, forming a copy of U(1) inside U(n). The standard action on C^n preserves the Hermitian form and thus preserves the associated inner product and norm.
A useful set of basic facts concerns eigenvalues and diagonalizability. Every U ∈ U(n) is diagonalizable by a unitary change of basis, and all eigenvalues lie on the unit circle in the complex plane. This spectral property underpins many applications in physics and engineering, where unitary evolution preserves probabilities or energy norms.
The Lie algebra of U(n) is the set of skew-Hermitian matrices, commonly denoted u(n). Concretely, A ∈ u(n) satisfies A† = −A, and U(n) can be locally described via the exponential map U = exp(A) with A ∈ u(n). This exponential map is surjective onto the connected component of the identity, reflecting the fact that U(n) is connected and compact. The commutator subgroup of U(n) is SU(n), and the abelianization is isomorphic to U(1), reflecting the central role of scalar phase factors in the structure of the group.
Definition and basic properties
- Structure and definitions
- U(n) = { U ∈ M_n(C) : U† U = I }.
- The center Z(U(n)) ≅ U(1) consists of all scalar matrices e^{iθ} I.
- The determinant map gives a short exact sequence 1 → SU(n) → U(n) → U(1) → 1.
- Geometry and topology
- U(n) is a compact Lie group of real dimension n^2.
- Its Lie algebra u(n) comprises skew-Hermitian matrices (A† = −A).
- Every unitary matrix has a logarithm: U = exp(iH) with H Hermitian.
- Representation and functions
- The Peter–Weyl theorem ensures that every finite-dimensional representation of U(n) is unitary with respect to a suitable inner product.
- Irreducible representations of U(n) are classified by highest weights, and their dimensions are given by Weyl’s dimension formula.
- Characters of irreducible representations can be expressed in terms of Schur functions, tying the representation theory of U(n) to symmetric function theory.
Representations and structure
The representation theory of U(n) is a classical subject with rich connections to combinatorics, geometry, and mathematical physics. Irreducible representations of U(n) are labeled by sequences of integers (λ_1 ≥ λ_2 ≥ … ≥ λ_n), often interpreted as highest weights. The dimension of the irreducible representation with highest weight λ is given by Weyl’s dimension formula. The corresponding characters are given by Schur functions, linking the representation theory to the theory of symmetric functions and to the combinatorics of Young diagrams.
The relationship between U(n) and SU(n) is central in both mathematics and physics. SU(n) consists of unitary matrices with determinant 1 and has real dimension n^2 − 1. SU(n) is a semisimple (indeed, simple for n ≥ 2) Lie group, while U(n) combines SU(n) with the abelian factor U(1) arising from determinant scaling. This decomposition is reflected in the representation theory: irreducible representations of U(n) can be built from those of SU(n) together with a U(1) charge, subject to compatibility conditions.
In the continuous and discrete settings where unitary symmetries appear, two foundational results are especially important: - The Weyl character formula provides explicit expressions for characters of irreducible representations of compact Lie groups, including U(n). - The Weyl dimension formula gives the dimension of each irreducible representation in terms of its highest weight.
Sources of structure and applications
- Physics and quantum theory
- Time evolution in quantum mechanics is represented by unitary operators, so the unitary group governs dynamics in finite-dimensional quantum systems. Quantum gates in quantum computing are modeled as unitary operations, making U(n) the mathematical underpinning of quantum circuits.
- Gauge theories often employ unitary groups as gauge groups, with U(n) and SU(n) playing roles in models of fundamental interactions.
- Signal processing
- Many transforms, including the discrete Fourier transform (often normalized to be unitary), preserve energy, a property expressed by unitarity. This perspective situates unitary groups at the heart of Fourier analysis and orthogonal signal processing.
- Geometry and random matrix theory
- The Haar measure on U(n) provides a canonical notion of a uniform random unitary matrix, which is a central object in random matrix theory and its applications to statistics and physics.
- Linear and multilinear algebra
- The spectral theorem for normal matrices implies that unitary similarity preserves eigenvalues and that normal matrices are unitarily diagonalizable.
Examples and special cases
- U(1) is the group of complex numbers with modulus 1, identified with the unit circle in the complex plane. It acts by phase multiplication on C, and represents the simplest nontrivial unitary group.
- U(n) contains SU(n) as a normal subgroup of index equal to the order of U(1) in the determinant map, reflecting how determinant constraints separate the unitary group into a special linear block and a phase factor.
- The representations of U(n) include those built from partitions that fit inside an (n × k) rectangle, with k unbounded in the finite-dimensional setting; in practice this connects to the combinatorics of Young diagrams and symmetric functions.