Unitary representation

A complex representation is unitary when every representing matrix preserves a positive-definite Hermitian inner product.

The diagonal element $\mathrm{diag}(m,m^{-1})$ conjugates $U(t)$ to $U(m^{2}t)=U(t{)}^{m^2}$. In a finite-dimensional unitary representation, conjugacy therefore makes its eigenvalue multiset invariant under every square power. Square-power rigidity of a finite nonzero spectrum makes the image of every upper unipotent the identity. Its kernel is a normal subgroup; elementary unipotent generators of SL2R then force the entire group representation to be trivial. This algebraic argument does not require continuity.

For a strongly continuous unitary representation, integration against $f\in L^1(G)$ defines an operator satisfying $\Vert \pi (f)\Vert \le \Vert f{\Vert }_1$. Fubini's theorem gives $\pi (f\ast g)=\pi (f)\pi (g)$, and the group-algebra involution gives $\pi (f^{\ast })=\pi (f{)}^{\ast }$. On an Abelian group, the operator-norm closure is a commutative C-star algebra. Its algebra characters produce continuous unitary group characters, allowing the Commutative Gelfand--Naimark theorem and Riesz-Markov-Kakutani representation theorem to turn a positive vector functional into a finite spectral measure.

A strongly continuous unitary representation of the circle has integer spectral weights: $H$ is the orthogonal sum of spaces on which $U_z=z^m$. Positive energy means that these weights are bounded below, equivalently its self-adjoint generator has spectrum bounded below. Multiplication by an integer character shifts the lower bound to zero. A unitary operator with a nonzero rotation weight cannot exist in its operator algebra: repeated application of it or its inverse would move a nonzero vector to arbitrarily low energy.

A strongly continuous unitary representation of the circle group splits as an orthogonal sum of character subspaces. The projections are $P_n={\int }_{\mathbb{T}}{z}^{-n}{U}_{z}dz$. Character orthogonality makes them mutually orthogonal, and Fejér sum shows their ranges span the represented Hilbert space. The self-adjoint generator has integer eigenvalues, with domain given by ${\sum }_n{n}^2\Vert P_n\xi {\Vert }^2<\infty$.

For a compact group, the matrix coefficients of its finite-dimensional irreducible unitary representations span a dense subspace of $C(G)$ and form a complete orthogonal decomposition of $L^2(G)$ with Haar measure. If $K$ is a closed subgroup and $W$ is a finite-dimensional unitary $K$-representation, the space of L2 sections of the associated homogeneous bundle has multiplicity $\dim {\mathrm{Hom}}_K(E{|}_K,W)$ for each irreducible $G$-representation $E$. This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.

Suppose finite-dimensional continuous group representations separate points of a compact Lie group. Choose an identity neighborhood $U$ containing no nontrivial subgroup, using Lie groups have no small subgroups. For each $g\notin U$, choose a representation whose matrix at $g$ is not the identity. The corresponding open nonidentity sets cover the compact complement of $U$, so finitely many representations suffice there. Their direct sum has kernel inside $U$ and therefore trivial kernel. Unitarization of a compact-group representation makes it a finite-dimensional faithful unitary representation.

A continuous symmetric approximate identity on a compact group gives a compact self-adjoint convolution operator $T$ on $L^2(G)$ which maps into $C(G)$ and commutes with left translations. Its nonzero eigenspaces are finite-dimensional invariant spaces of continuous functions, hence consist of matrix coefficients of finite-dimensional unitary representations. If $S_N$ projects onto increasing finite sums of these eigenspaces, $S_{N}Tf\to Tf$ in $L^2$, and the bound $\Vert Th{\Vert }_{\infty }\le \Vert k{\Vert }_2\Vert h{\Vert }_2$ gives $T{S}_{N}Tf\to T^{2}f$ uniformly. Choosing the approximate identity narrow makes $T^{2}f$ uniformly close to $f$. This supplies uniform approximation without confusing an $L^2$ limit with a uniform limit.

For a compact group $G$, closed subgroup $K$ and finite-dimensional unitary $K$-representation $W$, induced sections are functions satisfying $f(gk)=k^{-1}f(g)$. A $K$-intertwiner $\varphi :E\to W$ produces the $G$-intertwiner $v\mapsto (g\mapsto \varphi (g^{-1}v))$. Evaluation at the identity reverses this construction on each finite-dimensional continuous representation summand. Thus the multiplicity of an irreducible $E$ in the induced section space is $\dim {\mathrm{Hom}}_K(E{|}_K,W)$.

A unitary irreducible representation is both unitary and irreducible. Such representations are the building blocks in the harmonic analysis of groups and in quantum symmetry classifications.

Average a positive-definite Hermitian form over a finite group and choose an orthonormal basis for the averaged form. In that basis every representing matrix is unitary.

For a compact group with normalized Haar measure, averaging any positive-definite Hermitian form,produces an invariant positive-definite Hermitian form.

A finite-dimensional complex representation of a compact group is a direct sum of irreducible representations. Average a positive Hermitian inner product over normalized Haar measure. The orthogonal complement of an invariant subspace is then invariant, so induction on dimension gives the decomposition. The same argument works for finite-dimensional real representations with a positive real inner product.

For a continuous real representation $\rho :G\to GL(V)$ of a compact group, average any positive-definite real inner product against normalized Haar measure:The result is positive definite and $G$-invariant. An orthonormal basis for it conjugates the representation into an orthogonal group.

Every finite-dimensional continuous complex representation of $S^1$ is unitary and decomposes into one-dimensional weight spaces. Its continuous one-dimensional characters are $z\mapsto z^m$ for $m\in \mathbb{Z}$.

For the circle group parameterized by $e^{i\theta }$ with period $2\pi$, a one-dimensional representation acts as $e^{i\theta }\mapsto e^{iQ\theta }$, and its integer label $Q$ is its charge. Periodicity forces $Q\in \mathbb{Z}$. A tensor product representation adds U(1) charges, a dual representation negates them, and restriction to the finite cyclic group generated by $e^{2\pi i/n}$ retains only $Q\mathrm{mod}n$. Integer normalization must be fixed before discussing a largest finite subgroup left by a condensate.

If a central subgroup is isomorphic to $S^1$, its weight spaces are invariant under the entire group, because every representing operator commutes with the central circle action.