Symmetry Decompositions and Multiplicity Spaces

A symmetry decomposition separates inequivalent irreducible transformation laws and records how many copies of each law occur. The multiplicity spaces are essential: an operator can commute with the symmetry while mixing equivalent copies. This page states the compact-group decomposition and derives its bounded commutant. The Hilbert direct-sum construction itself belongs to Direct Sums and Superselection.

Required background. Direct Sums and Superselection distinguishes a decomposition from a physical restriction. Familiarity with irreducible finite-dimensional representations and Schur intertwiners is assumed.

Separate irreducible type from multiplicity

Let $G$ be a compact Hausdorff group and $U:G\to\mathcal U(\mathcal H)$ a strongly continuous unitary representation on a separable Hilbert space. The compact-group decomposition theorem gives

$$\mathcal H\simeq \bigoplus_{\lambda} \bigl(V_\lambda\otimes\mathcal M_\lambda\bigr),$$ $$U(g)\simeq \bigoplus_\lambda \bigl(D_\lambda(g)\otimes I_{\mathcal M_\lambda}\bigr).$$

Here $D_\lambda$ are mutually inequivalent finite-dimensional irreducible representations. The multiplicity space $\mathcal M_\lambda$ counts equivalent copies and may be infinite-dimensional. Only countably many nonzero summands occur for separable $\mathcal H$.

The full summand of one type is called an isotypic component. It is determined by the representation, whereas a basis or splitting of its multiplicity space into individual copies is generally not canonical.

Compactness is substantive. A general noncompact-group representation need not decompose as a countable direct sum of finite-dimensional irreducibles. Direct-integral theory and further hypotheses are then needed.

Symmetry-invariant operators act on multiplicities

Let $A$ be bounded and commute with every $U(g)$. Its block from type $\mu$ to type $\lambda$ intertwines the corresponding representations. Schur’s lemma eliminates blocks between inequivalent types and makes the action on each irreducible factor scalar. With multiplicities retained, the result is

$$U(G)'= \left\{ \bigoplus_\lambda I_{V_\lambda}\otimes B_\lambda: B_\lambda\in\mathcal B(\mathcal M_\lambda),\ \sup_\lambda\|B_\lambda\|<\infty \right\}.$$

One can see the multiplicity part by choosing basis vectors in each $\mathcal M_\lambda$. Each matrix block between two equivalent copies is an intertwiner on $V_\lambda$, hence a scalar multiple of its identity. The collection of those scalars forms the operator $B_\lambda$.

The norm bound is required for the direct sum to be a bounded operator. Saying only that each block is bounded is insufficient when infinitely many types occur.

Character projections identify the type

Let $\chi_\lambda$ be the character of $D_\lambda$ and $d_\lambda=\dim V_\lambda$. Normalized Haar measure gives the isotypic projection

$$P_\lambda= d_\lambda\int_G \overline{\chi_\lambda(g)}\,U(g)\,dg.$$

The integral can be defined through its action on each vector. Character orthogonality shows that it is identity on the $\lambda$ isotypic component and zero on inequivalent components. This formula locates the entire type without choosing a multiplicity basis.

Kowalski’s Corollary 5.4.2 gives the decomposition for general compact groups, and Theorem 5.5.1 gives character orthogonality and this projection formula. Hall’s Part III develops the compact Lie-group setting. These results make the distinction between type and copy explicit.

Two spins illustrate an invariant observable algebra

For two spin-$1/2$ systems, the simultaneous $SU(2)$ action decomposes as

$$\mathbb C^2\otimes\mathbb C^2 \cong V_0\oplus V_1.$$

Each type occurs once. Therefore every bounded operator commuting with all rotations has the form $A=a_0P_0+a_1P_1$. A rotationally invariant Hamiltonian in this four-dimensional model assigns one energy to the singlet and one to the triplet.

This restriction applies to rotationally invariant operators. It does not assert that every physically available measurement is rotationally invariant. A measurement with a specified laboratory axis can connect the singlet to the triplet; for example, the first spin’s $z$ component sends the singlet to a nonzero multiple of the triplet with magnetic number zero.

Thus a symmetry decomposition is not automatically a superselection rule. That conclusion would require restricting the entire allowed observable algebra, not just the Hamiltonian or a selected family of scalar observables.

Equivalent copies need not be separate sectors

For two equivalent spin-$1/2$ copies arranged as a direct sum, the representation is $V_{1/2}\otimes\mathbb C^2$, with the second factor labeling copies. The commutant contains $I_{V_{1/2}}\otimes X$, which exchanges them while preserving rotational symmetry.

The projection onto one chosen copy is therefore not central in this commutant. Treating the two copies as distinct superselection sectors would discard symmetry-invariant mixing operators without justification.

For unbounded invariant Hamiltonians, the safe condition is that their spectral projections commute with $U(G)$. Formal commutation on a common test domain does not by itself establish the decomposition of a self-adjoint operator and its domain.

Exercise: count invariant parameters

Suppose a compact-group representation is $(V_\alpha\otimes\mathbb C^2)\oplus V_\beta$, where $\alpha$ and $\beta$ are inequivalent. How many real parameters specify a self-adjoint operator commuting with the group?

Solution

It has the form $(I_{V_\alpha}\otimes B)\oplus bI_{V_\beta}$, where $B$ is a Hermitian $2\times2$ matrix and $b$ is real. The matrix contributes four real parameters and $b$ contributes one, for five total. Assuming one scalar for each of the three individual copies would miss the two real off-diagonal mixing parameters between the equivalent $\alpha$ copies.

References

• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Graduate Texts in Mathematics 222, Springer, 2015, Part III — compact Lie groups and their representations.
• E. Kowalski, Representation Theory, ETH Zürich lecture notes, version of September 23, 2011, Corollary 5.4.2 and Theorem 5.5.1 — decomposition, character orthogonality, and isotypic projections for compact groups.
• K. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Fundamental Theories of Physics 188, Springer, 2017, Chapter 5 — representation theory and quantum symmetry.