CLAIM-385: If a symmetric operator on ℝⁿ commutes with an orthogonal representation of a compact group, each eigenspace is a sum of real irreducibles and each multiplicity a sum of multiples of their dimensions, so SO(2) and ℤ/n for n ≥ 3 force paired eigenvalues; conversely the spectrum fixes neither the group nor a basis inside an eigenspace

The claim

Let G be a compact group (finite groups included) and ρ a real representation of G on ℝⁿ, orthogonal for some inner product (any real representation of a compact group is, for the averaged inner product). Let A be symmetric for that inner product, with Aρ(g) = ρ(g)A for every g. Then:

1. Eigenspaces are G-invariant and split into real irreducibles.
2. Multiplicities count irreducible dimensions. On an isotypic component W ≅ V^m, with V irreducible of real dimension d, every eigenspace of A meets W in a subspace of dimension kd for some k. So each multiplicity is a sum of multiples of the real-irreducible dimensions.
3. Rotation groups pair eigenvalues over ℝ. For SO(2), and for ℤ/n with n ≥ 3, every real irreducible that is not one-dimensional is a rotation plane, and A is a scalar on each such plane, so its eigenvalue there is at least doubled.
4. The converse fails. The spectrum of A determines neither G nor a basis inside a degenerate eigenspace.

Proof

(1) If Av = λv then Aρ(g)v = ρ(g)Av = λρ(g)v, so the λ-eigenspace E_λ is invariant. ρ is orthogonal, so the orthogonal complement of an invariant subspace is invariant; induction on dimension splits E_λ into irreducibles (complete reducibility).

(2) Decompose ℝⁿ into isotypic components W_i, the sums of all irreducible subspaces isomorphic to V_i. For non-isomorphic irreducibles V, V′, Schur’s lemma gives Hom_G(V, V′) = 0. A commutes with ρ, so its restriction to an irreducible subspace of W_i is an intertwiner into ℝⁿ, whose components in the other W_j vanish; hence A W_i ⊆ W_i (Kondor & Trivedi’s Lemma 6 is this step, LIT-305). E_λ ∩ W_i is invariant, so by (1) it is a direct sum of irreducibles, each in W_i and so isomorphic to V_i: its dimension is k·d_i. The multiplicity of λ is Σ_i k_i d_i. (End_G(V_i) is a real division algebra, ℝ, ℂ or ℍ, by Schur and Frobenius; the count above does not need which.)

(3) The real irreducibles of SO(2) are the trivial representation and the planes on which θ acts by rotation through jθ, j ≥ 1. For ℤ/n they are the trivial and, for even n, the sign character, and the planes on which the generator rotates by 2πj/n with 0 < j < n/2. On a rotation plane P with generator J (J² = −I, Jᵀ = −J), the commutant is {aI + bJ}, since a matrix commuting with every rotation of the plane is a rotation-scaling. If the plane occurs once, A restricted to P lies in that commutant and is symmetric, so b = 0 and A = aI on P, a doubled eigenvalue. If it occurs m times, (2) gives multiplicities that are multiples of 2. The record’s checks agree: a symmetric 7 × 7 circulant has three pairs and one singleton, and a symmetric matrix averaged over SO(2) acting with weights (1, 1, 2, 3) on ℝ⁸ has four pairs.

(4) Let A have distinct eigenvalues with multiplicities m_1, …, m_r. A commutes with every orthogonal map preserving each eigenspace, that is, with O(m_1) × … × O(m_r) acting block-wise, and so with every subgroup of that product. The spectrum is therefore consistent with every one of those groups, and no procedure that reads only the spectrum can return a unique group. O’Donnell’s cube gives a concrete pair (LIT-346): the Laplacian of the n-cube has an eigenvalue proportional to k with multiplicity C(n, k), for k = 0, …, n, and each level is irreducible under the hyperoctahedral group but splits under the symmetric group S_n (level 1 is the trivial representation plus the standard one). Within a degenerate eigenspace every orthonormal basis is an eigenbasis, so the spectrum fixes no basis there: the Walsh characters of a level are one choice among many, and a degeneracy test finds the pairs of LIT-362’s rotation planes but not their frequencies.

What it does not say

• That degeneracy detects symmetry. (4) says the reverse. What a spectrum carries is the isotypic dimension count of whatever group is present, not the group.
• That the sources state it. Peter & Weyl build representations as eigenspaces of an invariant kernel (LIT-329), and Kondor & Trivedi prove that intertwiners preserve isotypic components (LIT-305); neither states (2)–(4). The reading of the sources, with the real-scalar correction from LIT-362, stays in THEORY-019.
• Anything about noisy operators. The statement is exact. Near- degeneracy and the thresholds a test on estimated operators would need are open.
• That multiplicities are intrinsic in a further sense. They are unitarily invariant, which is THEORY-017’s sense of intrinsic; the basis inside a multiplet is extra data.