Source
- Peter & Weyl (1927), LIT-329, §§3–4, as read in NOTE-305.
- Kondor & Trivedi (2018), LIT-305, Lemmas 3–10, as read in NOTE-282, including its dated correction.
- Murota, Kanno, Kojima & Kojima, LIT-352, Props 3.2–3.5 and Thm 3.1(C), as read in NOTE-276.
- Cohen & Welling (2014), LIT-362, §2.2 and §3.3, as read in NOTE-307.
- O’Donnell, LIT-346, Props 2.37 and 2.47 and Ex. 1.30, as read in NOTE-291.
- Bronstein et al., LIT-319, pp. 37 and 54, as read in NOTE-274.
What was actually shown
Forward direction. Peter–Weyl prove that the eigenspaces of an invariant Hermitian kernel z(st⁻¹) carry representations of the group (§3). They then split those representations into irreducibles (§4). So an eigenvalue’s multiplicity is at least the dimension of an irreducible, and it can be more when several copies occur (NOTE-305). Kondor & Trivedi’s Lemma 6 is the general form: an equivariant linear map preserves isotypic components. Neither paper states “symmetric commuting operator ⇒ multiplicities are sums of irrep dimensions”. That step, from Lemma 6 plus Schur II, is the readers’ (NOTE-282).
Over the reals, abelian groups still pair eigenvalues. NOTE-282 first said that abelian symmetries give no degeneracy. The reading of LIT-362 corrected this (NOTE-307). The real irreducibles of SO(2), and of ℤ/n for n ≥ 3, are 2-D rotation planes. Their commutant is {aI + bJ}, whose only symmetric members are scalars, so each plane carries a doubled eigenvalue. The claim holds over ℂ and for groups with real characters, such as (ℤ/2)ⁿ. Murota et al.’s Thm 3.1(C) fails for the same reason: it assumes every real irreducible is of real type (NOTE-276).
I re-checked this when filing. A symmetric 7 × 7 circulant has three pairs and one singleton. Averaging a random symmetric 8 × 8 matrix over SO(2) with weights (1, 1, 2, 3) gives four pairs. Averaging a 4 × 4 matrix over the quaternion group acting on ℝ⁴ gives one fourfold eigenvalue, the ℍ-type case.
The converse fails in three ways.
- The group is not identified. The cube’s Laplacian has eigenvalue k with multiplicity C(n,k). Each level is irreducible under the hyperoctahedral group but reducible under S_n (NOTE-291). One spectrum is consistent with both groups.
- The pieces inside a multiplet are invisible. The Walsh characters in a level form one basis of the eigenspace among many (NOTE-291). A degeneracy test finds the pairs but not their frequencies (NOTE-307). GDL assumes degeneracy away so that the Fourier basis is well defined, and notes that eigenvectors are unstable (LIT-319, pp. 37, 54).
- Degeneracy can be accidental (NOTE-305).
For a single operator, degeneracy is a symmetry only in a trivial sense. Any k-fold eigenspace is rotated by O(k) while the operator stays fixed, so the multiplicity says nothing about a group acting on the data. Structure beyond that needs several operators and the joint commutant. Murota et al. compute it without a group supplied, and it gives central projectors and multiplicities, not a group (NOTE-276). This paragraph is the filer’s own inference.
What this does not say
- That spectral degeneracy is useless as a diagnostic. Eigenvalue multiplicities are unitarily invariant, and so intrinsic in THEORY-017’s sense. What they carry is the isotypic dimension count, not a group.
- That any of these papers detects an unknown symmetry. LIT-305, LIT-314 and LIT-319 impose G, and LIT-362 and LIT-363 need paired inputs and an assumed group or transformation (record/curation.d/2026/09/29/233704.md).
- Anything about noisy operators. Every source is exact-arithmetic. Near- degeneracy and the thresholds a test would need are open (NOTE-276, NOTE-274).
Connections
Presupposes THEORY-017’s sense of intrinsic: the multiplet is intrinsic, and the basis inside it is extra data. LIT-322’s weekday circles are real ℤ/7 irreducibles (NOTE-273). A degeneracy test would see them as pairs only. Wedderburn’s structure theorem (THEORY-652) gives the algebra-side half of the real Schur corollary: each simple real block is a full matrix algebra over a real division algebra. It does not give the step from a representation’s commutant to such a block; the note of 2026-10-29 gives it. Pontryagin duality (THEORY-645) is the abelian case, where the irreducibles over ℂ are characters.
Note of 2026-10-20: the assembled result is a claim
Promotion audit, batches 042 and 052. The forward direction, the pairing over ℝ and the failure of the converse re-derive from Schur’s lemma, complete reducibility and Frobenius’s theorem, and no held source states them together, so they are the record’s own result. They are filed as CLAIM-385, demonstrated, with the proof. This account keeps what the sources say and do not say, the readers’ real-scalar correction, and the promote_when, which a held representation-theory text would meet.
Note of 2026-10-29: the real Schur corollary, proved
The twelfth status review proved the corollary the promote_when asked a held source for, and checked it numerically.
Let V be a real irreducible of G of dimension d, with a G-invariant inner product, and let 𝔽 = End_G(V). By Schur’s lemma every nonzero element of 𝔽 is invertible, so 𝔽 is a finite-dimensional real division algebra, and by Frobenius’s theorem it is ℝ, ℂ or ℍ. On an isotypic component W ≅ V^m an intertwiner is an m × m array of intertwiners V → V, so End_G(W) = M_m(𝔽). Transposition maps 𝔽 to itself, because ρ is orthogonal. A symmetric s in 𝔽 has a real eigenvalue λ, and s − λI is in 𝔽 and singular, so it is zero: the symmetric elements of 𝔽 are the real scalars, and transposition is 𝔽‘s conjugation. A symmetric operator commuting with ρ is therefore, on W, a Hermitian matrix over 𝔽. Each of its eigenspaces in W is G-invariant, so it is a sum of copies of V and has real dimension a multiple of d; generically there are m eigenvalues, each of multiplicity exactly d.
The check (numpy, run by the reviewing session): for S₃ on three copies of its 2-dimensional representation (ℝ type), ℤ/5 on three copies of a rotation plane (ℂ type) and the quaternion group on two copies of ℍ (ℍ type), the commutant has dimension 9, 18 and 16, which is m² dim 𝔽, and the symmetric commutant 6, 9 and 6, the dimension of the Hermitian m × m matrices over 𝔽. A group-averaged random symmetric matrix has multiplicities (2, 2, 2), (2, 2, 2) and (4, 4), with G-invariant eigenspaces. As controls, a random symmetric matrix, and an SO(2)-averaged one perturbed by 10⁻³ in a non-commuting direction, have simple spectra. The 3-cube’s Laplacian, with spectrum 0, 2, 2, 2, 4, 4, 4, 6, commutes with the hyperoctahedral group B₃, under which each level has a one-dimensional commutant, and with S₃, under which levels 1 and 2 have a two-dimensional one: one spectrum, two groups.