Is a spectral symmetry class literally a reality type?

Object. Two trichotomies that keep landing on the same objects. First: the Frobenius–Schur indicator of an irreducible representation, +1 / 0 / −1 according as it is real, complex or quaternionic — computed directly, the standard representation of S₃ gives +1, a nontrivial character of ℤ/3 gives 0, the 2-dimensional irreducible of Q₈ gives −1. Second: the GOE / GUE / GSE classification of spectral statistics, and the orthogonal / unitary / symplectic symmetry types into which families of L-functions sort at large conductor.

Presenting problem (the friction). The correspondence is stated everywhere as a correspondence and used as if it were an identity, and the two sides are invariants of different things.

The Frobenius–Schur indicator is an invariant of a single representation of a single group: whether the representation admits an invariant bilinear form, and if so whether it is symmetric or alternating. A spectral symmetry type is an invariant of a family: the monodromy group in whose Haar measure the family's low-lying zeros equidistribute, or equivalently the ensemble whose eigenvalue statistics the spectrum reproduces. One is representation-theoretic and pointwise; the other is measure-theoretic and asymptotic.

In the classical cases they agree, which is why nobody presses. Nothing yet says they must, and the same trichotomy also organizes Arf invariants and quadratic refinements, which are neither.

Differential. 1. Identity, via a theorem: the symmetry type of a family is the reality type of the representation whose Frobenius conjugacy classes the family samples, and the equidistribution results are that theorem in analytic clothing. 2. Parallel classifications with a common cause, the cause being that both count the ways an invariant bilinear form can exist on a space with a structure map, but with no map from either statement to the other. 3. Coincidence of arity: three is the number of options for a structure map squaring to ±1, so any classification by such a map has three classes, and matching the labels is matching the arity rather than the content. The same coincidence would explain the Arf and quadratic-refinement appearances without any of them being the same object. 4. The identity holds for families arising from algebraic monodromy and fails outside them, which would make it a theorem with a hypothesis nobody has been stating.

Discriminating tests. Take families whose symmetry type is known and compute Frobenius–Schur indicators for the representations actually sampled, checking whether the assignment is forced or merely consistent. Then look for a case designed to separate them: a family whose sampled representation is real but whose statistics are unitary, or the reverse. Branch 3 predicts such a case exists and is easy to build; branch 1 predicts it cannot exist; branch 4 predicts it exists only outside algebraic monodromy.

Stakes. This trichotomy has been picked up at least four times under four names — Arf invariant, quadratic refinement, spectral symmetry type, reality of a representation — and recognized as one thing only on the fourth pass. If branch 1 holds, one invariant has been doing all that work and the earlier appearances were the same computation. If branch 3 holds, the unification is a pun on the number three, and three separate results have just been merged that should not have been.