The algebraicity threshold for orthogonal tensor multiplicities

Summary

This page explains, in order: what is being counted, what "algebraic" and "D-finite" mean for a counting series, what this release proves, why the answer stops at dimension three, and what has and has not been checked.

What is being counted. Symmetry groups act on spaces, and the simplest way to build bigger representations of a group is to take tensor powers: combine $N$ copies of the basic representation. The result breaks up into irreducible pieces, the "symmetry types". Here the group is the complex orthogonal group $O(m)$, the symmetries of $m$-dimensional space that preserve lengths, acting on its defining representation $V=\mathbb C^m$. Fix one symmetry type $W$ and record, for each $N$, how many copies of $W$ appear inside the $N$-fold tensor power. Packaging these counts as the coefficients of a power series gives a single object, the counting series of $W$.

Why the type of series matters. Counting series come in tiers of complexity. An algebraic series satisfies a polynomial equation, so it can be written with square roots and similar radicals; for example, in dimension two the invariant series is $\tfrac12\big(1+1/\sqrt{1-4t^2}\big)$. A D-finite series satisfies a linear differential equation with polynomial coefficients. Every algebraic series is D-finite, but the D-finite class is much larger. Knowing which tier a family belongs to tells researchers what kind of closed formula is worth looking for.

What was found. For the orthogonal group there is a clean dividing line. Every fixed nonzero genuine representation $W$ has a D-finite counting series in every dimension, and an algebraic one exactly when $m$ is one, two or three. From dimension four upwards no fixed genuine type is algebraic, so choosing a different type cannot restore algebraicity. The special case $W=$ trivial answers a question about the symmetric determinantal quotients (algebras built from symmetric matrices of rank at most $m$) recorded in the catalogue entry AIM-REPRESENTATION_THEORY-0022: the original proposed algebraicity assertion fails starting at rank four, and never returns.

Why it stops at three. In tensor degree $N=2n$ the counts grow like $m^{2n}$ times a power $n^{-\alpha}$ with $\alpha=m(m-1)/4$, a number fixed by the dimension of the group. When $\alpha$ is a whole number the series must contain a logarithm; when it is a half-integer, the exact leading constant involves a power of $\pi$ that no algebraic series can produce. Both cases rule out algebraicity once $m\ge4$.

Status. An unrefereed written classification candidate, assembled from explicitly attributed classical inputs. The scalar asymptotics and the rank-three formula have exact classical antecedents, and historical priority of the full classification is not established. The finite computer checks test formulas and conventions; they do not prove the universal theorem.

Summary for specialists

Let $V=\mathbb C^m$ be the defining representation and let $W\ne0$ be a fixed finite-dimensional algebraic representation of $O(m,\mathbb C)$, with $m\ge1$. Put

The candidate proves that $P_W$ is D-finite for every $m$, and algebraic over $\mathbb C(t)$ if and only if $m\le3$ (Section 9 of the paper). The quantifier includes reducible genuine representations, but excludes virtual subtraction and a representation varying with $N$.

For the symmetric determinantal quotient, put

where $I_{m+1}$ is generated by the $(m+1)$-minors of the generic symmetric matrix, so $m$ is the allowed rank. The invariant specialization gives algebraicity of its symmetric-sequence ordinary Hilbert series exactly for $m\le3$ (including the trivial $m=0$ case). This is not a claim about the usual finite-variable commutative Hilbert series, which is rational.

Technical account

From invariants to an integral. The classical first and second fundamental theorems for the orthogonal group identify the degree-$d$ piece of $B_m$ with the $O(m)$-invariant tensors in $V^{\otimes d}$. Averaging over the compact group $K=O(m,\mathbb R)$ then turns each count into a moment of the trace, $a_m(n)=\int_K(\operatorname{tr}g)^{2n}\,dg$. Covariants of a fixed type $W$ are handled the same way, with the character of $W$ inserted into the integral.

The low-dimensional side. The paper gives explicit formulas for all types of $O(1)$, $O(2)$ and $O(3)$. In dimension two the integral reduces to a constant term and gives $\tfrac12\binom{2n}{n}$. In dimension three the counts are walks governed by the Clebsch–Gordan rule. The half-line tensor-product recurrence must retain the exceptional boundary at spin zero; a parity projection then distinguishes the two extensions from $SO(3)$. The rank-three invariant formula is a specialization of Almkvist–Dicks–Formanek, Example 5.10.

The obstruction. Character integration over the full compact orthogonal group isolates the two maximal trace points, $I$ and $-I$. If the central parity of $W$ is $\varepsilon$, the compatible coefficients satisfy

where

The scalar constant is already supplied by Regev's work. The manuscript also derives the normalization directly, from an exact volume formula for the full, disconnected group, and explains why the two maximal points add rather than cancel on the compatible parity. A radial form of the classical asymptotic obstruction to algebraicity (often called Flajolet's criterion) then finishes the argument. When $m\equiv0,1\pmod4$ the exponent is an integer, and integer exponents force logarithmic singularities after differentiation. When $m\equiv2,3\pmod4$ the exponent is a half-integer; the leading amplitude is then a nonzero algebraic number times a positive integral power of $1/\pi$, which is incompatible with algebraicity above dimension three because $\pi$ is transcendental. Mixed central parity is handled by extraction.

D-finiteness. Each fixed covariant module is finitely generated and has bounded row support, so the Sam–Snowden theorem makes its exponential Hilbert series D-finite; a Hadamard product with $\sum_N N!\,t^N$ converts this to the ordinary series. Finite generation is verified in the paper rather than assumed.

Why "genuine" matters. The genuine-representation qualification is substantive. In dimension four the trivial and determinant multiplicities are $(C_n^2+C_n)/2$ and $(C_n^2-C_n)/2$. Their virtual difference is $C_n$, so subtraction cancels the high-rank obstruction. This illustration is not advertised as a new Catalan identity.

Evidence, assurance and limitations

• Proof. The universal claims rest on the written argument in the thirteen-page integrated manuscript and on named classical inputs: the orthogonal fundamental theorems and restriction identity, Schur–Weyl duality, Newton–Puiseux, the transcendence of $\pi$, and the cited D-finiteness theorem and closure properties.
• Finite checks. Producer replay compares 463 tableau shapes, checks 841 weight and 841 spin multiplicities, verifies parity and dimension identities, and rejects a deliberately corrupted hook divisor under normal and optimized Python. Fresh extraction and Linux CI validate the shipped finite diagnostics. They do not prove the universal theorem. No formal verification or independent reimplementation is claimed.
• Review. The package contains a precise source comparison, response matrix, preserved substantive reviews and five confirmation reports. Internal editorial confirmation accepted the scoped candidate with attribution and priority qualifications; it does not constitute independent journal peer review or externally established significance.
• Scope. The theorem does not classify arbitrary modules over twisted commutative algebras, nonreduced thickenings, growing representation types or all virtual characters. It excludes $W=0$ and positive characteristic. D-finiteness alone supplies neither a useful differential-equation order nor an efficient coefficient algorithm.
• Source record. The catalogue excerpt behind the original question and its provenance are included in the package; the live AIM source has not been independently recovered.

Relationship to earlier work

Regev (1981), Theorem 3.3 and Formula 4.5.1, already gives the scalar asymptotic including its constant. Almkvist, Dicks and Formanek (1985), Theorem 2.1, gives the relevant tensor-algebra character integral; their Example 5.10 gives the $SO(3)$ invariant radical whose even part is the orthogonal invariant series. The package records the published erratum. General fixed-type tensor asymptotics also have classical antecedents, including the Biane result discussed by Tate and Zelditch.

The presented contribution is the unified all-fixed-genuine-type orthogonal classification, with explicit low-dimensional formulas, parity treatment and the no-restoration consequence. It is not a claim to have originated the scalar asymptotic, the rank-three radical, character integration or the classical algebraicity obstruction. The supplied Regev and Almkvist–Dicks–Formanek papers were directly inspected. Cohen–Regev (1988) was unavailable and is explicitly marked uninspected; no claim about its detailed contents is made. The bounded comparison does not establish historical priority.

Who should care, and why

Why the problem matters

Algebraic series form a much narrower class than D-finite series. A sharp classification explains why a successful radical formula in low dimension cannot simply be extended by choosing a different genuine symmetry type in higher dimension. It also identifies cancellation, rather than an improved invariant formula, as the issue when virtual representations are allowed. These are mathematical consequences, not measured practical or productivity benefits.

How to inspect or reproduce the recorded checks

Download the versioned ZIP, compare its SHA-256 with SHA256SUMS, and extract it into a fresh directory. With Python 3.10 or later and no third-party library:

python3 replay.py
python3 -O replay.py

Both must report PASS; the receipt identifies the exact checks and their finite boundary. The manifest is checked before the diagnostics. A PASS covers scalar hook and corner dimensions, rank-three walks, low-rank formulas, optimized-mode mutation rejection and the package manifest; it does not prove the all-rank theorem.

To read the mathematics, start with Section 1 of the paper (statement and explicit formulas), then Sections 4–5 for the obstruction and Sections 9–12 for the extension to every fixed type. AI_INDEX.md maps each claim to its evidence, and CITATION_AUDIT.md gives the theorem-by-theorem prior-work comparison.

The most valuable next projects

1. Obtain an unaffiliated proof audit concentrating on compact-group normalization, the singularity obstruction and finite generation of fixed covariants.
2. Determine explicit annihilating differential equations and useful order bounds for selected higher-dimensional types.
3. Classify virtual character combinations whose singularities cancel; the genuine theorem does not settle this problem.

Who might contribute

Useful scrutiny would combine orthogonal representation theory, analytic combinatorics and twisted commutative algebra. Separate historical work could assess priority beyond the directly inspected antecedents. No endorsement by the cited authors is implied.

What is in the evidence package

The immutable candidate release includes the PDF and Markdown manuscript, Python verifiers and semantic controls, complete manifest, replay receipt, claim index, source audit, review history, editorial decision, citation metadata and component licences. Original prose is CC0; original code is MIT. Supplied full third-party papers are not redistributed. Historical paths in preserved reports are provenance, not dependencies needed to run the package. The same files are in the GitHub release and the Zenodo record.

@misc{symmetricdeterminantalhilbertthreshold2026,
  title        = {The algebraicity threshold for orthogonal tensor multiplicities},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22660867},
  url          = {https://doi.org/10.5281/zenodo.22660867},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/symmetric-determinantal-hilbert-threshold/}
}