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  "slug": "symmetric-determinantal-hilbert-threshold",
  "title": "The algebraicity threshold for orthogonal tensor multiplicities",
  "shortTitle": "Orthogonal algebraicity stops at three",
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  "oneLine": "A uniform classification candidate: every fixed genuine orthogonal representation has an algebraic tensor-multiplicity series exactly in dimensions one, two and three.",
  "abstract": "For every fixed nonzero finite-dimensional algebraic representation W of O(m,C), the ordinary series counting copies of W in tensor powers of the defining representation is D-finite, and algebraic exactly when m is at most three. Explicit low-dimensional formulas and parity-resolved character asymptotics give a uniform proof route. The invariant specialization classifies symmetric determinantal Hilbert series. Scalar asymptotics, low-rank invariant formulas and general tensor-asymptotic methods are attributed to prior work; historical priority of the full classification is not established.",
  "datePublished": "2026-09-08",
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  "keywords": [
    "orthogonal groups",
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    "algebraic generating functions",
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    "No genuine fixed type restores algebraicity in dimension four or above; virtual subtraction can cancel the obstruction.",
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    {
      "citation": "Regev (1981), Asymptotic values for degrees associated with strips of Young diagrams: Theorem 3.3 and Formula 4.5.1 already supply the scalar asymptotic.",
      "url": "https://doi.org/10.1016/0001-8708(81)90012-8",
      "doi": "10.1016/0001-8708(81)90012-8"
    },
    {
      "citation": "Almkvist, Dicks and Formanek (1985), Hilbert series of fixed free algebras and noncommutative classical invariant theory: Theorem 2.1 and Example 5.10.",
      "url": "https://doi.org/10.1016/0021-8693(85)90183-8",
      "doi": "10.1016/0021-8693(85)90183-8"
    },
    {
      "citation": "Rains (1998), Increasing subsequences and the classical groups, Section 3.",
      "url": "https://doi.org/10.37236/1350",
      "doi": "10.37236/1350"
    },
    {
      "citation": "Sam and Snowden (2018), Hilbert series for twisted commutative algebras, Theorem 6.1 and Example 7.11.",
      "url": "https://alco.centre-mersenne.org/articles/10.5802/alco.9/",
      "doi": null
    },
    {
      "citation": "Tate and Zelditch (2004), Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers, Theorem 8.",
      "url": "https://doi.org/10.1016/j.jfa.2004.01.004",
      "doi": "10.1016/j.jfa.2004.01.004"
    },
    {
      "citation": "Bostan, Salvy and Singer (2025), On deciding transcendence of power series, discussion of classical asymptotic obstructions.",
      "url": "https://arxiv.org/abs/2504.16697",
      "doi": "10.48550/arXiv.2504.16697"
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