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Tensor saturation for even spin groups
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Abstract
We prove the saturation conjecture for $\mathop{\mathrm{Spin}}\nolimits (2n)$, n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.
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Related prior literature mentioned by others
- Joshua Kiers, On the saturation conjecture for Spin(2n) (arXiv:1804.09229, revised 2018-09-11). Proves root-lattice saturation for Spin(10) and Spin(12), the low-rank cases highlighted as prior work in AIProofWatch’s record of this manuscript.
Possibly relevant literature (found by the agent, not checked)
- Shrawan Kumar, Additive Eigenvalue Problem (a survey), with appendix by M. Kapovich (arXiv:1305.4697, May 2013). Survey of the eigencone and the saturated tensor semigroup with a section on the saturation problem, background for the paper's framing of saturation as real (eigencone) feasibility plus the root-lattice congruence; the bibliography cites Kumar's 2010 ICM article, not this survey.
- Thomas J. Haines (appendix with M. Kapovich and J. Millson), Equidimensionality of convolution morphisms and applications to saturation problems (arXiv:math/0501504; Adv. Math. 207 (2006)). Applies Mirković–Vilonen convolution geometry to saturation for structure constants of Hecke and representation rings, an alternative to the Kapovich–Millson [KM08] building and path methods the paper uses to get real paths.
- Velleda Baldoni, Michèle Vergne, Michael Walter, Quiver subrepresentations and the Derksen-Weyman saturation property (arXiv:2506.23633, June 2025). Gives a short, direct proof of Derksen–Weyman quiver saturation via Schofield's general subrepresentations, both of which the paper cites in its type-A background and its quiver/preprojective zero-tail step.
- Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties (arXiv:math/9912012, 1999-12-02). Its lattice-point formulas for tensor product multiplicities provide a possible comparison framework for the manuscript’s Section 5 construction of nonzero tensor intertwiners from integral schedules.
Digests and expositions
- Gradient Damage, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More (video compilation, 2026-10-14). Chapter 29, beginning at 9:43, covers the manuscript’s claimed tensor saturation for Spin(2n) in a compilation whose animations are described as illustrations rather than proofs.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 204
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Tensor-Saturation-for-Even-Spin-Groups-September-24-2026,
author = {{OpenAI}},
title = {{Tensor saturation for even spin groups}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026.pdf}{OAI:Tensor-Saturation-for-Even-Spin-Groups-September-24-2026}},
year = {2026}
}Family 204: Tensor saturation for even spin groups
Proves saturation factor one for $\mathop{\mathrm{Spin}}\nolimits (2n)$, n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.
This is the only paper in this family.