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The Kapovich-Millson saturation conjecture for simply laced groups, type D: tensor saturation for Spin(2n)\mathrm{Spin}(2n)

Let GG be a simply connected complex semisimple group with root lattice QQ, and V(λ)V(\lambda) the irreducible representation of highest weight λ\lambda. For dominant λ,μ,ν\lambda,\mu,\nu with λ+μ+ν∈Q\lambda+\mu+\nu\in Q, saturation asks whether (V(Nλ)⊗V(Nμ)⊗V(Nν))G≠0(V(N\lambda)\otimes V(N\mu)\otimes V(N\nu))^G\ne0 for some positive integer NN forces (V(λ)⊗V(μ)⊗V(ν))G≠0(V(\lambda)\otimes V(\mu)\otimes V(\nu))^G\ne0. Knutson and Tao proved this for type A. Kapovich and Millson conjectured it for all simply laced groups. Known before this work: Spin(8)\mathrm{Spin}(8) (Kapovich-Kumar-Millson), Spin(10)\mathrm{Spin}(10) and Spin(12)\mathrm{Spin}(12) (Kiers), and saturation factors two or four for classical groups (Kapovich-Millson, Belkale-Kumar, Sam, Hong-Shen), which do not give the factor-one statement. Does saturation with factor one hold for Spin(2n)\mathrm{Spin}(2n) for every nn?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Representation theory: tensor product multiplicities and saturation
Posed by
M. Kapovich and J. J. Millson, Structure of the tensor product semigroup, Asian J. Math. 10 (2006), Conjecture 1.4(1); see also Kapovich-Kumar-Millson (2009), Conjecture 1.5
Year posed
2006
Years open
20y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for G=Spin(2n)G=\mathrm{Spin}(2n), n≥2n\ge2, and dominant integral λ,μ,ν\lambda,\mu,\nu with λ+μ+ν∈Q(Dn)\lambda+\mu+\nu\in Q(D_n), and every positive integer NN, (V(λ)⊗V(μ)⊗V(ν))G≠0(V(\lambda)\otimes V(\mu)\otimes V(\nu))^G\ne0 if and only if (V(Nλ)⊗V(Nμ)⊗V(Nν))G≠0(V(N\lambda)\otimes V(N\mu)\otimes V(N\nu))^G\ne0. This settles the type-D part of the simply laced saturation conjecture, extending the known cases Spin(8),Spin(10),Spin(12)\mathrm{Spin}(8),\mathrm{Spin}(10),\mathrm{Spin}(12). Not shown: the exceptional types E6,E7,E8E_6,E_7,E_8, non-simply-laced groups, or any statement about multiplicities beyond nonvanishing.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Kapovich-Millson Conjecture 1.4(1); the theorem states factor-one saturation for Spin(2n)\mathrm{Spin}(2n), n≥2n\ge2, under the root-lattice congruence, including half-integral spin weights. The proof was not refereed. It constructs integral schedules from real Littelmann-type paths, deforms switch coefficients to integrality, and produces tensor intertwiners through exterior and spin columns, with a preprojective-module step removing zero blocks. Exceptional simply laced types are not addressed. No Lean formalization is supplied for this family.

Sources

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