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

Theorem 1.1: for $G=\mathrm{Spin}(2n)$ , $n\ge2$ , and dominant integral $\lambda,\mu,\nu$ with $\lambda+\mu+\nu\in Q(D_n)$ , and every positive integer $N$ , $(V(\lambda)\otimes V(\mu)\otimes V(\nu))^G\ne0$ if and only if $(V(N\lambda)\otimes V(N\mu)\otimes V(N\nu))^G\ne0$ . This settles the type-D part of the simply laced saturation conjecture, extending the known cases $\mathrm{Spin}(8),\mathrm{Spin}(10),\mathrm{Spin}(12)$ . Not shown: the exceptional types $E_6,E_7,E_8$ , non-simply-laced groups, or any statement about multiplicities beyond nonvanishing.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

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Registry verification: unreviewed · announcement · partial

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

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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 $\mathrm{Spin}(2n)$ , $n\ge2$ , 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.

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

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Kapovich and Millson, Structure of the tensor product semigroup (arXiv:math/0508186)

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

Kapovich and Millson, Structure of the tensor product semigroup (arXiv:math/0508186) evidence for this event

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

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