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.

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.