The nonnegative expansion of a reduced Kronecker coefficient

Expanding Brion's pairing. Insert the four expansions of [F2] into the pairing of [F1]: $\sigma [XY]={\sum }_{\rho }{s}_{\rho }[X]s_{\rho }[Y]$, and by the coproduct identity at the sum $XY+(X+Y)$, $s_{\gamma }[XY+X+Y]={\sum }_{\mu ,\nu }{c}_{\mu \nu }^{\gamma }{s}_{\mu }[XY]s_{\nu }[X+Y]$, with $s_{\mu }[XY]={\sum }_{\lambda ,\eta }{g}_{\lambda \eta }^{\mu }{s}_{\lambda }[X]s_{\eta }[Y]$ and $s_{\nu }[X+Y]={\sum }_{\theta ,\pi }{c}_{\theta \pi }^{\nu }{s}_{\theta }[X]s_{\pi }[Y]$. Multiplying out and pairing with $s_{\alpha }[X]s_{\beta }[Y]$ using orthonormality [F4] extracts $⟨s_{\alpha },s_{\rho }{s}_{\lambda }{s}_{\theta }{⟩}_H=c_{\rho ,\lambda ,\theta }^{\alpha }$ and $⟨s_{\beta },s_{\rho }{s}_{\eta }{s}_{\pi }{⟩}_H=c_{\rho ,\eta ,\pi }^{\beta }$ by [F3], so ${\bar{g}}_{\alpha ,\beta }^{\gamma }={\sum }_{\mu ,\nu ,\lambda ,\eta ,\rho ,\theta ,\pi }{c}_{\mu \nu }^{\gamma }{c}_{\theta \pi }^{\nu }{g}_{\lambda \eta }^{\mu }{c}_{\rho \lambda \theta }^{\alpha }{c}_{\rho \eta \pi }^{\beta }$.

Collapsing the middle index. By the iterated identity of [F3], ${\sum }_{\nu }{c}_{\mu \nu }^{\gamma }{c}_{\theta \pi }^{\nu }=c_{\mu ,\theta ,\pi }^{\gamma }$; substituting into step 1.1 and relabelling the summation indices gives the displayed expansion. The threefold coefficients are symmetric in their three subscripts by their definition as coefficients of the commutative product $s_{\mu }{s}_{\theta }{s}_{\pi }$, so the displayed index pattern is the same as the source's equation (8) with the four factors assigned to the four faces of the index tetrahedron in a different order. All coefficients are nonnegative integers, so no cancellation occurs in the sum.

Consequences (2) and (4). A nonzero summand has all four coefficients positive (they are nonnegative integers), and then the degree equations of (2) follow from the size constraints in [F4]: $\mathrm{\mid }\gamma \mathrm{\mid }=\mathrm{\mid }\mu \mathrm{\mid }+\mathrm{\mid }\theta \mathrm{\mid }+\mathrm{\mid }\pi \mathrm{\mid }$, $\mathrm{\mid }\alpha \mathrm{\mid }=\mathrm{\mid }\rho \mathrm{\mid }+\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\theta \mathrm{\mid }$, $\mathrm{\mid }\beta \mathrm{\mid }=\mathrm{\mid }\rho \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\pi \mathrm{\mid }$ and $\mathrm{\mid }\mu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }\eta \mathrm{\mid }$; conversely, any summand with all four coefficients positive contributes positively, giving the criterion (1) since the total is a sum of nonnegative integers.

Consequence (3), finiteness. If $c_{\rho ,\lambda ,\theta }^{\alpha }>0$ then, by the iterated form of [F3], there is a partition $\varphi$ with $c_{\rho ,\lambda }^{\varphi }>0$ and $c_{\varphi ,\theta }^{\alpha }>0$; by [F4] this forces $\rho \subseteq \varphi \subseteq \alpha$, $\lambda \subseteq \varphi \subseteq \alpha$ and $\theta \subseteq \alpha$, so $\rho ,\lambda ,\theta$ range over the finitely many subpartitions of $\alpha$. Similarly $c_{\rho ,\eta ,\pi }^{\beta }>0$ forces $\rho ,\eta ,\pi \subseteq \beta$, and $g_{\lambda ,\eta }^{\mu }>0$ forces $\mathrm{\mid }\mu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }$, so $\mu$ ranges over the finitely many partitions of the bounded number $\mathrm{\mid }\lambda \mathrm{\mid }$. Hence only finitely many six-tuples contribute, and the expansion is a finite sum; the proof uses finite expansions and no choice principle.