The reduced (stable) Kronecker coefficient ${\bar{g}}_{\alpha ,\beta }^{\gamma }$

For partitions $\alpha ,\beta ,\gamma$ define the reduced (stable) Kronecker coefficient ${\bar{g}}_{\alpha ,\beta }^{\gamma }$ to be the eventual value of the padded sequence $n\mapsto g_n(\alpha ,\beta ,\gamma )$ furnished by Murnaghan stability for Kronecker coefficients; equivalently ${\bar{g}}_{\alpha ,\beta }^{\gamma }:={\lim }_{n\to \mathrm{\infty }}{g}_{\alpha [n],\beta [n]}^{\gamma [n]},$ where the sequence is defined for $n\ge \max \{\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1,\mathrm{\mid }\beta \mathrm{\mid }+{\beta }_1,\mathrm{\mid }\gamma \mathrm{\mid }+{\gamma }_1\}$ and set to $0$ below by the convention in Murnaghan stability for Kronecker coefficients (the size-indexed coefficients of Padded Kronecker coefficient sequences themselves are defined only on that common domain; see also Kronecker coefficients and the internal tensor product).

Well-definedness. The sequence is eventually constant by the Murnaghan stability theorem, and the eventual value of an eventually constant sequence of nonnegative integers is a unique nonnegative integer; no choice is involved. Two immediate conventions: (i) ${\bar{g}}_{\alpha ,\beta }^{\gamma }={\bar{g}}_{\beta ,\alpha }^{\gamma }$ for all $\alpha ,\beta ,\gamma$, because $V_{\mu }\otimes V_{\nu }\cong V_{\nu }\otimes V_{\mu }$ and the Kronecker coefficients are symmetric in their two lower indices; (ii) for fixed $\alpha ,\beta$ only finitely many $\gamma$ have ${\bar{g}}_{\alpha ,\beta }^{\gamma }\mathrm{\ne }0$, because the finite set $S(\alpha ,\beta )$ of the stability theorem carries the whole support. The values used later and verified in the source's introduction for $\alpha =\beta =(2)$ are ${\bar{g}}_{(2),(2)}^{(2)}=2$ and ${\bar{g}}_{(2),(2)}^{(4)}=1$; they are read off from the displayed stable row and are not used as logical inputs anywhere on this page.