Littlewood--Richardson coefficients stabilise with rank

Suppose $c_{\lambda \mu }^{\nu }\mathrm{\ne }0$ and let $U$ be an LR tableau of shape $\nu \mathrm{/}\lambda$ and content $\mu$. The containment $[\lambda ]\subseteq [\nu ]$ and the size identity $\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$ are part of [F1]. For the first row, read from right to left: the reading word of $U$ begins with the entries $b_1\ge b_2\ge \cdots \ge b_k$ of the first row (weak increase becomes weak decrease read right to left), where $k={\nu }_1-{\lambda }_1$ is the number of boxes of the first row of the skew diagram. If $k>0$, the lattice condition at the first letter forces $b_1=1$: otherwise that prefix has one $b_1$ and no $b_1-1$. Hence $b_j=1$ for every $j$; if $k=0$, the desired inequality holds immediately, so all first-row entries equal $1$ and $k\le {\mu }_1$ because $U$ contains only ${\mu }_1$ copies of $1$. Hence ${\nu }_1-{\lambda }_1\le {\mu }_1$, i.e. ${\nu }_1\le {\lambda }_1+{\mu }_1$.

Row bound. Suppose $\nu$ has a box in row $i>\mathrm{\ell }(\lambda )+\mathrm{\ell }(\mu )$. Since $i>\mathrm{\ell }(\lambda )$, we have ${\lambda }_j=0$ for every $j>\mathrm{\ell }(\lambda )$; since row $i$ occurs in the partition $\nu$, we also have ${\nu }_j\ge 1$ for every $j\le i$. Thus each row $j=\mathrm{\ell }(\lambda )+1,\dots ,i$ contributes a box in column $1$ to $\nu \mathrm{/}\lambda$, giving at least $i-\mathrm{\ell }(\lambda )$ boxes in that column. Strict increase down the column makes their entries distinct, and all entries lie in $\{1,\dots ,\mathrm{\ell }(\mu )\}$ because the tableau has content $\mu$ [F1]; hence $i-\mathrm{\ell }(\lambda )\le \mathrm{\ell }(\mu )$, contradicting $i>\mathrm{\ell }(\lambda )+\mathrm{\ell }(\mu )$. Therefore $\mathrm{\ell }(\nu )\le \mathrm{\ell }(\lambda )+\mathrm{\ell }(\mu )$.

Part (ii) for the tensor product is exactly [F3], applied at each rank $r\ge \max (1,\mathrm{\ell }(\lambda ),\mathrm{\ell }(\mu ))$; the coefficients appearing are the rank-independent tableau counts $c_{\lambda \mu }^{\nu }$ of Littlewood--Richardson tableaux and coefficients, so they are the same for every such $r$. If $r\ge \mathrm{\ell }(\lambda )+\mathrm{\ell }(\mu )$, then every $\nu$ with $c_{\lambda \mu }^{\nu }\mathrm{\ne }0$ satisfies $\mathrm{\ell }(\nu )\le r$ by step 1.2, so no coefficient disappears when the rank is lowered to $r$ from a larger rank; equivalently no coefficient visible at a larger rank is lost.

The character identity is the character form of the decomposition in [F3], with the convention $s_{\nu }(x_1,\dots ,x_r)=0$ for $\mathrm{\ell }(\nu )>r$; it holds for every $r\ge \max (1,\mathrm{\ell }(\lambda ),\mathrm{\ell }(\mu ))$ by [F3]; once $r\ge \mathrm{\ell }(\lambda )+\mathrm{\ell }(\mu )$, the set of partitions with nonzero coefficients and those coefficients are independent of $r$ by step 2.1. The Schur polynomials themselves are evaluated in the rank-dependent variables $x_1,\dots ,x_r$.