I'm reading through the two chapters in Fulton and Harris on the representation theory of $\mathfrak{sl}(3,\mathbb{C})$, in preparation for lecturing on them this week. I'll use F&H's notation, so that $\Gamma_{a,b}$ is the irreducible representation of highest weight $a e_1 - b e_3$ (with $e_i$ the character taking a diagonal matrix to its $(i,i)$ entry).

In the discussion of the multiplicities of the weights occurring in $\Gamma_{a,b}$, it seems to me that there's a gap -- so I am wondering what I'm missing here (or if perhaps there really is a gap). The issue arises at the sentence "To begin with..." on page 184, so let me give a quick summary of the argument to that point. Let $V$ be the standard representation of $\mathfrak{sl}(3,\mathbb{C})$ on $\mathbb{C}^3$, and $V^*$ its dual. The weights of ` $W :=\mathrm{Sym}^a V \otimes \mathrm{Sym}^b V^*$ lie on shrinking concentric hexagons $H_0,H_1,\ldots$ (with $H_0$ the outermost hexagon, and with $H_i$ degenerating to a triangle for $i$ sufficiently large). The multiplicities of $W$ are constant along each hexagon. For the argument to go through it suffices to show that the only highest vectors of $W$ are the unique ones (up to scaling) that occur at the unique dominant vertices of each $H_i$ up to $i = \textrm{min}(a,b)$, each of which contributes a $\Gamma_{a-i,b-i}$ to $W$.

The claim in the sentence "To begin with..." is that this follows for essentially combinatorial reasons: if there were a highest weight vector of weight $\alpha$, where $\alpha$ lies on $H_i$ but is not the dominant vertex, then "the multiplicity of $\alpha$ in $W$ would be strictly greater than [the multiplicity of the dominant vertex of $H_i$]." But why? Obviously this is correct if one considers only the contributions from highest weight vectors lying on $H_i$ or in its interior. But it seems a priori possible that some $\Gamma_{a-j,b-j}$ with $j < i$ (i.e., the irreducible constituent coming from one of the higher weight vectors on a hexagon lying outside $i$) could contribute a larger multiplicity to the dominant vertex of $H_i$ than to $\alpha$. If you haven't yet proved that the multiplicities of $\Gamma_{a,b}$ are also constant along the hexagons, I don't immediately see how this argument is going to give an inductive proof of that claim. Am I missing something elementary here, or does one have to work harder than F&H claim in order to finish the argument?