= Solution
Use <Dynkin labels> $(a,b)$ for the <highest weight> $a\omega_1+b\omega_2$ of the complex <special linear Lie algebra> $\mathfrak{sl}_3$. The <A2 root system> has $\alpha_1=(2,-1)$ and $\alpha_2=(-1,2)$ in these coordinates. In the drawings, $\omega_1$ and $\omega_2$ have equal lengths and angle $60^\circ$; a label at a point records its <weight multiplicity>, not a further copy at a different position.
The defining <fundamental representation> has the three <weights>
$$
\boxed{\Gamma_{1,0}:\quad(1,0),\ (-1,1),\ (0,-1),\quad\text{each of multiplicity }1.}
$$
For $\Gamma_{2,1}$, lower from its <highest weight> by the <simple roots>, retaining multiplicities. One convenient way to calculate them is the <sl3 interlacing character formula>: for shape $(3,1,0)$ the integer patterns satisfy $1\le p\le3$, $0\le q\le1$, $q\le r\le p$, and contribute the <weight>
$$
(r-(p+q-r),\ (p+q-r)-(4-p-q)).
$$
Enumerating these patterns gives the <weight diagram>
$$
\begin{array}{c|l}
\text{multiplicity}&\text{weights of }\Gamma_{2,1}\\\hline
2&(1,0),\ (-1,1),\ (0,-1)\\
1&(2,1),\ (3,-1),\ (2,-2),\ (1,-3),\ (-1,-2),\ (-2,0),\ (-3,2),\ (-2,3),\ (0,2).
\end{array}
$$
Its dimension is $3\cdot2+9=15$. The diagram below draws all twelve distinct positions, with the three inner multiplicities equal to two. The extra panel gives the <symmetric square> used in the calculation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-2-input-weight-diagrams.png]
{title=A2 weight diagrams for Gamma(2,1), the defining Gamma(1,0), and its symmetric square, with every weight multiplicity}
{height=620}
The six symmetric monomials in the defining <basis> give $S^2\Gamma_{1,0}=\Gamma_{2,0}$, with <weights>
$$
(2,0),\ (0,1),\ (1,-1),\ (-2,2),\ (-1,0),\ (0,-2),
$$
each occurring once. Thus \b[the tensor product has dimension]
$$
\boxed{\dim V=15\cdot6=90.}
$$
In a <tensor product of Lie algebra representations>, <weights> add and their multiplicities multiply. In terms of <formal characters>, $\operatorname{ch}V=\operatorname{ch}\Gamma_{2,1}\operatorname{ch}\Gamma_{2,0}$. Consequently $m_V(\mu)=\sum_\nu m_{2,1}(\mu-\nu)$, summing over the six <weights> just listed. To show the indicated dominant multiplicities explicitly, the contributions in that order are
$$
\begin{array}{c|rrrrrr|r}
\mu& (2,0)&(0,1)&(1,-1)&(-2,2)&(-1,0)&(0,-2)&m_V(\mu)\\\hline
(4,1)&1&0&0&0&0&0&1\\
(2,2)&1&1&0&0&0&0&2\\
(3,0)&2&1&1&0&0&0&4\\
(0,3)&1&1&0&1&0&0&3\\
(1,1)&2&2&1&1&1&0&7
\end{array}
$$
The <tensor-product weight diagram> below includes every position, and highlights these dominant <weights>. It also records the zero-weight multiplicity nine; that multiplicity is not a count of trivial summands.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-2-tensor-weight-diagram.png]
{title=All weights of the ninety-dimensional sl3 tensor product Gamma(2,1) tensor Sym2 Gamma(1,0), with dominant weights highlighted and multiplicities labelled}
{height=800}
Apply the <Weyl complete reducibility theorem> and subtract irreducible <formal characters> in decreasing <dominance order>. The multiplicities at these five dominant positions in the potential summands are
$$
\begin{array}{c|rrrrr|r}
& (4,1)&(2,2)&(3,0)&(0,3)&(1,1)&\text{dimension}\\\hline
\Gamma_{4,1}&1&1&2&1&2&35\\
\Gamma_{2,2}&0&1&1&1&2&27\\
\Gamma_{3,0}&0&0&1&0&1&10\\
\Gamma_{0,3}&0&0&0&1&1&10\\
\Gamma_{1,1}&0&0&0&0&1&8
\end{array}
$$
These entries can be obtained by the same interlacing enumeration or by <weight strings>. Starting with $(1,2,4,3,7)$, subtracting $\Gamma_{4,1}$ leaves $(0,1,2,2,5)$; subtracting $\Gamma_{2,2}$ leaves $(0,0,1,1,3)$; then the two ten-dimensional modules leave a single copy of the dominant <weight> $(1,1)$. This is <highest-weight character subtraction>. Therefore
$$
\boxed{V\cong\Gamma_{4,1}\oplus\Gamma_{2,2}\oplus\Gamma_{3,0}\oplus\Gamma_{0,3}\oplus\Gamma_{1,1}.}
$$
Every summand occurs once. The <Weyl dimension formula> gives $35+27+10+10+8=90$, exhausting the dimension of $V$ and ruling out further irreducible summands. Computing the complete <formal character> also leaves no residual <weight multiplicities>.
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