= Solution
The ambient <group ring of a weight lattice> is
$$
\mathbb Z[\Lambda_W]=\left\{\sum_{\lambda\in\Lambda_W}a_\lambda e(\lambda):
a_\lambda\in\mathbb Z,\text{ with finite support}\right\},
\qquad e(\lambda)e(\mu)=e(\lambda+\mu).
$$
Addition is coefficientwise, and multiplication uses addition in the <weight lattice>. For a finite-dimensional <Lie algebra representation> $V=\bigoplus_\lambda V_\lambda$, its <formal character> is
$$
\boxed{\operatorname{char}V=\sum_\lambda(\dim V_\lambda)e(\lambda).}
$$
The <formal character> records the <dimensions> of all <weight spaces>. The ambient lattice ring should be distinguished from the <representation ring of a semisimple Lie algebra>, which identifies with its Weyl-invariant subring.
For the $\mathfrak{sl}_3$ module $\Gamma_{1,2}$, $L_1+L_2+L_3=0$ and the <highest weight> is $L_1-2L_3=\omega_1+2\omega_2$. The decomposition $V\otimes\operatorname{Sym}^2V^*=\Gamma_{1,2}\oplus V^*$ gives
$$
\operatorname{char}\Gamma_{1,2}
=\left(\sum_i e(L_i)\right)\left(\sum_{j\leq k}e(-L_j-L_k)\right)-\sum_i e(-L_i).
$$
Collecting equal <weights> yields
$$
\boxed{\operatorname{char}\Gamma_{1,2}
=\sum_{i\ne j}e(L_i-2L_j)+\sum_{i=1}^3e(2L_i)+2\sum_{i=1}^3e(-L_i).}
$$
The six first <weights> and three second <weights> have multiplicity one; each of the three <weights> $-L_i$ has multiplicity two. Their total is $6+3+6=15$, as required.
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