= A2 defining tensor crystals
{c}
For the defining representation of the <special linear Lie algebra> $\mathfrak{sl}_3$, the <crystal basis> is $1\xrightarrow{1}2\xrightarrow{2}3$, with weights $\varepsilon_i$ restricted to trace-zero diagonal matrices. Its dual is $\bar3\xrightarrow{2}\bar2\xrightarrow{1}\bar1$, with weights $-\varepsilon_i$. Using the <crystal tensor-product rule> with lowering on the first factor when $\varphi_i(a)>\varepsilon_i(b)$, the tensor square has highest vertices $1\otimes1$ and $1\otimes2$, of <highest weights> $2\omega_1,\omega_2$, generating components of six and three vertices. The defining tensor dual has highest vertices $1\otimes\bar3$ and $1\otimes\bar1$, generating components of eight and one vertices and weights $\omega_1+\omega_2,0$. These encode the <symmetric square> plus <exterior square>, and the trace-free endomorphisms plus scalar identity, respectively. A crystal vertex labelled by a tensor pair need not represent that same undeformed ordinary simple tensor; the actual invariant vector is $\sum_i e_i\otimes e_i^*$.
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