= B2 adjoint crystal
{c}
{title2=$B(2\Lambda_2)$}
With long-first <simple roots> $\alpha_1=\varepsilon_1-\varepsilon_2$, $\alpha_2=\varepsilon_2$, the <Adjoint representation> has <highest weight> $\varepsilon_1+\varepsilon_2=2\Lambda_2$. Its <crystal basis> has one vertex $x_\beta$ for every root and two distinct zero-weight vertices $z_1,z_2$. The color-one strings are $x_{\varepsilon_1}\to x_{\varepsilon_2}$, $x_{\alpha_1}\to z_1\to x_{-\alpha_1}$, and $x_{-\varepsilon_2}\to x_{-\varepsilon_1}$. The color-two strings are $x_{\varepsilon_1+\varepsilon_2}\to x_{\varepsilon_1}\to x_{\varepsilon_1-\varepsilon_2}$, $x_{\varepsilon_2}\to z_2\to x_{-\varepsilon_2}$, and $x_{-\varepsilon_1+\varepsilon_2}\to x_{-\varepsilon_1}\to x_{-\varepsilon_1-\varepsilon_2}$. All other edges vanish. One construction takes the component generated by $1\otimes2$ in the tensor square of the <crystal of the defining odd-orthogonal representation>; the <crystal tensor-product rule> produces exactly these ten vertices. The equivariant isomorphism $\Lambda^2\mathbb C^5\cong\mathfrak{so}_5$ identifies its highest weight and representation.
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