= Crystal of the defining odd-orthogonal representation
The defining representation of $\mathfrak{so}_{2n+1}$ has <weights> $\varepsilon_1,\ldots,\varepsilon_n,0,-\varepsilon_n,\ldots,-\varepsilon_1$. Its <crystal basis> is a chain on these vertices, with lowering-edge colors $1,\ldots,n-1,n,n,n-1,\ldots,1$. The short-root string has length two through zero. With the <crystal tensor-product rule>, for $n\ge2$ the tensor square has highest vertices $1\otimes1$, $1\otimes2$ and $1\otimes\bar1$, corresponding to the traceless <symmetric square>, the <exterior square> and the <trivial Lie algebra representation>. Their <dimensions> are $(2n+1)(n+1)-1$, $n(2n+1)$ and one.
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