 Solution

For type $B_n$, the spinor representation has highest weightand its weights are the $2^n$ sign vectorsEach weight has multiplicity one. Along the simple root ${\alpha }_j={\epsilon }_j-{\epsilon }_{j+1}$, the Kashiwara operator ${\tilde{e}}_j$ can raise a weight exactly when $(i_j,i_{j+1})=(-1,+1)$, when it replaces that pair by $(+1,-1)$. For the short root ${\alpha }_n={\epsilon }_n$, ${\tilde{e}}_n$ replaces a final $-1$ by $+1$. This proves the stated crystal by the root-string property of a crystal.

For $n=3$, the complete list of raising edges isThe tensor product of crystals has four highest-weight connected components, of highest weightsConsequently, for the eight-dimensional spin representation $S$ of ${\mathfrak{so}}_7$,with dimensionsEquivalently these summands are ${\Lambda }^k({\mathbb{C}}^7)$ for $0\le k\le 3$.