= Solution
For type $B_n$, the <spinor representation> has highest weight
$$
\omega_n=\frac12(\varepsilon_1+\cdots+\varepsilon_n)
$$
and its weights are the $2^n$ sign vectors
$$
\frac12\sum_{j=1}^n i_j\varepsilon_j,\qquad i_j\in\{\pm1\}.
$$
Each weight has multiplicity one. Along the simple root $\alpha_j=\varepsilon_j-\varepsilon_{j+1}$, the <Kashiwara operator> $\widetilde e_j$ 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 $\alpha_n=\varepsilon_n$, $\widetilde e_n$ 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 is
$$
\begin{gathered}
(-,-,-)\xrightarrow{3}(-,-,+)
\xrightarrow{2}(-,+,-),\\
(-,+,-)\xrightarrow{1}(+,-,-),\qquad
(-,+,-)\xrightarrow{3}(-,+,+),\\
(+,-,-)\xrightarrow{3}(+,-,+),\qquad
(-,+,+)\xrightarrow{1}(+,-,+),\\
(+,-,+)\xrightarrow{2}(+,+,-)
\xrightarrow{3}(+,+,+).
\end{gathered}
$$
The <tensor product of crystals> has four highest-weight connected components, of highest weights
$$
0,\qquad\omega_1,\qquad\omega_2,\qquad2\omega_3.
$$
Consequently, for the eight-dimensional spin representation $S$ of $\mathfrak{so}_7$,
$$
S\otimes S
\cong
\mathbb C\oplus V(\omega_1)\oplus V(\omega_2)\oplus V(2\omega_3),
$$
with dimensions
$$
64=1+7+21+35.
$$
Equivalently these summands are $\Lambda^k(\mathbb C^7)$ for $0\leq k\leq3$.
Back to article page
