Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 第 281 页 · Chapter 6，Section 6.2.2，Proposition 6.2.3
Proposition 6.2.3. ($\dim V = 2l$) The half-spin representations $\pi^\pm$ of $\mathfrak{so}(V,\beta)$ are irreducible with highest weights $\varpi_\pm = (\varepsilon_1 + \dots + \varepsilon_{l-1} \pm \varepsilon_l)/2$. The weights are $(\pm\varepsilon_1 \pm \dots \pm \varepsilon_l)/2$ (an even number of minus signs for $\pi^+$ and an odd number of minus signs for $\pi^-$), and each weight has multiplicity one.