Let $G$ be a simple complex Lie group with Borel subgroup $B$. A subset $\Delta_P$ of the set of simple roots $\Delta = \{\alpha_1, \dots, \alpha_r\}$ determines a parabolic subgroup $B \subset P \subset G$. For example, if $G = SL(4)$ with simple roots $\{\alpha_1, \alpha_2, \alpha_3\}$ and $\Delta_P = \{ \alpha_1, \alpha_3\}$, then we get $P$ such that $G/P = Gr(2,4)$.

If we identify $H^2(G/P)$ with the lattice generated by the fundamental weights $\{w_{\alpha} \mid \alpha \in \Delta \setminus \Delta_P \}$ (by considering the first Chern class of the line bundle of given weights), the anticanonical class $-K$ of $G/P$ is known to be the sum of the positive roots not contained in $\mathbb{Z} \Delta_P$.

Is there some meaning of the coefficients of $-K$ with respect to fundamental weights?

If $P=B$, then $\Delta_P$ is empty and $-K = \sum (\text{positive roots}) = 2\sum w_{\alpha}$, so all coefficients are two. If we take $G=SL(4)$ and $\Delta_P = \{\alpha_3\}$, then we get $-K = 2w_{\alpha_1} + 3w_{\alpha_2}$.

I could not find rules even for simple examples. But since fundamental weights form a natural basis for $H^2(G/P)$, I wonder if there is some meaningful formula.


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