Canonical class of partial flag variety

Question

Let $F(d_1,d_2,\ldots,d_k;n )$ be the variety of all flags $\mathbb A^{d_1} \subset\mathbb A^{d_2}\subset\ldots\subset \mathbb A^{d_k}\subset \mathbb A^{n}$. This variety has the natural maps to Grassmann varieties: $$ f_i:F(d_1,d_2,\ldots,d_k; n) \to F(d_i;n), $$ and pull-backs $f_i^*\mathcal O(1)$ from Grassmannians give a basis of $\mathrm{Pic}(F(d_1,\ldots,d_k;n))$. Or, combinatorially, the pullbacks of the unique Shubert divisors on Grassmannians are Shubert divisors on $F(d_1,\ldots,d_k;n)$ and they generate the Picard group.

How can one calculate the canonical class of $F(d_1,\ldots, d_k; n)$ in these terms?

Or, one can ask more general question.

Let $G/P$ be a projective homogeneous space. Here $P\subset G$ is parabolic and $G/P$ can be described by a system of roots (for $G$) and its subsystem (for $P$). How can one calculate the canonical class of $G/P$ in this terms?

Answer 1

If $G$ is simply connected (so, $SL_n$ in your less general question), then every line bundle $\mathcal L$ is uniquely of the form $G \times^B \mathbb C_\lambda$ where $\mathbb C_\lambda$ is the $T$-irrep with weight $\lambda$. From there, you can figure out $\lambda$ from $\mathcal L$ by looking at the $T$-weight over the basepoint.

The tangent space at the basepoint is $\mathfrak g/\mathfrak p$, and the $T$-weights are thus $\Delta^G_- \setminus \Delta^L_-$, where $L$ is the Levi subgroup of $P$. The $T$-weight on the fiber of the canonical bundle is thus the sum of $\Delta^G_+ \setminus \Delta^L_+$.

In the $G=SL_n$ case (I haven't worked out other cases), the coefficient in the anticanonical class of the $i$th Schubert divisor (here $i$ is a simple root not in $L$) is $$1+\max \{ |[j,k] \subseteq [1,n-1]| : \text{ the only simple root in $[j,k]$ in $L$ is }i\}.$$ For example, in $SL_n/B$ this interval is $[i,i]$ for each $i$ and so each appears with coefficient $2$. (Which fits with $\sum \Delta^G_+ = 2\rho = 2\sum_{i=1}^{n-1} [X_{r_i}]$.) On $Gr(k,n)$ this interval is $[1,n-1]$ for $i=k$ and so the anticanonical class is $\mathcal O(n)$.