For the context for this post, see yesterday's.

### Index

To compute the index we use lemma 2.19 and remark 2.20 of the Kuznetsov–Polishchuk paper. Combined they say the following.

Lemma Let $\beta$ be the simple root corresponding to the chosen maximal parabolic subgroup $P$, and $\xi$ the associated fundamental weight. Let $\overline{\beta}$ be the maximal root of the same length as $\beta$ such that the coefficient of $\beta$ in the expression of $\overline{\beta}$ is 1. Then the index of $G/P$ equals \begin{equation*} i_{G/P}=\frac{(\rho,\beta+\overline{\beta})}{(\xi,\beta)}. \end{equation*}

The Sage code for this is

By accident I ran into these indices in the literature, they are also available in the proposition on page 124 of Akhiezer's Lie group actions in complex analysis.

$P_1$ $P_2$ $P_3$ $P_4$ $P_5$ $P_6$ $P_7$ $P_8$ 2 3 4 4 5 5 6 6 6 7 7 7 8 8 8 8 ... 3 4 5 4 6 7 6 5 8 9 8 7 6 10 11 10 9 8 7 12 13 12 11 10 9 8 14 ... 6 5 8 7 6 8 10 9 8 7 10 12 11 10 9 8 12 ... 12 11 9 7 17 14 11 8 10 13 18 23 17 13 9 11 14 19 29 8 5 7 11 5 3

Maybe in the future I have something useful to say about these derived categories, who knows.

It's not quite true that $\mathrm{B}_i=\mathrm{C}_i$ for the index, one has to reorder the indices a bit, but I didn't want to change the layout of the table compared to the previous ones. The table is given for $\mathrm{B}_i$. For $\mathrm{C}_i$ one has to put the entry for $P_i$ at the beginning, and shift all other entries to the right.