# Index of partial flag varieties

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.

$P_1$ | $P_2$ | $P_3$ | $P_4$ | $P_5$ | $P_6$ | $P_7$ | $P_8$ | |
---|---|---|---|---|---|---|---|---|

$\mathrm{A}_1$ | 2 | |||||||

$\mathrm{A}_2$ | 3 | |||||||

$\mathrm{A}_3$ | 4 | 4 | ||||||

$\mathrm{A}_4$ | 5 | 5 | ||||||

$\mathrm{A}_5$ | 6 | 6 | 6 | |||||

$\mathrm{A}_6$ | 7 | 7 | 7 | |||||

$\mathrm{A}_7$ | 8 | 8 | 8 | 8 | ||||

... | ||||||||

$\mathrm{B}_2=\mathrm{C}_2$ | 3 | 4 | ||||||

$\mathrm{B}_3=\mathrm{C}_3$ | 5 | 4 | 6 | |||||

$\mathrm{B}_4=\mathrm{C}_4$ | 7 | 6 | 5 | 8 | ||||

$\mathrm{B}_5=\mathrm{C}_5$ | 9 | 8 | 7 | 6 | 10 | |||

$\mathrm{B}_6=\mathrm{C}_6$ | 11 | 10 | 9 | 8 | 7 | 12 | ||

$\mathrm{B}_7=\mathrm{C}_7$ | 13 | 12 | 11 | 10 | 9 | 8 | 14 | |

... | ||||||||

$\mathrm{D}_4$ | 6 | 5 | ||||||

$\mathrm{D}_5$ | 8 | 7 | 6 | 8 | ||||

$\mathrm{D}_6$ | 10 | 9 | 8 | 7 | 10 | |||

$\mathrm{D}_7$ | 12 | 11 | 10 | 9 | 8 | 12 | ||

... | ||||||||

$\mathrm{E}_6$ | 12 | 11 | 9 | 7 | ||||

$\mathrm{E}_7$ | 17 | 14 | 11 | 8 | 10 | 13 | 18 | |

$\mathrm{E}_8$ | 23 | 17 | 13 | 9 | 11 | 14 | 19 | 29 |

$\mathrm{F}_4$ | 8 | 5 | 7 | 11 | ||||

$\mathrm{G}_2$ | 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.