1
$\begingroup$

I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. https://www.ias.ac.in/article/fulltext/pmsc/100/01/0045-0052.

There are a few notions and notations in the paper that I don't understand. My questions are :

(1) In page 3 of the paper (printed page 47), in Section 2, the authors define certain parabolic subgroups $Q_i$ inductively as $Q_1=P_1$ and $Q_{i+1}=Q_i \cap P_{i+1}$ . What I don't understand is: What are these $P_i$ s ?

(2) In Theorem 2.1, they use the notion of "equidimensionality" of the restriction of projection maps. What does this being equidimensional mean ?

(3) In Theorem 2.1, what does $W(Q_i)$ mean ? Is it the Weyl subgroup corresponding to $Q$ i.e. is $W(Q_i) \cong N_{Q_i} (T)/T$ ?

It would be highly appreciated if someone could clarify my doubts. Thanks.

$\endgroup$
17
  • 2
    $\begingroup$ I would guess that the P_i’s are maximal parabolics. $\endgroup$ May 6, 2019 at 20:09
  • 3
    $\begingroup$ @NajibIdrissi In some versions of the English language (notably in India), "doubt" just means "question". $\endgroup$ May 6, 2019 at 20:09
  • 1
    $\begingroup$ I think they’re the “standard” maximal parabolics corresponding to the nodes of the Dynkin diagram. $\endgroup$ May 6, 2019 at 20:16
  • 3
    $\begingroup$ The survey paper Lakshmibai, Musili, and Shesadri - Geometry of $G/P$ is more careful. It says: (1) $P_i$ is the maximal parabolic corresponding to the fundamental weight $\pi_i$ (p. 433); (2) $W_Q = N_Q(T)/T$ (p. 432), as you guess. It does not mention equi-dimensionality, but an algebraic-geometer colleague says that the unusual useage probably suggests that there is some $d$ such that every component of every fibre has dimension $d$. $\endgroup$
    – LSpice
    May 7, 2019 at 21:25
  • 1
    $\begingroup$ I think that $P_i$ should be the block upper-triangular matrices of the form $\begin{pmatrix} A & X \\ 0 & B \end{pmatrix}$, where $A$ is $i \times i$ and $B$ is $(n - i) \times (n - i)$. $\endgroup$
    – LSpice
    May 7, 2019 at 21:42

1 Answer 1

1
$\begingroup$

(3) My guess would be that $W(Q_i)$ is the Weyl group of the Levi part of $Q_i$.

(2) My guess would be that it means that the domain and codomain have the same dimension.

(1) I have no guess.

Bottom line: This is bad writing.

$\endgroup$
2
  • 1
    $\begingroup$ thanks for your view ... it might be bad writing, but every single paper or monograph on smoothness or singular loci of Schubert varieties that I've seen, does refer to this paper, and I'm really trying to understand this paper :-( $\endgroup$
    – user102248
    May 6, 2019 at 20:03
  • 1
    $\begingroup$ $\DeclareMathOperator\Int{Int}$Note that there is no difference between the Weyl group of a parabolic $Q$, in the sense proposed by @user102248, and the Weyl group of a Levi component. Indeed if $Q = M U$ is a Levi decomposition, and if $m \in M$ and $u \in U$ are such that $m u$ normalises $T$, then the image of $\Int(m u)T = T$ in $Q \to Q/U$ is $\Int(m)T = T$, so $m \in N_M(T)$; so then $u \in N_U(T)$, but, if $u \ne 1$, then there is some $t \in T$ that doesn't centralise $u$, so $\Int(u)t^{-1} = t^{-1}\cdot[t, u] \in T$ has non-trivial component $[t, u] \in U$, which is a contradiction. $\endgroup$
    – LSpice
    May 7, 2019 at 22:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.