Parabolic subgroups and Automorphism groups of Schubert varieties

Pinakinath Saha; S.Senthamarai Kannan

REVIEW 3 major objections 6 minor 12 references

Parabolic subgroups and Automorphism groups of Schubert varieties

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every proper parabolic subgroup of a simple algebraic group is the identity component of the automorphism group of some Schubert variety.

desk verdict Every proper parabolic in adjoint G is Aut0 of some Schubert variety—a new and plausible theorem, but the G2 cohomology step needs a fix and the descent citation needs to be checked. read the letter →

arxiv 1908.04768 v1 pith:DCAFW7I2 submitted 2019-08-13 math.AG math.COmath.GR

S.Senthamarai Kannan , Pinakinath Saha This is my paper · ORCID

classification math.AGmath.COmath.GR MSC 14M1514L3020G15

keywords SchubertvarietiesautomorphismgroupsparabolicsubgroupsflagalgebraicWeylgroupcohomologyofhomogeneousvectorbundlesadjointtype

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that, for any simple algebraic group $G$ of adjoint type over the complex numbers, every parabolic subgroup $P$ strictly containing a fixed Borel subgroup $B$ can be realized as the identity component of the algebraic automorphism group of some Schubert variety $X(w)$ inside the flag variety $G/B$. This answers an open question from earlier work on smooth Schubert varieties, where the equality was known exactly when a certain root condition holds. The proof constructs the needed Weyl group element $w$ explicitly and reduces the problem to a known classification of flag varieties with maximal automorphism group, except for three small cases that are checked by direct computation. The result shows that the family of Schubert varieties in $G/B$ is rich enough to encode every parabolic subgroup geometrically.

What carries the argument

The load-bearing object is the Schubert variety $X(w)$ in the full flag variety $G/B$, together with its identity component of automorphisms $\mathrm{Aut}_0(X(w))$. The main mechanism is a birational morphism $\pi: X(w) \to G/P'$, where $P'$ is a parabolic subgroup determined by $J' = -w_0(J)$; Brion's corollary turns this into an injective homomorphism $\pi_*: \mathrm{Aut}_0(X(w)) \to \mathrm{Aut}_0(G/P')$. When $\mathrm{Aut}_0(G/P') = G$, this injection forces $\mathrm{Aut}_0(X(w))$ to be a closed subgroup of $G$ containing $P$, and since $P$ is the stabilizer of $X(w)$, equality follows. For the three exceptional parabolics, the mechanism is instead a direct weight-space computation, using Demazure's lemma to show $H^0(w, \mathfrak{g}/\mathfrak{b}) = \mathfrak{g}$, which pins down the Lie algebra of the automorphism group.

What would settle it

Compute $\mathrm{Aut}_0(X(w))$ directly for the explicit $w$ given in one of the three exceptional cases — for example, $G$ of type $G_2$ with $w = s_1s_2s_1s_2$ — and check whether the group is strictly larger than $P_{\alpha_1}$; the theorem says it is exactly $P_{\alpha_1}$. Alternatively, find any parabolic $P$ for which the constructed $\pi_*$ fails to be injective on $\mathrm{Aut}_0(X(w))$; that would break the proof even if the theorem still held.

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Extended reading notes

Core claim

The central claim, Theorem 2.1, states: if $G$ is a simple algebraic group of adjoint type over $\mathbb{C}$, $B$ is a Borel subgroup, and $P$ is a parabolic subgroup with $B \subsetneq P$, then there exists $w \in W$ such that $P = \mathrm{Aut}_0(X(w))$, the identity component of the algebraic automorphism group of the Schubert variety $X(w)$. For $P = P_I$ with $\emptyset \neq I \subsetneq S$, the proof sets $J = S \setminus I$ and takes $w = (w_0^J)^{-1}$, the inverse of the minimal representative of the longest element $w_0$ in the parabolic Weyl group $W_J$. Then $P$ is precisely the stabilizer of $X(w)$ in $G$. A natural homomorphism $P \to \mathrm{Aut}_0(X(w))$ is injective because $w^{-1}(\alpha_0) < 0$ for the highest root $\alpha_0$. The authors then show $\mathrm{Aut}_0(X(w))$ cannot be larger than $P$: a birational morphism from $X(w)$ to a suitable partial flag variety $G/P'$ induces an injective homomorphism $\mathrm{Aut}_0(X(w)) \to \mathrm{Aut}_0(G/P')$, and for all but three explicitly listed pairs $(G,P)$, the known classification gives $\mathrm{Aut}_0(G/P') = G$, forcing $\mathrm{Aut}_0(X(w))$ to be a parabolic subgroup of $G$ contained in the stabilizer of $X(w)$, hence equal to $P$. The three remaining cases are handled by writing down explicit $w$ and computing the Lie algebra of $\mathrm{Aut}_0(X(w))$ via Demazure's cohomology lemma, obtaining $\mathrm{Aut}_0(X(w)) = P$ directly.

Load-bearing premise

The proof depends on the assertion that a birational morphism $\pi: X(w) \to G/P'$ with $\pi_*(\mathcal{O}_{X(w)}) = \mathcal{O}_{G/P'}$ induces an injective homomorphism $\pi_*: \mathrm{Aut}_0(X(w)) \to \mathrm{Aut}_0(G/P')$; if this injectivity fails for some constructed $w$, the conclusion that $\mathrm{Aut}_0(X(w))$ equals the stabilizer $P$ does not follow.

Editorial extensions

If this is right

Every proper parabolic subgroup of a simple algebraic group of adjoint type over $\mathbb{C}$ occurs as $\mathrm{Aut}_0(X(w))$ for some Schubert variety in the full flag variety.
The proof gives an explicit $w$ for each parabolic: $w = (w_0^J)^{-1}$ in the generic case, with explicit elements supplied in the three exceptional cases.
For Schubert varieties in type-$A$ Grassmannians $G/P_{\hat{\alpha}_r}$, the automorphism group is computed exactly as a parabolic subgroup $P_{J(w)}$, where $J(w)$ is read off from the gaps in the sequence indexing $w$.
In projective space $\mathbb{P}^n$, no proper parabolic subgroup of $\mathrm{PSL}(n+1,\mathbb{C})$ is the automorphism group of a Schubert variety; only the whole group occurs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit construction suggests that $w = (w_0^J)^{-1}$ may be the minimal-length Weyl group element realizing $P$; testing whether any shorter element works is a direct extension.
The three exceptional cases ($B_n$ with $P_{\alpha_n}$, $C_n$ with $P_{\alpha_1}$, $G_2$ with $P_{\alpha_1}$) stand apart because the partial flag variety $G/P'$ itself has a larger automorphism group than $G$; a uniform argument avoiding case analysis might expose why these parabolics are special.
Since the statement is formulated over $\mathbb{C}$ as an algebraic-geometric realization, the same result is likely to hold over other algebraically closed fields of characteristic zero by base change, though the cohomological arguments would need adaptation in positive characteristic.
The type-$A$ Grassmannian computation gives a precise combinatorial rule for $\mathrm{Aut}_0$ of a Schubert variety; this rule could be tested against known examples of non-smooth Schubert varieties, where the automorphism group is not predicted by smoothness alone.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The theorem is real. Section 3 plus Proposition 5.1 give a credible affirmative answer to the question Kannan posed in [10], and the partial flag results in Section 6 are a useful bonus. Most of the proof reduces to known classification results; the three exceptional cases are handled with actual weight-space computations, which is where the paper earns its keep.

What I like: Theorem 2.1 is exactly the statement the literature was missing. For B_n, C_n, and G2 the authors pick explicit w and compute H^0(w,g/b) by Demazure-style induction. Those computations are parameter-free and mostly checkable. Proposition 6.2, characterizing Aut0 of Grassmannian Schubert varieties in type A, is also new and follows the same pattern. No circularity: [10] is used for injectivity of P -> Aut0(X(w)), and that is an independent published result.

Now the soft spots, in order.

The G2 case has a real gap. In Case 3 of Proposition 5.1, the authors compute H^0(s2s1s2,g/b) = g and then write "Thus we have H^0(w,g/b)=H^0(s1,g)=g." The first equality is not justified. H^0(s1,-) applied to g as a B1-module is not obviously g; it needs the same kind of explicit check they did for the other braid steps. It may be true, and I suspect it is true, but the paper has to show it. This is the strongest reason for a conditional verdict.

The stress-test note about Section 3 is also fair. The paper asserts that [5, Cor. 2.2] turns the birational morphism pi: X(w) -> G/P' into a homomorphism pi_*: Aut0(X(w)) -> Aut0(G/P') just from pi_*O_X = O_Y. The corollary's hypotheses are not stated, and birational morphisms do not generally induce maps on automorphism groups. Blanchard's lemma or a descent statement may well apply here, but the reader cannot check it without hunting down a 2011 paper that is not quoted. Since this step is the bridge that makes Aut0(X(w)) a subgroup of G, it is load-bearing. I would ask the authors to quote the corollary and verify its hypotheses explicitly.

The black-box use of [1, Theorem 2] for Aut0(G/P')=G is standard in the area, so I would not insist on a proof, but a precise statement would help.

Bottom line: the main theorem is almost certainly correct. The strategy is sound, the computations are explicit, and the gaps are local and repairable. This deserves a serious referee and probably a conditional accept with a request to fix the G2 step and expand the two citations. I would bring it to reading group and would cite it if I worked on automorphism groups of Schubert varieties.

Referee Report

3 major / 6 minor

Summary. The paper proves that for a simple algebraic group G of adjoint type over C and any parabolic subgroup P of G properly containing a Borel subgroup B, there exists a Schubert variety X(w) in G/B whose identity component of the algebraic automorphism group equals P. The proof reduces the general case to three exceptional pairs via known results on automorphism groups of flag varieties and on smooth Schubert varieties, and then treats the three cases by explicit weight-space computations of H^0(w,g/b) using Demazure's lemma. A final section gives partial results for Schubert varieties in Grassmannians of type A_n, including a characterization of the parabolic subgroups arising in that setting.

Significance. If correct, Theorem 2.1 settles a natural converse question to earlier work of the first author, showing that every proper standard parabolic subgroup of G occurs as Aut0 of a Schubert variety. The proof is constructive: the element w is given explicitly in each case, and the cohomological arguments are concrete and parameter-free. The paper also clarifies the contrast with partial flag varieties, where the analogous statement fails. The main weakness is that two external results are used in a load-bearing way without their hypotheses being stated or verified, and one step in the G2 case is under-justified.

major comments (3)

[Section 3] The proof that a birational morphism π:X(w)→G/P' induces a homomorphism π_*:Aut0(X(w))→Aut0(G/P') cites [5, Corollary 2.2, p.45] after verifying π_*O_X=O_G/P'. This verification is not sufficient for an arbitrary birational morphism to induce a homomorphism on automorphism groups: for example, a birational contraction with several exceptional divisors admits automorphisms of the source that move one exceptional divisor to another and do not descend to the base. Since the injectivity of π_* is the only step that embeds Aut0(X(w)) into G, the authors should state the exact form of [5, Corollary 2.2] and verify that every hypothesis (e.g., any condition on fibers or equivariance) holds for the restriction of G/B→G/P' to X(w). Without this, the main proof in the non-exceptional cases is incomplete.
[Proposition 5.1, Case 3] The final step asserts H^0(w,g/b)=H^0(s1,g)=g for w=s1s2s1s2. The preceding computation determines H^0(s2s1s2,g/b) only as a T-module (by listing its weight spaces); it is not shown that this B-module is isomorphic to g as a B-module, which is what would be needed to apply H^0(s1,−) to g. Although the associated bundle L(g) on G/B is trivial and hence H^0(s1,g)=g holds for the B-module g, the argument as written needs an explicit justification that H^0(s2s1s2,g/b) carries the B-module structure of g, or a direct computation of H^0(s1,H^0(s2s1s2,g/b)). This is a load-bearing step in one of the three exceptional cases.
[Sections 3 and 5] The injectivity of φ_w:P→Aut0(X(w)) is justified by [10, Theorem 4.2(2), p.772], but the introduction states that theorem for smooth Schubert varieties in simply-laced groups. The proof of Theorem 2.1 applies it to arbitrary types and to possibly singular Schubert varieties (Remark 5.3 explicitly considers a singular example). Please quote the precise statement of [10, Theorem 4.2(2)] and confirm that its hypotheses are satisfied in the present generality; otherwise an alternative proof of the faithfulness of the P-action on X(w) should be supplied.

minor comments (6)

[Abstract] The phrase 'the connected component, containing the identity element of the group of all algebraic automorphisms' should be 'the identity component of the group of all algebraic automorphisms'.
[Introduction] The sentence 'If P=B, there is no such Schubert variety in G/B' would benefit from a one-line explanation, since the preceding discussion does not make the obstruction explicit.
[Section 5, Case 1] The case analysis for ⟨β,γ⟩=0 (and the conclusion that β+γ∈R− and γ=αn) is terse; a few more details of how Lemma 4.2 is being applied in each subcase would improve readability.
[Section 5, Case 2] In the display for H^0(sr...sn,g/b), the condition 'µ runs over all positive roots in ∑_{i=r}^n Z_{≥0}α_i' is ambiguous for type Cn because the long roots have coefficient 2; please specify the root system convention.
[Global] There are several typographical errors: 'deﬁnd' should be 'defined' in Section 4; the French résumé contains 'Weil group' for 'Weyl group'; the running title has 'V A RIETIES'.
[Proposition 6.2] The verification of P_J(w)⊆P_w would be easier to check if the two alternatives for w^{-1}(α) were spelled out explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the construction of w and the cohomology computations are independent of the conclusion, and self-citations are used as published lemmas rather than as assumed outputs.

full rationale

I walked the derivation chain. Theorem 2.1 is proved by choosing w from the stabilizer computation R+(w^{-1}) ∩ S = I, then showing that Aut0(X(w)) is a closed subgroup of G containing P. The injectivity of φ_w is cited to [10, Theorem 4.2(2)], a parameter-free published theorem from the first author's prior work; it supplies the same implication (w^{-1}(α0) < 0 implies injective) but does not assume the theorem under proof. The descent homomorphism π_* uses [5, Corollary 2.2], an external result, and the classification input [1, Theorem 2] is also external. The exceptional cases in Section 5 are handled by direct, parameter-free Demazure cohomology computations (H^0(w, g/b) = g) that do not rely on the target conclusion. Section 6 uses the same pattern with [10, Lemma 3.5(3)] for a restriction-map surjectivity; again this is independent support, not an assumption of the conclusion. No equation is defined in terms of the target, no fitted parameter is relabeled as a prediction, and no self-citation chain forces the result. Concerns about unstated hypotheses of [5, Corollary 2.2] or the scope of [10, Theorem 4.2(2)] are correctness risks, not circularity; they do not amount to a reduction of the claims to their own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The derivation relies on standard theorems from algebraic groups and one ad hoc assertion flagged in red flags.

assumptions (6)

standard math Akhiezer's classification of flag varieties G/P with Aut0(G/P)=G ([1, Theorem 2, p.75]).
Used in Section 3 to conclude Aut0(G/P')=G for all but three pairs (G,P).
standard math Injectivity of the homomorphism pi_* induced by a birational morphism with pi_*O_X=O_Y ([5, Corollary 2.2, p.45]).
Used in Section 3 to embed Aut0(X(w)) into Aut0(G/P').
standard math Criterion w^{-1}(alpha0)<0 implies phi_w: P -> Aut0(X(w)) is injective ([10, Theorem 4.2(2), p.772]).
Used to ensure the natural action of P on X(w) is faithful.
standard math Demazure's lemma computing H^j(w,lambda) and related cohomology (Lemma 4.1).
Foundational for the direct computations in Proposition 5.1.
domain assumption G simple adjoint over C, B an opposite Borel, standard structure theory of algebraic groups.
Sets the entire framework of Schubert varieties and parabolic subgroups.
ad hoc to paper The equality H^0(s1,g)=g used in Case 3 of Proposition 5.1.
Asserted without proof; a possible gap for the G2 case.

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Cite this review

Pith. "Pith review of Parabolic subgroups and Automorphism groups of Schubert varieties." pith.science (2026). https://pith.science/paper/DCAFW7I2

@misc{pith2026190804768,
  author       = {Pith},
  title        = {Pith review of: Parabolic subgroups and Automorphism groups of Schubert varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DCAFW7I2}},
  note         = {Machine review of arXiv:1908.04768}
}

abstract

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. In this article we show that given any parabolic subgroup $P$ of $G$ containing $B$ properly, there is an element $w\in W$ such that $P$ is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w).$

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Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Senthamarai Kannan, On the automorphism group of a smoo th Schubert Variety, Algebr

S. Senthamarai Kannan, On the automorphism group of a smoo th Schubert Variety, Algebr. Rep- resent. Theory 19 (2016), no 4, 761-782

work page 2016

[2] [1]

D. N. Akhiezer, Lie group actions in complex analysis, Aspects of M athematics E 27 Vieweg, Braun- schweig/Wiesbaden, 1995

work page 1995

[3] [2]

Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springe r-Verlag, Berlin Heidelberg, New York, 2002

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springe r-Verlag, Berlin Heidelberg, New York, 2002

work page 2002

[4] [3]

Balaji, S

V. Balaji, S. Senthamarai Kannan, K.V. Subrahmanyam, Cohomo logy of line bundles on Schubert varieties-I, Transformation Groups 9 (2004), no.2, 105-131

2004

[5] [4]

Brion, S

M. Brion, S. Kumar, Frobenius splitting methods in geometry and r epresentation theory, Progress in Mathematics, Vol. 231, Birkh¨ auser, Boston, Inc., Boston, MA, 2005

work page 2005

[6] [5]

Brion, On automorphism groups of ﬁber bundles

M. Brion, On automorphism groups of ﬁber bundles. Publ.Mat.Urug .12 (2011), 39-66

work page 2011

[7] [6]

Demazure, A very simple proof of Bott’s theorem, Invent

M. Demazure, A very simple proof of Bott’s theorem, Invent. Ma th. 33 (1976), 271-272

1976

[8] [7]

Humphreys, Introduction to Lie algebras and Representat ion theory, Springer-Verlag, Berlin Heidelberg, New York, 1972

J.E. Humphreys, Introduction to Lie algebras and Representat ion theory, Springer-Verlag, Berlin Heidelberg, New York, 1972

1972