{"id":"6f1c8ba1-652e-4d6b-b565-8a32640e25cb","arxiv_id":"1908.04768","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every proper parabolic subgroup P of a simple algebraic group G, there is a Schubert variety X(w) in G/B whose connected automorphism group Aut0(X(w)) equals P.","lead":"Mathematicians show that every parabolic subgroup of a simple algebraic group appears as the identity component of the automorphism group of some Schubert variety. The result answers a natural question left open in earlier work by the first author.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof hinges on an unstated hypothesis of [5, Cor. 2.2] for the descent homomorphism π_*; the paper should state and verify the corollary's hypotheses, since a birational morphism need not induce a homomorphism on automorphism groups without them.","rationale":"The reader's weakest_assumption identifies the same step that I consider most load-bearing: the existence and injectivity of π_* via the unstated Corollary 2.2. Without this, the proof cannot place Aut0(X(w)) inside G. The paper's one-line citation of [5] leaves this unresolved, and the same is true for [1, Theorem 2]. I do not think this is a fatal mathematical error — Blanchard's lemma plausibly supplies the needed homomorphism, and injectivity follows from birationality — but the manuscript does not say so, and the reader cannot verify the claim without looking up the cited texts. The external references may well be correct, but the proof as written is conditional on them. The three explicit cases in Section 5 also rely on the unstated validity of [10, Theorem 4.2(2)] for injectivity of φ_w; if that theorem is only stated for smooth Schubert varieties, its use in non-smooth cases would be a further gap. The intended theorem is likely true, but the presentation leaves enough unverified external dependencies that the conditional verdict is appropriate.","tokens_in":10224,"tokens_out":48724,"duration_ms":486736,"concrete_test":"Locate [5, Corollary 2.2] (Brion, Publ. Mat. Urug. 12 (2011), 39–66) and verify its precise hypotheses, then check them for the morphism π: X(w) → G/P' with w = (w0^J)^{-1}, J = S∖I, and P' = P_{-w0(J)}. In particular, confirm whether the corollary applies to an arbitrary proper morphism with π_*O_X = O_Y, or only to locally trivial fiber bundles; if it requires local triviality, run the check on the G2 case w = s1s2s1s2, where π has positive-dimensional fibers over the exceptional locus, and see whether the claimed homomorphism Aut0(X(w)) → Aut0(G/P') actually exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, the authors assert (after verifying π_*O_X = O_Y) that [5, Cor. 2.2] gives a homomorphism π_*: Aut0(X(w)) → Aut0(G/P'), and that π_* is injective because π is birational. This step is load-bearing: it is the only bridge that embeds Aut0(X(w)) into G and allows the conclusion that Aut0(X(w)) is a parabolic subgroup of G containing P. If the cited corollary actually requires stronger hypotheses than the paper states — for example, that π is a locally trivial fiber bundle, or that every automorphism of the source preserves the fibers of π — then the application fails, because the birational morphism π: X(w) → G/P' is generally not locally trivial and may have positive-dimensional exceptional fibers. The paper does not reproduce the statement of Corollary 2.2, nor does it explicitly verify the hypotheses beyond π_*O_X = O_Y. While injectivity from birationality is plausible (the general fiber is a point, so a nontrivial kernel would fix a dense open subset pointwise), the existence of the homomorphism is the nontrivial part. Blanchard's lemma would supply this if its hypotheses are satisfied; the manuscript's terse citation makes this impossible to check without the reference. A second, related external dependence is the unquoted [1, Theorem 2], used to conclude Aut0(G/P') = G in the non-exceptional cases. Neither citation is stated, so the central claim is not self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10571,"tokens_out":16187,"duration_ms":157905,"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":[{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Proposition 5.1, Case 3"},{"comment":"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.","section":"Sections 3 and 5"}],"minor_comments":[{"comment":"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'.","section":"Abstract"},{"comment":"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":"Introduction"},{"comment":"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":"Section 5, Case 1"},{"comment":"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.","section":"Section 5, Case 2"},{"comment":"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'.","section":"Global"},{"comment":"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.","section":"Proposition 6.2"}],"recommendation":"major_revision","confidential_remarks":"The result is attractive and the exceptional-case computations are largely explicit. My main concerns are the unverified use of [5, Corollary 2.2] and the missing justification in Case 3 of Proposition 5.1; both are likely repairable by quoting the relevant statements and adding short arguments. I recommend a major revision rather than rejection, and would be willing to look at a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nNow the soft spots, in order.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":11079,"tokens_out":3106,"would_cite":true,"duration_ms":32283,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14L30","20G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every proper parabolic subgroup of a simple algebraic group is the identity component of the automorphism group of some Schubert variety.","keywords":["Schubert varieties","automorphism groups","parabolic subgroups","flag varieties","algebraic groups","Weyl group","cohomology of homogeneous vector bundles","adjoint type"],"falsifier":"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.","tokens_in":10014,"feed_emoji":"📐","tokens_out":8528,"duration_ms":68386,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every proper parabolic is an automorphism group","feed_subtitle":"Schubert varieties in a flag variety realize every parabolic subgroup as their identity-component automorphism group.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the criterion $w^{-1}(\\alpha_0) < 0$ guaranteeing that the natural homomorphism $P \\to \\mathrm{Aut}_0(X(w))$ is injective.","marker":"[10]"},{"why":"Corollary 2.2 supplies the induced homomorphism on automorphism groups from a birational morphism, used to obtain injectivity of $\\pi_*$.","marker":"[5]"},{"why":"Theorem 2, p.75 classifies flag varieties $G/P'$ whose automorphism group is $G$, reducing the main proof to three exceptional cases.","marker":"[1]"},{"why":"Provides the sheaf-theoretic results (Theorem 3.3.4(a) and Lemma 3.3.3(b)) used to establish $\\pi_*(\\mathcal{O}_{X(w)}) = \\mathcal{O}_{G/P'}$.","marker":"[4]"},{"why":"Gives that $\\mathrm{Aut}_0(X(w))$ is an algebraic group and that its Lie algebra is $H^0(X(w), T_{X(w)})$, foundational for the Lie-algebra computations.","marker":"[12]"},{"why":"Demazure's lemma is the cohomology tool used in the three exceptional cases to compute $H^0(w, \\mathfrak{g}/\\mathfrak{b})$.","marker":"[6]"}],"fun_headline_variants":["Every parabolic subgroup is an Aut0 of some Schubert variety","Parabolics realized as identity components of Schubert automorphisms","Schubert varieties carry all parabolic automorphism groups","Each parabolic is an automorphism group of a Schubert variety","Aut0 of Schubert varieties covers every parabolic subgroup"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every parabolic subgroup is an Aut0 of some Schubert variety","Parabolics realized as identity components of Schubert automorphisms","Schubert varieties carry all parabolic automorphism groups","Each parabolic is an automorphism group of a Schubert variety","Aut0 of Schubert varieties covers every parabolic subgroup"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1777,"prompt_tokens":1061,"completion_tokens":716,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":677,"completion_tokens_details":{"reasoning_tokens":633}},"tokens_in":677,"tokens_out":716,"duration_ms":7033,"temperature":1.0,"reasoning_tokens":633,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:33:01.496515+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Senthamarai Kannan, On the automorphism group of a smoo th Schubert Variety, Algebr","cited_arxiv_id":null,"evidence_quote":"Provides the criterion $w^{-1}(\\alpha_0) < 0$ guaranteeing that the natural homomorphism $P \\to \\mathrm{Aut}_0(X(w))$ is injective."},{"cited_title":"Brion, On automorphism groups of fiber bundles","cited_arxiv_id":null,"evidence_quote":"Corollary 2.2 supplies the induced homomorphism on automorphism groups from a birational morphism, used to obtain injectivity of $\\pi_*$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Theorem 2, p.75 classifies flag varieties $G/P'$ whose automorphism group is $G$, reducing the main proof to three exceptional cases."},{"cited_title":"Brion, S","cited_arxiv_id":null,"evidence_quote":"Provides the sheaf-theoretic results (Theorem 3.3.4(a) and Lemma 3.3.3(b)) used to establish $\\pi_*(\\mathcal{O}_{X(w)}) = \\mathcal{O}_{G/P'}$."},{"cited_title":"Matsumura, F","cited_arxiv_id":null,"evidence_quote":"Gives that $\\mathrm{Aut}_0(X(w))$ is an algebraic group and that its Lie algebra is $H^0(X(w), T_{X(w)})$, foundational for the Lie-algebra computations."}],"review_version":1}