{"id":"df7c5b21-2ca1-4d0e-b1d6-7f1b3b2cd80f","arxiv_id":"1908.05595","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For F4, the BSDH variety from a Coxeter-element power is rigid exactly when a1 is not 3 or a2 is not 2; for G2 it is never rigid.","lead":"This paper classifies which Bott-Samelson-Demazure-Hansen varieties, built from reduced expressions of the longest Weyl group element, are rigid for the exceptional Lie types F4 and G2. It gives a complete answer for the reduced expressions that are powers of a Coxeter element, and shows rigidity depends on the chosen expression.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-rigid proof in Theorem 7.1 covers only the canonical reduced expression of c=s3s4s2s1; Lemma 3.2 also allows other reduced expressions, which are not handled.","rationale":"I read the paper in good faith. The main theorems are new, and the long computations may well be correct; the paper gives a systematic case analysis and the G2 result is complete because G2 has only two reduced expressions of w0. However, the F4 theorem has a coverage gap inside the Coxeter-element family itself, not just outside it. The reader's stated weakest assumption was that the F4 classification is restricted to Coxeter-element powers and says nothing about other reduced expressions of w0. My concern is more specific and more serious: even among the concatenations admitted by Lemma 3.2, the non-rigid direction of Theorem 7.1 is proved only for the canonical reduced expression s3s4s2s1 of the exceptional Coxeter element. Lemma 3.2 explicitly allows arbitrary reduced expressions of c in each of the six blocks, and the proof does not touch the other reduced expressions, such as s3s2s4s1. This is not a cosmetic omission, because the displayed LES and the use of Lemma 6.2 depend on (3,4,2) being a prefix of the word. Without an argument for the other reduced expressions, the 'if and only if' claim is unproved for a concrete class of inputs. I therefore recommend keeping a conditional verdict, but with the added condition that the authors either prove the non-rigid direction for every reduced expression of c=s3s4s2s1 or explicitly restrict the statement to the canonical expression and explain the scope. The proposed test would settle which revision is needed.","tokens_in":55630,"tokens_out":32871,"duration_ms":303908,"concrete_test":"Take i to be six repetitions of the reduced expression (3,2,4,1) of the exceptional Coxeter element c, and run the paper's LES recursion to compute H^1(Z(w0,i),T) for this word. The recursion is finite and can be scripted from Lemmas 2.2 and 2.3 together with the weight-space computations in Sections 4 and 5. If the final H^1 is nonzero, the theorem may survive but needs a revised proof; if it vanishes, Theorem 7.1's 'if and only if' statement is false for the set of i allowed by Lemma 3.2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3.2(2) states that any concatenation i=(i1,...,i6), with each ir a reduced expression of a Coxeter element c, is a reduced expression of w0. Theorem 7.1 then claims rigidity depends only on the parameters a1,a2 of c. The non-rigid direction is proved only for the canonical word c=s3s4s2s1: the proof chooses u=s3s4s2 and invokes Lemma 6.2, whose surjectivity H^1(Z(w0,i)) -> H^1(Z(u,(3,4,2))) holds only when (3,4,2) is a prefix of i. In the allowed non-canonical reduced expression c=s3s2s4s1, obtained by commuting s4 past s2, the first block is (3,2,4,1), and (3,4,2) is not a prefix of the word. The proof contains no replacement for Lemma 6.2 in this case, so the claimed 'if and only if' is not established for the full set of i admitted by Lemma 3.2. This is the load-bearing step: if H^1 vanishes for such an i, the classification in Theorem 7.1 is false; if it does not vanish, the argument still needs a new prefix or a different surjectivity argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the vanishing of the cohomology modules H^j(Z(w0,i), T_{(w0,i)}) for Bott-Samelson-Demazure-Hansen varieties associated to the longest element w0 of the Weyl group in types F4 and G2. For F4, Theorem 7.1 asserts that, for every reduced expression i of w0 obtained by concatenating six reduced expressions of a Coxeter element c, all higher cohomology of the tangent bundle vanishes if and only if c is not the Coxeter element s3s4s2s1. For G2, Theorem 8.2 asserts that neither of the two reduced expressions of w0 gives a rigid BSDH variety. The proof is based on the Demazure short exact sequence, reductions to cohomology of line bundles, and a long series of explicit weight-space computations.","tokens_in":55850,"tokens_out":7608,"duration_ms":73681,"significance":"If the statements are correct in their full generality, the paper gives the first rigidity classification for BSDH varieties in a non-simply-laced group of rank 4, and it exhibits a clear dependence of H^1 on the choice of reduced expression. The paper is careful in its weight-space computations and does not introduce free parameters or assume the desired conclusion. The G2 result is complete and convincing. The F4 result, however, is proved only for one canonical word per Coxeter element, whereas the theorem as stated covers every concatenation of reduced expressions of c; this gap is load-bearing for the F4 classification.","major_comments":[{"comment":"The nonvanishing direction is proved only for the canonical word c = s3s4s2s1 and for the prefix u = s3s4s2. Lemma 6.2 supplies the surjectivity H^1(Z(w0,i), T) -> H^1(Z(u,(3,4,2)), T) only when (3,4,2) is a prefix of i. Lemma 3.2(2) admits i whose first block is any reduced expression of c, e.g. (3,2,4,1) obtained by commuting s2 and s4. For such an i, (3,4,2) is not a prefix and the argument in the paper does not apply. Since Theorem 7.1 is stated for all choices of reduced expressions of c in each block, the 'only if' direction is not established for the full set of i admitted by the theorem.","section":"Section 7, proof of (⇒) in Theorem 7.1"},{"comment":"The vanishing direction is also carried out case by case for one particular reduced expression of w0 in each case, namely the word obtained by repeating the displayed canonical expression of c (for example, Case 1 uses v6 = [1,4]^6 and Case 7 uses i' = (4,3,4,2,3,4,l3,1,2,1)). The LES arguments and the surjectivity lemmas in Section 6 are tied to these specific words. The theorem, however, claims rigidity for every reduced expression i = (i1,...,i6) with each ir a reduced expression of c. No argument is given that H^1(Z(w0,i), T) is independent of the choice of reduced expression within each block, nor is an additional case analysis supplied for words such as those obtained by commuting commuting simple reflections inside a block. Thus the 'if' direction is incomplete as stated.","section":"Section 7, proof of (⇐) in Theorem 7.1"}],"minor_comments":[{"comment":"In the proof of Lemma 8.1(1), the line 'we have c6(ωi) = -ωi' should refer to c^3, since the conclusion is that c^3 = -identity.","section":"Lemma 8.1"},{"comment":"There are duplicated summation symbols, for example '⊕⊕C' in Lemma 5.7(2) and Corollary 5.6(2), and several misspellings such as 'twi dimensional' and 'irreduible'. These should be corrected.","section":"Several displayed formulas in Sections 4-5"},{"comment":"The abstract refers to 'Theorem 8.1' for the rigidity result, but the F4 rigidity theorem is numbered Theorem 7.1; the numbering should be reconciled.","section":"Abstract and Introduction"},{"comment":"The statement of Theorem 7.1 does not explicitly define the word i. It should state that i = (i1,...,i6) with each ir a reduced expression of the Coxeter element c, as is used in the proof and in Lemma 3.2(2).","section":"Theorem 7.1"}],"recommendation":"major_revision","confidential_remarks":"The F4 theorem as stated is stronger than what the proof establishes. The missing noncanonical block cases may be manageable by braid/commutation arguments or by restricting the statement to the canonical repeated word, but the current text does not do so. The G2 result is solid and the computational core of the paper appears sound for the canonical words."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper extends earlier rigidity computations for Bott-Samelson-Demazure-Hansen varieties from types B and C to F4 and G2, using the same LES/weight-space machinery. The F4 criterion (Theorem 7.1) and the G2 non-rigidity theorem are new, and the proofs are detailed, with exhaustive case-by-case weight-space arithmetic. I found no fatal error in the parts I checked; the G2 argument is complete, and the F4 computation appears sound for the canonical reduced expressions of the Coxeter element.\n\nThe main soft spot is scope. Lemma 3.2 says any six-fold concatenation of reduced expressions of a Coxeter element c gives a reduced expression of w0, and the theorem is stated for that family. But the non-rigid direction of Theorem 7.1 is proved only for the specific expression c = s3s4s2s1. The same Coxeter element also has the reduced expression s3s2s4s1 (s2 and s4 commute), and the chosen prefix u = s3s4s2 is not a prefix of that word. Lemma 6.2 gives surjectivity only for prefixes of the actual reduced expression, so the argument does not cover the commuted word unless one adds an isomorphism of BSDH varieties under commutation of commuting factors. The paper doesn't state or prove such an isomorphism. If the theorem is meant to cover all reduced expressions of c, that is a genuine gap; if it is meant only for the canonical block, the abstract and Lemma 3.2 oversell the scope. Either way, a referee should ask for a clarifying statement and the missing commutation argument.\n\nThere are also numerous typos in displayed equations (c^6 for c^3 in Lemma 8.1, a missing '=0' in Lemma 6.9, duplicated plus signs). They don't appear to change the mathematics, but they slow down any independent check.\n\nBottom line: this is a legitimate, careful extension of previous work with real computational content. The F4 classification is not fully established for the family the authors seem to claim, but the fix is likely small. I'd send it to a knowledgeable referee and ask for the scope to be made precise and the commutation case handled.","headline":"A careful computational extension of earlier BSDH rigidity results to F4 and G2, with a genuine scope gap in the F4 non-rigidity proof if it is meant to cover all reduced expressions of the Coxeter element.","tokens_in":56435,"tokens_out":11612,"would_cite":true,"duration_ms":110505,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14F05","14D15","20G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For F4, the Bott-Samelson desingularization of the longest Schubert variety is rigid exactly when the Coxeter word avoids $a_1=3$ and $a_2=2$; in G2 no such desingularization is rigid.","keywords":["Bott-Samelson-Demazure-Hansen variety","Schubert variety","rigidity","tangent bundle cohomology","Coxeter element","F4 root system","G2 root system","deformation theory"],"falsifier":"Compute $H^1(Z(w_0,\\underline{i}),T)$ for the F4 reduced expression $\\underline{i}=(3,4,2,1)^6$: the theorem predicts a nonzero class of weight $\\alpha_2+\\alpha_3$, visible already in the prefix $u=s_3s_4s_2$. A direct calculation showing this module vanishes would refute Theorem 7.1; conversely, exhibiting any reduced expression with $a_1\\neq3$ or $a_2\\neq2$ whose $H^1$ is nonzero would show the classification is incomplete.","tokens_in":55375,"feed_emoji":"","tokens_out":7509,"duration_ms":70528,"temperature":0.7,"pith_summary":"The paper asks which Bott-Samelson–Demazure–Hansen (BSDH) desingularizations of the longest Schubert variety are rigid, meaning that the first cohomology of their tangent bundle vanishes. For the exceptional group F4, it claims a complete answer among the reduced expressions built by repeating one Coxeter element six times: the variety is rigid exactly when the canonical word $[a_1,4][a_2,a_1-1]\\cdots$ does not have both $a_1=3$ and $a_2=2$. For G2, it claims the opposite: both reduced expressions of the longest element give nonzero $H^1$, so the corresponding BSDH varieties are never rigid. Since higher tangent-bundle cohomology vanishes in general, rigidity is here equivalent to $H^1=0$, and the paper computes this module through a long exact sequence of $B$-modules. The result matters because it shows that rigidity of these desingularizations depends on the chosen reduced expression once the group is not simply laced.","feed_headline":"F4 rigidity turns on two entries of the Coxeter word","feed_subtitle":"In F4 the resolutions are rigid unless the Coxeter word starts 3, then 2; in G2, never.","key_machinery":"The load-bearing objects are the BSDH varieties $Z(w,\\underline{i})$, iterated $\\mathbb{P}^1$-bundles resolving Schubert varieties, and the $B$-module cohomology groups $H^j(w,\\alpha)$ of the relative tangent bundle for a simple root $\\alpha$. The proof runs on a long exact sequence that expresses $H^1(Z(w_0,\\underline{i}),T)$ in terms of these smaller modules; Demazure-type weight computations reduce each $H^0$ and $H^1$ to explicit one-dimensional weight spaces; and a surjectivity lemma ensures a surjection $H^1(Z(w_0,\\underline{i}),T)\\to H^1(Z(u,\\underline{j}),T)$ for prefixes $u$, so a single nonzero weight in a prefix forces non-rigidity of the whole variety. The Coxeter-element decomposition and the identity $w_0=c^6$ (for F4) or $w_0=c^3$ (for G2) supply the finite list of words to check, and Corollary 7.2 converts $H^1=0$ together with the general vanishing $H^j=0$ for $j\\ge2$ into absence of deformations.","core_discovery":"The central claim is Theorem 7.1: for a simple algebraic group of adjoint type with root system F4, if $w_0$ is the longest Weyl-group element and $\\underline{i}$ is the reduced expression obtained as six repetitions of a Coxeter element $c$ written as $c=[a_1,4][a_2,a_1-1]\\cdots[a_k,a_{k-1}-1]$ with $4\\ge a_1>\\cdots>a_k=1$, then $H^j(Z(w_0,\\underline{i}),T_{(w_0,\\underline{i})})=0$ for all $j\\ge1$ if and only if $a_1\\neq3$ or $a_2\\neq2$. The single exceptional Coxeter word is $c=s_3s_4s_2s_1$, which forces a nonzero class in $H^1$. Theorem 8.2 proves the G2 analogue with the opposite conclusion: for $w_0=(s_1s_2)^3$ and for $w_0=(s_2s_1)^3$, $H^1(Z(w_0,\\underline{i}_r),T)\\neq0$ for both reduced expressions, so no BSDH desingularization of the longest Schubert variety is rigid in type G2.","pith_inferences":["The F4 theorem's 'if and only if' is a classification only of reduced expressions of $w_0$ of the form $c^6$ with a single Coxeter element; the paper does not establish that every reduced expression of $w_0$ in F4 has this shape, so a full classification of all BSDH varieties of the longest Schubert variety in F4 remains open.","Because the obstruction in both G2 and the exceptional F4 case appears already in a short prefix of $w_0$, the same long-exact-sequence technique could test rigidity for arbitrary reduced expressions in other non-simply-laced types, such as $B_n$ or $C_n$.","The pattern suggests that in non-simply-laced groups, rigidity is controlled by how the asymmetry between long and short roots interacts with the ordering of simple reflections in the word; a testable extension would be to determine whether every rigid reduced expression of $w_0$ in F4 must be a Coxeter-power word with $a_1\\neq3$ or $a_2\\neq2$."],"forward_implications":["For the F4 reduced expressions covered by the theorem, exactly one Coxeter-element family, the one with $a_1=3$ and $a_2=2$, produces non-rigid BSDH varieties; all other Coxeter-type expressions are rigid and undeformed.","For G2, neither reduced expression of $w_0$ gives a rigid BSDH variety, so rigidity fails for every BSDH desingularization of the longest Schubert variety in that type.","The proofs identify explicit nonzero weights in $H^1$, for instance the weight $\\alpha_2+\\alpha_3$ in the bad F4 case, showing exactly which tangent directions obstruct rigidity.","Where the vanishing holds, the BSDH variety has no deformations, not merely no first-order ones, because the higher cohomology groups of the tangent bundle vanish as well."],"supporting_citations":[{"why":"Provides the long exact sequence (LES) connecting tangent-bundle cohomology of BSDH varieties to the relative-tangent modules $H^j(w,\\alpha)$, and the vanishing of $H^j$ for $j\\ge2$ that frames the whole computation.","marker":"[5, Proposition 3.1]"},{"why":"Supplies the vanishing results for $H^j(w,\\alpha)$ used repeatedly to strip prefixes of $w_0$ and to handle short-root cases.","marker":"[15, Corollary 5.6]"},{"why":"Gives the Coxeter-element power relation $w_0=c^6$ for F4 and $w_0=c^3$ for G2 via the integers $h(i,c)$.","marker":"[17, Proposition 1.3]"},{"why":"Yields the surjectivity of $H^1(Z(w_0,\\underline{i}),T)\\to H^1(Z(w,\\underline{i}),T)$ that lets a nonzero class in a prefix force non-rigidity of the full variety.","marker":"[4, Lemma 7.1]"},{"why":"Used in Corollary 7.2 to conclude absence of deformations from vanishing of all positive-degree tangent-bundle cohomology.","marker":"[13, Proposition 6.2.10]"},{"why":"Gives the Coxeter number $h=2|R^+|/\\mathrm{rank}$, used to compute $h(i,c)$ in Lemma 3.2.","marker":"[11, Proposition 3.18]"},{"why":"Supplies the length table $l(w_0)=|R^+|$ that justifies that the six-fold concatenation of a Coxeter-element word is reduced.","marker":"[9]"}],"fun_headline_variants":["F4 rigid unless Coxeter word starts 3,2; G2 never","F4 rigidity: single Coxeter word exception; G2 never rigid","Rigidity in F4 BSDH resolved; G2 always nonrigid","Coxeter word 3,2 breaks F4 rigidity; G2 fails always","F4: only 3,2-start fails rigidity; G2 never rigid"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For F4, the classification covers only the reduced expressions of $w_0$ built by repeating a single reduced expression of a Coxeter element six times; the paper does not prove that every reduced expression of $w_0$ has this form, so the theorem says nothing about rigidity for any other reduced expression of $w_0$ in F4.","fun_headline_variants_meta":{"raw":{"variants":["F4 rigid unless Coxeter word starts 3,2; G2 never","F4 rigidity: single Coxeter word exception; G2 never rigid","Rigidity in F4 BSDH resolved; G2 always nonrigid","Coxeter word 3,2 breaks F4 rigidity; G2 fails always","F4: only 3,2-start fails rigidity; G2 never rigid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1667,"prompt_tokens":1091,"completion_tokens":576,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":471}},"tokens_in":707,"tokens_out":576,"duration_ms":6467,"temperature":1.0,"reasoning_tokens":471,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:31:48.898047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $H^1(Z(w_0,\\underline{i}),T)$ for the F4 reduced expression $\\underline{i}=(3,4,2,1)^6$: the theorem predicts a nonzero class of weight $\\alpha_2+\\alpha_3$, visible already in the prefix $u=s_3s_4s_2$. A direct calculation showing this module vanishes would refute Theorem 7.1; conversely, exhibiting any reduced expression with $a_1\\neq3$ or $a_2\\neq2$ whose $H^1$ is nonzero would show the classification is incomplete.","supporting_citations":[{"cited_title":"Humphreys, Introduction to Lie algebras and Representat ion theory, Springer-Verlag, Berlin Heidelberg, New York, 1972","cited_arxiv_id":null,"evidence_quote":"Supplies the length table $l(w_0)=|R^+|$ that justifies that the six-fold concatenation of a Coxeter-element word is reduced."}],"review_version":1}