Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$

Pinakinath Saha; S.Senthamarai Kannan

REVIEW 2 major objections 4 minor 17 references

Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.05595 v1 pith:XU5TJLK6 submitted 2019-08-13 math.AG math.COmath.RT

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

classification math.AGmath.COmath.RT MSC 14M1514F0514D1520G05

keywords Bott-Samelson-Demazure-HansenvarietySchubertrigiditytangentbundlecohomologyCoxeterelementF4rootsystemG2deformationtheory

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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 $a_2\neq2$.

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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

Dear colleague,

The 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.

The 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.

There 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.

Bottom 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 7, proof of (⇒) in Theorem 7.1] 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 7, proof of (⇐) in Theorem 7.1] 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.

minor comments (4)

[Lemma 8.1] 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.
[Several displayed formulas in Sections 4-5] 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.
[Abstract and Introduction] 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.
[Theorem 7.1] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the F4/G2 rigidity criteria are derived by cohomology computations from standard prior results, not assumed.

full rationale

None of the enumerated circularity patterns occurs. The rigidity criterion in Theorem 7.1 is not used to define the Coxeter-element parameters a1,a2; rather, all cases are enumerated from the standard normal form c=[a1,4][a2,a1-1]..., and H^1 is then computed with the LES. The vanishing/non-vanishing is the output of those computations, not an input. Lemma 3.2 (w0=c^6) is cited from [17], an external result. The main inputs with overlapping authors are [5, Prop 3.1] (LES and H^j vanishing for j>=2), [15, Cor 5.6/6.4] (line-bundle cohomology for simple roots), and [4, Lemma 7.1] (surjectivity used as Lemma 6.2); these are stated as general theorems about BSDH and Schubert varieties, independent of the rigidity classification, and are not restatements of Theorems 7.1/8.2. The proof of Lemma 6.2 is deliberately deferred to [4], which is an omitted proof rather than a circular step. A possible completeness gap in the non-rigid direction of Theorem 7.1 (only the canonical word s3s4s2s1 is checked, while Lemma 3.2(2) permits other reduced expressions of the same Coxeter element) is a correctness or coverage concern, not circularity of the derivation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper makes no new ad hoc assumptions or entities; it imports standard results from the literature, including several from the authors' own earlier papers. The computations are parameter-free.

assumptions (4)

standard math Demazure's short exact sequences and the long exact sequence for tangent-bundle cohomology from [5, Section 3] hold as stated.
Used throughout Sections 4 to 7 to relate H^1(Z(w,i), T) to H^1(w, alpha_i) and weight spaces; the LES is quoted in Section 6 and the SES in Section 2.
domain assumption Vanishing results from [15, Cor 5.6]: H^j(w, alpha) = 0 for j >= 2 and H^1(w, alpha) = 0 for short roots alpha.
Invoked repeatedly (e.g., Sections 4-7, Lemma 3.3, Lemma 5.9, Lemma 6.3) to reduce computations to H^1(w, alpha_2).
standard math Coxeter-element facts from [17, Prop 1.3, 1.7] and [11, Prop 3.18]: h(i,c) + h(i*,c) = h and w0 = c^6 for F4, w0 = c^3 for G2.
Used in Lemma 3.2 and Lemma 8.1 to parameterize reduced expressions of w0 by Coxeter elements.
standard math The decomposition of indecomposable B_alpha-modules as tensor products from [1, Cor 9.1] (Lemma 2.4), and Lemma 2.3 on cohomology over P^1.
Basis for weight-space computations in Sections 4 and 5.

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Cite this review

Pith. "Pith review of Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$." pith.science (2026). https://pith.science/paper/XU5TJLK6

@misc{pith2026190805595,
  author       = {Pith},
  title        = {Pith review of: Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XU5TJLK6}},
  note         = {Machine review of arXiv:1908.05595}
}

abstract

Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that $Z(w_{0}, \underline{i})$ is rigid (see Theorem 8.1). Further, if $G$ is of type $G_{2},$ there is no reduced expression $\underline{i}$ of $w_{0}$ for which $Z(w_{0}, \underline{i})$ is rigid (see Theorem 8.2).

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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

work page 2004
[2]

Bott and H

R. Bott and H. Samelson, Applications of the theory of Morse to s ymmetric spaces, Amer. J. Math. 80 (1958), 964-1029

work page 1958
[3]

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
[4]

2, 435-468

B.Narasimha Chary, S.Senthamarai Kannan, Rigidity of Bott-Sam elson-Demazure-Hansen variety for P Sp(2n,C), Journal of Lie Theory, 27 (2017), no. 2, 435-468

work page 2017
[5]

B.Narasimha Chary, S.Senthamarai Kannan, A.J.Parameswaran, Automorphism group of a Bott- Samelson-Demazure-Hansen variety, Transformation Groups, 20 (2015), no.3, 665-698

work page 2015
[6]

Demazure, Desingularisation des varieties de schubert gener alisees

M. Demazure, Desingularisation des varieties de schubert gener alisees. Ann. Sci. Ecole Norm. Sup 7(1974), 53-88

work page 1974
[7]

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

work page 1976
[8]

Hansen, On cycles on ﬂag manifolds, Math

H.C. Hansen, On cycles on ﬂag manifolds, Math. Scand. 33 (1973), 269-274

work page 1973

Show all 17 references

[9]

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
[10]

Humphreys, Linear Algebraic Groups, Springer-Verlag, Be rlin Heidelberg, New York, 1975

J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, Be rlin Heidelberg, New York, 1975

1975
[11]

Humphreys, Reﬂection groups and Coxeter groups, Vol 2 9, Cambridge uni-versity press, 1992

J.E. Humphreys, Reﬂection groups and Coxeter groups, Vol 2 9, Cambridge uni-versity press, 1992. 46 S.SENTHAMARAI KANNAN AND PINAKINATH SAHA

1992
[12]

Humphreys, Conjugacy classes in semisimple algebraic group s, Math

J.E. Humphreys, Conjugacy classes in semisimple algebraic group s, Math. Surveys Monographs, vol.43, Amer. Math. Soc., 1995

1995
[13]

Huybrechts, ”Complex Geometry: An Introduction”, Sprin ger-Verlag, Berlin Heidelberg, New York, 2005

D. Huybrechts, ”Complex Geometry: An Introduction”, Sprin ger-Verlag, Berlin Heidelberg, New York, 2005

2005
[14]

Jantzen, Representations of Algebraic Groups, (Second Edition ), Mathematical Surveys and Monographs, Vol.107, 2003

J.C. Jantzen, Representations of Algebraic Groups, (Second Edition ), Mathematical Surveys and Monographs, Vol.107, 2003

2003
[15]

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

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

2016
[16]

S.Senthamarai Kannan, Pinakinath Saha, Rigidity of Bott-Same lson-Demazure-Hansen variety for P SO(2n + 1,C), preprint
[17]

Yang and A.Zelevinsky, Cluster algebras of ﬁnite type via Cox eter elements and principal minors, Transformation Groups, 13 (2008), no.3-4, 855-895

S.W. Yang and A.Zelevinsky, Cluster algebras of ﬁnite type via Cox eter elements and principal minors, Transformation Groups, 13 (2008), no.3-4, 855-895. Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Si ruseri, Kelambakkam, 603103, India. E-mail address : kannan@cmi...

2008

Pith tools

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

[1] [1]

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

work page 2004

[2] [2]

Bott and H

R. Bott and H. Samelson, Applications of the theory of Morse to s ymmetric spaces, Amer. J. Math. 80 (1958), 964-1029

work page 1958

[3] [3]

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

[4] [4]

2, 435-468

B.Narasimha Chary, S.Senthamarai Kannan, Rigidity of Bott-Sam elson-Demazure-Hansen variety for P Sp(2n,C), Journal of Lie Theory, 27 (2017), no. 2, 435-468

work page 2017

[5] [5]

B.Narasimha Chary, S.Senthamarai Kannan, A.J.Parameswaran, Automorphism group of a Bott- Samelson-Demazure-Hansen variety, Transformation Groups, 20 (2015), no.3, 665-698

work page 2015

[6] [6]

Demazure, Desingularisation des varieties de schubert gener alisees

M. Demazure, Desingularisation des varieties de schubert gener alisees. Ann. Sci. Ecole Norm. Sup 7(1974), 53-88

work page 1974

[7] [7]

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

work page 1976

[8] [8]

Hansen, On cycles on ﬂag manifolds, Math

H.C. Hansen, On cycles on ﬂag manifolds, Math. Scand. 33 (1973), 269-274

work page 1973