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Geometry of regular semisimple Lusztig varieties

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Pith's one-line read Regular semisimple Lusztig varieties degenerate flatly to Hessenberg varieties, proving a type A homeomorphism conjecture and generalizing it to all Lie types.

desk verdict Strong first half, but Theorem 5.13's flatness claim is not justified—worth peer review with a required fix. read the letter →

arxiv 2504.15868 v3 pith:UNCIUXKO submitted 2025-04-22 math.AG math.COmath.RT

Patrick Brosnan , Jaehyun Hong , Donggun Lee This is my paper · ORCID
classification math.AGmath.COmath.RT MSC 14M1514F1714B0514L3014D0614M17
keywords LusztigvarietiesHessenbergSchubertBott-SamelsonresolutionsFrobeniussplittingcohomologyvanishingGKMgraphsflatdegeneration
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 studies Lusztig varieties, subvarieties of the flag manifold defined by the condition that $g^{{-1}}$sg lies in a Bruhat double coset, for s regular semisimple. It claims these varieties are normal, Cohen-Macaulay, of pure dimension ℓ(w), with rational singularities, and that ample line bundles on them have no higher cohomology in any characteristic. The main structural result is stronger: there is a smooth projective family over a blowup of G whose fibers over regular semisimple elements are Lusztig varieties and whose fibers over the exceptional divisor are Hessenberg varieties. That flat degeneration makes Hessenberg varieties the linearized limits of Lusztig varieties, and it implies that two smooth elements of the Weyl group with the same Schubert tangent space at the identity give diffeomorphic Lusztig varieties. In type A this proves the conjectured homeomorphism, and the statement holds in arbitrary Lie type.

What carries the argument

The central mechanism is a deformation-to-normal-cone family built from the universal Lusztig family. One starts with the fiber product Y_w ⊂ G×B defined by $g^{{-1}}$xg ∈ BwB, pulls it back along the blowup bG of G at the identity, and blows up the diagonal inside B×B along the exceptional locus; the exceptional fiber becomes the projectivized normal cone, which is exactly the Hessenberg scheme X_{H_w}(s). The Hessenberg space H_w is $π_b^{{-1}}$(C_w), where C_w is the tangent cone of the Schubert variety X_w at the identity. On the cohomology side, the GKM graphs of Y_w(s) and X_{H_w}(s) coincide for smooth w, transferring the Weyl-group dot action from one variety to the other. The singularity results are carried by Bott-Samelson resolutions, whose boundary is a simple normal crossing anti-canonical divisor; in positive characteristic that divisor yields Frobenius splittings, and in characteristic zero it feeds a Kawamata-Viehweg vanishing argument.

What would settle it

Compute the exceptional fiber of bY_w → bG in a small singular case, such as G = GL_4 with w = 4231 where the tangent cone is a quadric cone, and compare it with the Hessenberg scheme X_{H_w}(s); if they differ, Theorem 5.13(2) fails. Independently, test whether the total space bY_w is Cohen-Macaulay: if not, the flatness criterion invoked at the end of Section 5.3 does not apply, and the diffeomorphism conclusion has no proof.

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Extended reading notes

Core claim

The paper's central discovery is that regular semisimple Lusztig varieties and regular semisimple Hessenberg varieties are not merely cohomologically parallel: they are the two ends of a single flat family. After blowing up G at the identity and deleting the proper transform of the non-regular locus, the paper constructs a smooth projective morphism bY_w → bG whose fiber over a regular semisimple element s is the Lusztig variety Y_w(s), and whose fiber over a point [s] in the exceptional divisor is the Hessenberg variety X_{H_w}(s), with H_w built from the tangent cone of the Schubert variety X_w at the identity. For smooth w this family is smooth, so a standard topological argument makes all fibers diffeomorphic; two smooth Weyl group elements with the same Schubert tangent space therefore yield diffeomorphic Lusztig varieties. That proves the type A homeomorphism conjecture and extends it to all Lie types. Along the way the paper establishes that regular semisimple Lusztig varieties are normal, Cohen-Macaulay, of pure dimension ℓ(w), with rational singularities, that their ample line bundles have vanishing higher cohomology in all characteristics, and that Lusztig cells and Deligne-Lusztig cells are affine.

Load-bearing premise

The load-bearing premise is that bY_w → bG is flat: the paper invokes a standard flatness criterion at the end of Section 5.3 that requires a Cohen-Macaulay hypothesis on the total space or fibers, and that hypothesis is not established for bY_w; if flatness fails, the degeneration and the diffeomorphism corollary do not follow.

Editorial extensions

If this is right

  • Smooth Weyl group elements w and w′ with the same Schubert tangent space at the identity yield diffeomorphic regular semisimple Lusztig varieties for any regular semisimple elements; in type A this is the conjectured homeomorphism.
  • Ample line bundles on every regular semisimple Lusztig variety have vanishing higher cohomology in arbitrary characteristic, and nef line bundles do too in characteristic zero or sufficiently large characteristic.
  • Lusztig cells are affine, as are Deligne-Lusztig cells, resolving an open question from the first papers on Deligne-Lusztig varieties.
  • For smooth w and dominant λ, the Weyl group acts on the weight spaces of H^0(Y_w(s), L_λ) with equal dimensions along each orbit.
  • Cohomology vanishing and restriction surjectivity transfer to all regular semisimple Hessenberg varieties in type A, extending earlier results for weak Fano cases.

Reading between the lines

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

  • One consequence the paper leaves implicit: the degeneration family is defined for all w, not only smooth ones, so the same construction may transfer intersection-cohomology or Hodge-theoretic invariants from Hessenberg schemes to singular Lusztig varieties.
  • The tangent-cone map w ↦ H_w is purely combinatorial; classifying which Hessenberg spaces arise from smooth Schubert varieties (the paper notes type C_3 exceptions) would give a concrete test of how sharp the degeneration theorem is.
  • If the flatness gap is repaired, the family should put the monodromy actions on both sides into a single local system over bG, making the dot-action comparison a statement about monodromy rather than about GKM graphs alone.
  • The affineness of Deligne-Lusztig cells may lead to new cohomological vanishing for the representations they carry; testing whether the same argument works for non-reduced words or other Frobenius twists is a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies regular semisimple Lusztig varieties Y_w(s) in a flag variety. The first half proves structural results: these varieties are normal, Cohen-Macaulay, of pure dimension l(w), have rational singularities, and enjoy vanishing theorems for ample and nef line bundles in arbitrary characteristic, via anti-canonical divisors, Kawamata-Viehweg vanishing, and Frobenius splitting. It also proves that Lusztig cells and Deligne-Lusztig cells are affine. The second half relates Lusztig varieties to Hessenberg varieties. For smooth w, the paper defines a Hessenberg space H_w from the tangent space of X_w at the identity, proves that Y_w(s) and X_{H_w}(s) have the same GKM graphs, and constructs a family bY_w over a blowup bG of G at the identity that is claimed to be a flat degeneration of the Lusztig family over G^rs to the Hessenberg family over P(g)^rs. From this degeneration, the paper derives a diffeomorphism result (Corollary 5.16) proving and generalizing the Abreu-Nigro conjecture.

Significance. If the main degeneration theorem is correct, the paper resolves a conjecture of Abreu and Nigro and extends it to all Lie types; it also provides a uniform geometric explanation for the known cohomological relationship between Lusztig and Hessenberg varieties. The first part of the paper is strong and largely self-contained: the singularity results in Section 2, the anti-canonical computation in Proposition 3.1, the vanishing arguments in Sections 3 and 4, and the affineness results for Lusztig and Deligne-Lusztig cells are substantial and, as far as I can see, correctly argued. The GKM comparison in Theorem 5.12 is a natural extension of the type-A results of Abreu-Nigro and is convincingly sketched. The degeneration construction itself is original and potentially important. However, the flatness of the degeneration family is asserted rather than proved, and the special-fiber computation is only sketched; since the diffeomorphism conclusion depends directly on these points, the central claim is not yet fully established.

major comments (2)
  1. [Section 5.3, proof of Theorem 5.13, equation (5.12)] The flatness of bY_w -> bG is not established. The final paragraph of Section 5.3 says: 'Since bG is smooth and all the fibers have the same dimension ..., (5.12) is flat.' This is not a valid flatness criterion. The standard Miracle Flatness theorem requires the total space bY_w (or, in some variants, the fibers) to be Cohen-Macaulay; bY_w is constructed as a fiber product of blowups in diagram (5.11), so its Cohen-Macaulayness is not automatic and is not proved. The parenthetical claim that the family is smooth when w is smooth also requires flatness (or a proof that bY_w is smooth and then an application of Miracle Flatness); smooth fibers alone do not imply that a morphism is smooth. Since Corollary 5.16 applies Ehresmann's theorem to this family, and since Conjecture 1.7 is derived from that corollary, this gap is load-bearing. Please prove flatness directly, for example by proving that bY_w is Cohen-Macaulay or smooth, or by showing that the fibers have constant Hilbert polynomial over the connected base bG.
  2. [Section 5.3, Lemma 5.14] The identification of the special fiber over bE is only sketched. The proof asserts several nontrivial identifications: that the map from bE×B to P_{bE×B}(pr_1^*O_{bE}(-1) ⊕ pr_2^*T_B) factors through pr_2^*T_B, that the relevant preimage of N_{\Delta(B)/B×B} is C_{\Delta(B)/X_w} ≅ G×^B H_w/b, and that the fiber is therefore the fiber product (5.2) with H=H_w. These claims are not derived; the phrase 'one can easily check, for example using one-parameter families' is not a substitute for a normal-cone computation. Because this lemma identifies the closed fiber of the degeneration as the Hessenberg variety X_{H_w}(s), it is essential for the Ehresmann comparison in Corollary 5.16. I recommend that the proof of Lemma 5.14 be expanded into a complete computation.
minor comments (4)
  1. [Abstract and Section 3.3, Corollary 3.12] There are typographical errors: 'Deline-Lusztig' should be 'Deligne-Lusztig' in the abstract and in the heading of Corollary 3.12.
  2. [Section 2.3, proof of Theorem 2.12] The word 'Gorentein' should be 'Gorenstein'.
  3. [Section 2.3, Remark 2.8(1)] The notation Y_e(s) = {wB : w∈W} uses the symbol w both for the fixed Weyl element and for the running variable; this is confusing and should be rephrased, for example as {vB : v∈W}.
  4. [Section 5.2, Theorem 5.12] The proof of Theorem 5.12 compares T-invariant curves, but the conclusion is an isomorphism of graded W-representations. Please state explicitly that the GKM graphs are compared in the labeled sense, with vertices labeled by elements of W and with the natural W-action on the graph; the underlying unlabeled graph alone would not determine the dot action.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the central degeneration and GKM comparison are proved from independent inputs, and the flagged flatness gap is a correctness issue, not a circular step.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The vanishing theorems rest on Bott-Samelson resolutions, adjunction, Kawamata-Viehweg vanishing, and Mehta-Ramanathan Frobenius splitting; the positive-characteristic results are proved directly, not imported from the authors' own prior work. Section 5's GKM comparison is a genuine proof: Theorem 5.12 computes the T-invariant curves of Y_w(s) directly using (5.7), and compares them with the known GKM graph of X_{H_w}(s) from [2, Proposition 8.2]. The Hessenberg space H_w is defined independently from the tangent cone of X_w, and the equality of GKM graphs is then established, not assumed. The degeneration family in Theorem 5.13 is constructed from blowups and deformation-to-normal-cone data, and Lemma 5.14 identifies the special fiber; no parameter is fitted to the cohomology or diffeomorphism conclusions. Citations to the authors' own work ([14], [15]) are contextual and not load-bearing, and the Abreu-Nigro conjecture is proved by an independent degeneration argument rather than by citing it. The one flagged concern is in the final paragraph of Section 5.3: 'Since bG is smooth and all the fibers have the same dimension ..., (5.12) is flat' invokes an incomplete criterion (Miracle Flatness requires a Cohen-Macaulay total space, which is not proved). This is an unsupported inference and a genuine correctness risk for the Ehresmann step in Corollary 5.16, but it is not circular: it does not assume the conclusion or rename an input as a prediction. Hence the circularity score is 0.

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

No free parameters are fitted; this is a proof-based paper. The axioms are standard tools or domain conditions. One ad hoc assumption is flagged: the unstated Cohen-Macaulay hypothesis in the flatness argument for Theorem 5.13. No new entities are postulated.

assumptions (6)
  • standard math Kawamata-Viehweg vanishing theorem
    Used in Section 3.4 to derive H^i vanishing for nef and ample line bundles on Y_w(s) in characteristic zero; the paper states the version it uses.
  • standard math Frobenius splitting machinery of Mehta-Ramanathan and Ramanathan
    Used in Section 4 to prove Y_w(s) is Frobenius split and to obtain vanishing and surjectivity in positive characteristic; cited from [44], [49], [50].
  • domain assumption Lusztig's transversality lemma for regular semisimple elements
    Assumes s in G^rs; guarantees Y_w^o(s) is smooth of dimension l(w) and the Bott-Samelson boundary is a simple normal crossing divisor; based on [39].
  • domain assumption Tangent space description of smooth Schubert varieties at the identity
    Lemma 5.4 relies on this description (citing [17], [47], [8]) to identify H_w with the tangent space; it is needed in Theorem 5.12 to compare GKM graphs.
  • standard math GKM theory and Tymoczko's dot action on cohomology
    Used in Theorem 5.12 to pass from equality of GKM graphs to isomorphism of cohomology rings with W-action; standard framework from [25], [56].
  • ad hoc to paper Miracle Flatness criterion applied without explicit Cohen-Macaulay verification
    The proof asserts flatness of bY_w to bG because bG is smooth and fibers have constant dimension; this criterion requires the source (or fibers) to be Cohen-Macaulay, which is not explicitly verified. This is the weakest premise.

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Pith. "Pith review of Geometry of regular semisimple Lusztig varieties." pith.science (2026). https://pith.science/paper/UNCIUXKO

@misc{pith2026250415868,
  author       = {Pith},
  title        = {Pith review of: Geometry of regular semisimple Lusztig varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNCIUXKO}},
  note         = {Machine review of arXiv:2504.15868}
}
abstract

Lusztig varieties are subvarieties in flag manifolds $G/B$ associated to an element $w$ in the Weyl group $W$ and an element $x$ in $G$, introduced in Lusztig's papers on character sheaves. We study the geometry of these varieties when $x$ is regular semisimple. In the first part, we establish that they are normal, Cohen-Macaulay, of pure expected dimension and have rational singularities. We then show that the cohomology of ample line bundles vanishes in positive degrees, in arbitrary characteristic. This extends to nef line bundles when the base field has characteristic zero or sufficiently large characteristic. Along the way, we prove that Lusztig varieties are Frobenius split in positive characteristic and that their open cells are affine. We also prove that the open cells in Deligne-Lusztig varieties are affine, settling a question that has been open since the foundational paper of Deligne and Lusztig. In the second part, we explore their relationship with regular semisimple Hessenberg varieties. Both varieties admit Tymoczko's dot action of $W$ on their (intersection) cohomology. We associate to each element $w$ in $W$ a Hessenberg space using the tangent cone of the Schubert variety associated with $w$, and show that the cohomology of the associated regular semisimple Lusztig varieties and Hessenberg varieties is isomorphic as graded $W$-representations when they are smooth. This relationship extends to the level of varieties: we construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties. In particular, this proves a conjecture of Abreu and Nigro on the homeomorphism types of regular semisimple Lusztig varieties in type $A$, and generalizes it to arbitrary Lie types.

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Works this paper leans on

57 extracted references · 51 canonical work pages

  1. [1]

    H. Abe, N. Fujita, and H. Zeng. Fano and weak Fano Hessenberg varieties.Michigan Math. J., 73(3):511–555, 2023

    work page 2023

  2. [2]

    T. Abe, T. Horiguchi, M. Masuda, S. Murai, and T. Sato. Hessenberg varieties and hyperplane arrangements.J. Reine Angew. Math., 764:241–286, 2020

    work page 2020

  3. [3]

    Abreu and A

    A. Abreu and A. Nigro. An update on Haiman’s conjectures.Forum Math. Sigma, 12:Paper No. e86, 15, 2024

    work page 2024

  4. [4]

    Abreu and A

    A. Abreu and A. Nigro. A geometric approach to characters of Hecke algebras.J. Reine Angew. Math., 821:53–114, 2025

    work page 2025

  5. [5]

    D. N. Akhiezer.Lie group actions in complex analysis. Aspects of Mathematics, E27. Friedr. Vieweg & Sohn, Braunschweig, 1995

    work page 1995

  6. [6]

    Bhaumik and P

    S. Bhaumik and P. Saha. Line bundles onG-Bott-Samelson-Demazure-Hansen vari- eties.J. Pure Appl. Algebra, 228(7):Paper No. 107640, 21, 2024

    work page 2024

  7. [7]

    Bia lynicki-Birula

    A. Bia lynicki-Birula. Some theorems on actions of algebraic groups.Annals of Math- ematics. Second Series, 98:480–497, 1973

    work page 1973

  8. [8]

    Billey and V

    S. Billey and V. Lakshmibai.Singular loci of Schubert varieties, volume 182 of Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2000

    work page 2000

Show all 57 references
  1. [9]

    Bj¨ orner and F

    A. Bj¨ orner and F. Brenti.Combinatorics of Coxeter groups, volume 231 ofGraduate Texts in Mathematics. Springer, New York, 2005

    2005

  2. [10]

    Bonnaf´ e and R

    C. Bonnaf´ e and R. Rouquier. Affineness of Deligne-Lusztig varieties for minimal length elements.J. Algebra, 320(3):1200–1206, 2008

    2008

  3. [11]

    M. Brion. Lectures on the geometry of flag varieties. InTopics in cohomological studies of algebraic varieties, Trends Math., pages 33–85. Birkh¨ auser, Basel, 2005

    2005

  4. [12]

    Brion and S

    M. Brion and S. P. Inamdar. Frobenius splitting of spherical varieties. InAlgebraic groups and their generalizations: classical methods (University Park, PA, 1991), vol- ume 56, Part 1 ofProc. Sympos. Pure Math., pages 207–218. Amer. Math. Soc., Providence, RI, 1994

    1991

  5. [13]

    Brion and S

    M. Brion and S. Kumar.Frobenius splitting methods in geometry and representation theory, volume 231 ofProgress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2005

    2005

  6. [14]

    Brosnan and T

    P. Brosnan and T. Y. Chow. Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties.Adv. Math., 329:955–1001, 2018

    2018

  7. [15]

    Brosnan, L

    P. Brosnan, L. Escobar, J. Hong, D. Lee, E. Lee, A. Mellit, and E. Sommers. Au- tomorphisms and deformations of regular semisimple Hessenberg varieties. Preprint, arXiv:2405.18313. GEOMETRY OF REGULAR SEMISIMPLE LUSZTIG VARIETIES 39

  8. [16]

    B˘ alibanu and P

    A. B˘ alibanu and P. Crooks. Perverse sheaves and the cohomology of regular Hessen- berg varieties.Transform. Groups, 29(3):909–933, 2024

    2024

  9. [17]

    J. B. Carrell. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and ra- tional smoothness of Schubert varieties. InAlgebraic groups and their generalizations: classical methods (University Park, PA, 1991), volume 56, Part 1 ofProc. Sympos. Pure Math., pages 53–61....

    1991

  10. [18]

    Clearman, M

    S. Clearman, M. Hyatt, B. Shelton, and M. Skandera. Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements.Electron. J. Combin., 23(2):Paper 2.7, 56, 2016

    2016

  11. [19]

    De Mari, C

    F. De Mari, C. Procesi, and M. A. Shayman. Hessenberg varieties.Trans. Amer. Math. Soc., 332(2):529–534, 1992

    1992

  12. [20]

    Deligne and G

    P. Deligne and G. Lusztig. Representations of reductive groups over finite fields.Ann. of Math. (2), 103(1):103–161, 1976

    1976

  13. [21]

    E. W. Ellers and N. Gordeev. Intersection of conjugacy classes with Bruhat cells in Chevalley groups.Pacific J. Math., 214(2):245–261, 2004

    2004

  14. [22]

    Esnault and E

    H. Esnault and E. Viehweg.Lectures on vanishing theorems, volume 20 ofDMV Seminar. Birkh¨ auser Verlag, Basel, 1992

    1992

  15. [23]

    Fuchs, A

    D. Fuchs, A. Kirillov, S. Morier-Genoud, and V. Ovsienko. On tangent cones of Schubert varieties.Arnold Math. J., 3(4):451–482, 2017

    2017

  16. [24]

    Fulton.Intersection theory, volume 2 ofErgebnisse der Mathematik und ihrer Grenzgebiete

    W. Fulton.Intersection theory, volume 2 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Math- ematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second ...

    1998

  17. [25]

    Goresky, R

    M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul du- ality, and the localization theorem.Invent. Math., 131(1):25–83, 1998

    1998

  18. [26]

    Guay-Paquet

    M. Guay-Paquet. A second proof of the Shareshian–Wachs conjecture, by way of a new Hopf algebra. Preprint, arXiv:1601.05498

    arXiv

  19. [27]

    M. Haiman. Hecke algebra characters and immanant conjectures.J. Amer. Math. Soc., 6(3):569–595, 1993

    1993

  20. [28]

    Harashita

    S. Harashita. On the affineness of distinguished Deligne-Lusztig varieties.J. Algebra, 390:290–297, 2013

    2013

  21. [29]

    Hartshorne.Algebraic geometry, volume No

    R. Hartshorne.Algebraic geometry, volume No. 52 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977

    1977

  22. [30]

    X. He. On the affineness of Deligne-Lusztig varieties.J. Algebra, 320(3):1207–1219, 2008

    2008

  23. [31]

    He and R

    X. He and R. La. Lusztig varieties for regular elements.J. Algebra, 686:845–853, 2026

    2026

  24. [32]

    He and G

    X. He and G. Lusztig. A generalization of Steinberg’s cross section.J. Amer. Math. Soc., 25(3):739–757, 2012

    2012

  25. [33]

    J. C. Jantzen.Representations of algebraic groups, volume 107 ofMathematical Sur- veys and Monographs. American Mathematical Society, Providence, RI, second edi- tion, 2003

    2003

  26. [34]

    Kiem and D

    Y.-H. Kiem and D. Lee. Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture.J. Combin. Theory Ser. A, 206:Paper No. 105884, 2024

    2024

  27. [35]

    S. Kumar. Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture.Invent. Math., 93(1):117–130, 1988

    1988

  28. [36]

    S. Kumar. The nil Hecke ring and singularity of Schubert varieties.Invent. Math., 123(3):471–506, 1996

    1996

  29. [37]

    Lauritzen and J

    N. Lauritzen and J. F. Thomsen. Line bundles on Bott-Samelson varieties.J. Alge- braic Geom., 13(3):461–473, 2004

    2004

  30. [38]

    Lazarsfeld.Positivity in algebraic geometry

    R. Lazarsfeld.Positivity in algebraic geometry. I, volume 48 ofErgebnisse der Math- ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys 40 PATRICK BROSNAN, JAEHYUN ...

    2004

  31. [39]

    G. Lusztig. On the reflection representation of a finite Chevalley group. InRepre- sentation theory of Lie groups, pages 325–337. Cambridge Univ. Press, Cambridge, 1979

    1979

  32. [40]

    G. Lusztig. Character sheaves. I.Adv. in Math., 56(3):193–237, 1985

    1985

  33. [41]

    G. Lusztig. Character sheaves. II, III.Adv. in Math., 57(3):226–265, 266–315, 1985

    1985

  34. [42]

    G. Lusztig. Character sheaves. IV.Adv. in Math., 59(1):1–63, 1986

    1986

  35. [43]

    G. Lusztig. Character sheaves. V.Adv. in Math., 61(2):103–155, 1986

    1986

  36. [44]

    V. B. Mehta and A. Ramanathan. Frobenius splitting and cohomology vanishing for Schubert varieties.Ann. of Math. (2), 122(1):27–40, 1985

    1985

  37. [45]

    V. B. Mehta and A. Ramanathan. Schubert varieties inG/B×G/B.Compositio Math., 67(3):355–358, 1988

    1988

  38. [46]

    Orlik and M

    S. Orlik and M. Rapoport. Deligne-Lusztig varieties and period domains over finite fields.J. Algebra, 320(3):1220–1234, 2008

    2008

  39. [47]

    P. Polo. On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar.Indag. Math. (N.S.), 5(4):483–493, 1994

    1994

  40. [48]

    Precup and E

    M. Precup and E. Sommers. Perverse sheaves, nilpotent Hessenberg varieties, and the modular law.Pure Appl. Math. Q., 21(1):495–540, 2025

    2025

  41. [49]

    Ramanan and A

    S. Ramanan and A. Ramanathan. Projective normality of flag varieties and Schubert varieties.Invent. Math., 79(2):217–224, 1985

    1985

  42. [50]

    Ramanathan

    A. Ramanathan. Equations defining Schubert varieties and Frobenius splitting of diagonals.Inst. Hautes ´Etudes Sci. Publ. Math., (65):61–90, 1987

    1987

  43. [51]

    Shareshian and M

    J. Shareshian and M. L. Wachs. Chromatic quasisymmetric functions.Adv. Math., 295:497–551, 2016

    2016

  44. [52]

    T. A. Springer.Linear algebraic groups. Modern Birkh¨ auser Classics. Birkh¨ auser Boston, Inc., Boston, MA, second edition, 2009

    2009

  45. [53]

    The Stacks Project Authors.Stacks project.https://stacks.math.columbia.edu, 2025

    2025

  46. [54]

    R. P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph.Adv. Math., 111(1):166–194, 1995

    1995

  47. [55]

    Steinberg

    R. Steinberg. Regular elements of semisimple algebraic groups.Inst. Hautes ´Etudes Sci. Publ. Math., (25):49–80, 1965

    1965

  48. [56]

    J. S. Tymoczko. Permutation actions on equivariant cohomology of flag varieties. In Toric topology, volume 460 ofContemp. Math., pages 365–384. Amer. Math. Soc., Providence, RI, 2008

    2008

  49. [57]

    J. S. Tymoczko. Permutation representations on Schubert varieties.Amer. J. Math., 130(5):1171–1194, 2008. Department of Mathematics, University of Maryland, College Park, MD USA Email address:pbrosnan@math.umd.edu Center for Complex Geometry, Institute for Basic Science (IBS),...

    2008

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.