pith. sign in

arxiv: 2606.07142 · v1 · pith:WMDY3GP5new · submitted 2026-06-05 · 🧮 math.AG · math.AC· math.CO· math.KT

Polyhedral models for K-theory of toric and flag varieties

Leonid Monin , Evgeny Smirnov This is my paper

Pith reviewed 2026-06-27 21:10 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.COmath.KT
keywords K-theorytoric varietiesflag varietiesvirtual polytopesGelfand-Zetlin polytopesSchubert varietiesequivariant K-theory
0
0 comments X

The pith

The K-theory of toric and flag varieties arises as K-rings from linear families of virtual polytopes.

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

The paper extends polyhedral constructions previously used for cohomology rings to the setting of K-theory. It defines an abstract K-ring for any linear family of virtual polytopes by taking a suitable quotient of the group algebra of a free abelian group, then specializes to two families: integer polytopes sharing a fixed normal fan, and Gelfand-Zetlin polytopes. These abstract rings are shown to coincide with the K-theory rings of toric varieties and flag varieties, respectively, producing explicit relations and identifying the classes of structure sheaves of orbit closures and Schubert varieties. The same identifications hold after passing to the T-equivariant K-theory.

Core claim

We study Frobenius algebras obtained as quotients of the group algebra of a free abelian group. We apply this construction to define a K-ring associated to a linear family of virtual polytopes. For the family of integer polytopes with fixed normal fan and for the family of Gelfand-Zetlin polytopes, these K-rings realize the K-theory of toric varieties and flag varieties. This yields natural sets of relations in the K-rings and descriptions of the classes of structure sheaves of toric orbit closures and of Schubert varieties in type A flag varieties. The results hold in the T-equivariant setting as well.

What carries the argument

The K-ring of a linear family of virtual polytopes, obtained as a quotient of the group algebra of a free abelian group carrying a Frobenius algebra structure.

If this is right

  • The K-theory rings of toric varieties and flag varieties admit explicit presentations coming from the chosen polytope families.
  • Natural relations among generators in these K-rings follow directly from the linear family structure.
  • The classes of structure sheaves of toric orbit closures are explicitly realized inside the associated K-ring.
  • The classes of structure sheaves of Schubert varieties in type A flag varieties are explicitly realized inside the associated K-ring.
  • All of the above identifications and relations continue to hold after base change to the T-equivariant K-theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The polytope models could be used to perform explicit multiplication tables in K-theory by counting lattice points or using volume formulas.
  • Similar quotient constructions might supply combinatorial models for K-theory of other varieties that admit polyhedral descriptions.
  • The equivariant versions may interact with localization sequences or fixed-point formulas in a purely combinatorial way.

Load-bearing premise

The abstract K-ring constructed from the linear family of virtual polytopes is identical to the geometric K-theory ring of the corresponding toric or flag variety.

What would settle it

A direct calculation of the product of two generators in the polytope-based K-ring that differs from the known product in the K-theory ring of a concrete toric surface or a small flag variety.

Figures

Figures reproduced from arXiv: 2606.07142 by Evgeny Smirnov, Leonid Monin.

Figure 1
Figure 1. Figure 1: Gelfand–Zetlin polytope in dimension 3 The following proposition is immediate. Proposition 6.2. For a given n, all Gelfand–Zetlin polytopes have the same normal fan. The Ehrhart polynomial of GZ(λ) is equal to the Weyl polynomial of GL(n): Ehr(GZ(λ)) = FGL(n) = Y i<j λi − λj − i + j j − i . We denote the lattice of (possibly virtual) integer Gelfand–Zetlin polytopes by PGZ. Theo￾rem 5.9 together with Propo… view at source ↗
Figure 2
Figure 2. Figure 2: Six-term relations Remark 6.8. The linear parts of these relations are exactly the linear relations in the Pukhlikov– Khovanskii ring of a Gelfand–Zetlin polytope, see [KST12, Prop. 3.2]. This corresponds to replacing the operator of shift by eij by taking the directional derivative along the same vector. Example 6.9. Let n = 3. In this case, all “six-term” relations contain only four or two terms; they ar… view at source ↗
Figure 3
Figure 3. Figure 3: Diagrams of Kogan faces 22 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
read the original abstract

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov-Khovanskii type presentation to the case of K-theory of toric and flag varieties. First, we study the Frobenius algebras obtained as quotients of the group algebra of free abelian group (possibly of infinite rank). Then we apply this construction to define a K-ring associated to a linear family of (virtual) polytopes. We study in detail two examples of such families: the family of integer (virtual) polytopes with a fixed normal fan and the family of (virtual) Gelfand-Zetlin polytopes. We show that the K-theory of toric and flag varieties can be realized as K-rings of the above families and use this to get natural set of relations in the above K-rings. Further, we describe the classes of structure sheaves of toric orbit closures and Schubert varieties in type A flag varieties. Finally, we show that our results also hold true in T-equivariant setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper extends Pukhlikov-Khovanskii and Kaveh constructions from cohomology to K-theory by defining K-rings as quotients of group algebras of free abelian groups (possibly infinite rank) equipped with Frobenius algebra structures induced by linear families of virtual polytopes. It applies this to two families: integer virtual polytopes with fixed normal fan (for toric varieties) and Gelfand-Zetlin polytopes (for type A flag varieties). The central claims are that these abstract K-rings realize the geometric K-theory rings of the varieties (including T-equivariant versions), yielding natural relations, and that the classes of structure sheaves of toric orbit closures and Schubert varieties can be described explicitly in these models.

Significance. If the ring isomorphisms hold, this supplies combinatorial polyhedral models for K-theory rings, generalizing the cohomology results and providing explicit descriptions of geometric classes together with relations in the rings. The general development of Frobenius algebras from group-algebra quotients is a reusable tool. The equivariant extension is a positive feature. Significance is conditional on verification that the abstract multiplication matches the geometric tensor product.

major comments (3)
  1. [Main theorems on realization for toric and flag varieties (sections developing the two families)] The main theorems asserting that the K-ring of the fixed-normal-fan family realizes K(X) for a toric variety X (and likewise for the Gelfand-Zetlin family and flag varieties) rest on direct comparison with known presentations of the K-rings. This comparison must be shown to preserve the multiplicative structure: the Frobenius product induced by the linear family of virtual polytopes must coincide with the tensor product of coherent sheaves. Without an explicit argument or computation verifying agreement on a basis (e.g., classes of structure sheaves), the claimed ring isomorphism is not established, undermining the extraction of relations and the descriptions of orbit-closure and Schubert classes.
  2. [Section describing classes of structure sheaves] The descriptions of the classes of structure sheaves of toric orbit closures and of Schubert varieties in type A flag varieties are derived inside the abstract K-ring and transferred to geometry via the asserted isomorphism. Because the isomorphism is load-bearing for the ring structure, these descriptions require the same verification that the product matches the geometric one; otherwise the classes may not satisfy the correct multiplication relations in K(X).
  3. [Equivariant setting] The T-equivariant extension asserts analogous realizations and descriptions. The same requirement applies: the equivariant Frobenius product must be shown to match the equivariant tensor product, again beyond matching additive groups or generators.
minor comments (2)
  1. [Abstract] The abstract refers to a 'natural set of relations' without indicating their explicit form; a brief description or example would aid readability.
  2. [General construction of abstract K-rings] Notation for the group algebra, the quotient, and the induced Frobenius structure should be introduced once and used consistently across the general theory and the two examples.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to explicitly verify that the multiplicative structures agree. We address each major comment below and will revise the manuscript to include the requested checks.

read point-by-point responses

  1. Referee: [Main theorems on realization for toric and flag varieties (sections developing the two families)] The main theorems asserting that the K-ring of the fixed-normal-fan family realizes K(X) for a toric variety X (and likewise for the Gelfand-Zetlin family and flag varieties) rest on direct comparison with known presentations of the K-rings. This comparison must be shown to preserve the multiplicative structure: the Frobenius product induced by the linear family of virtual polytopes must coincide with the tensor product of coherent sheaves. Without an explicit argument or computation verifying agreement on a basis (e.g., classes of structure sheaves), the claimed ring isomorphism is not established, undermining the extraction of relations and the descriptions of orbit-closure and Schubert classes.

    Authors: We agree that an explicit verification of the multiplicative structure on a basis is necessary to fully establish the ring isomorphism. While the manuscript compares generators and relations with known presentations of the K-rings, we will add a new subsection with direct computations confirming that the Frobenius product induced by the linear families agrees with the geometric tensor product on the classes of structure sheaves, for both the toric and flag variety cases. revision: yes

  2. Referee: [Section describing classes of structure sheaves] The descriptions of the classes of structure sheaves of toric orbit closures and of Schubert varieties in type A flag varieties are derived inside the abstract K-ring and transferred to geometry via the asserted isomorphism. Because the isomorphism is load-bearing for the ring structure, these descriptions require the same verification that the product matches the geometric one; otherwise the classes may not satisfy the correct multiplication relations in K(X).

    Authors: The descriptions of the structure sheaf classes rely on the ring isomorphism. Once the explicit multiplicative verification is added as described above, the transferred classes will satisfy the correct relations. We will also insert a clarifying remark in this section noting the dependence on the verified isomorphism. revision: yes

  3. Referee: [Equivariant setting] The T-equivariant extension asserts analogous realizations and descriptions. The same requirement applies: the equivariant Frobenius product must be shown to match the equivariant tensor product, again beyond matching additive groups or generators.

    Authors: We will extend the new verification subsection to the equivariant setting, providing explicit checks that the equivariant Frobenius product coincides with the equivariant tensor product on the relevant basis, using the known presentations of equivariant K-rings. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external known presentations and non-author citations.

full rationale

The paper defines an abstract Frobenius algebra from quotients of group algebras attached to linear families of virtual polytopes, then asserts an isomorphism to geometric K-theory of toric and flag varieties by direct comparison to independently known presentations of those K-rings (extending cited Pukhlikov-Khovanskii and Kaveh constructions for cohomology). No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the cited prior works are by different authors, and the algebra-structure match is presented as following from the comparison rather than being presupposed. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted or verified from the provided text.

pith-pipeline@v0.9.1-grok · 5765 in / 1252 out tokens · 18901 ms · 2026-06-27T21:10:55.346978+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

99 extracted references · 32 canonical work pages

  1. [1]

    1916 , PAGES =

    Macaulay, Francis Sowerby , TITLE =. 1916 , PAGES =

    1916

  2. [2]

    Macaulay, Francis Sowerby , TITLE =. Proc. London Math. Soc. (2) , FJOURNAL =. 1927 , PAGES =. doi:10.1112/plms/s2-26.1.531 , URL =

    work page doi:10.1112/plms/s2-26.1.531 1927

  3. [3]

    Dupraz, Matthew and Gross, Andreas and Monin, Leonid , title =

  4. [4]

    and Knudsen, Finn Faye and Mumford, D

    Kempf, G. and Knudsen, Finn Faye and Mumford, D. and Saint-Donat, B. , TITLE =. 1973 , PAGES =

    1973

  5. [5]

    arXiv preprint arXiv:2601.22514 , year=

    Note on Euler characteristic of a toric vector bundle , author=. arXiv preprint arXiv:2601.22514 , year=

    Pith/arXiv arXiv

  6. [6]

    1995 , pages =

    Eisenbud, David , TITLE =. 1995 , PAGES =. doi:10.1007/978-1-4612-5350-1 , URL =

    work page doi:10.1007/978-1-4612-5350-1 1995

  7. [7]

    Intersection theory , Url =

    Fulton, William , Date-Added =. Intersection theory , Url =. 1998 , Bdsk-Url-1 =. doi:10.1007/978-1-4612-1700-8 , Edition =

    work page doi:10.1007/978-1-4612-1700-8 1998

  8. [8]

    Intersections of hypersurfaces and ring of conditions of a spherical homogeneous space , Url =

    Kaveh, Kiumars and Khovanskii, Askold , Date-Added =. Intersections of hypersurfaces and ring of conditions of a spherical homogeneous space , Url =. SIGMA Symmetry Integrability Geom. Methods Appl. , Pages =. 2020 , Bdsk-Url-1 =. doi:10.3842/SIGMA.2020.016 , Fjournal =

    work page doi:10.3842/sigma.2020.016 2020

  9. [9]

    Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in

    Khovanskii, Askold , journal=. Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in. 1986 , publisher=

    1986

  10. [10]

    Duke Math

    Kaveh, Kiumars , TITLE =. Duke Math. J. , FJOURNAL =. 2015 , NUMBER =. doi:10.1215/00127094-3146389 , URL =

    work page doi:10.1215/00127094-3146389 2015

  11. [11]

    Algebraic topology , Url =

    Hatcher, Allen , Date-Added =. Algebraic topology , Url =. 2002 , Bdsk-Url-1 =

    2002

  12. [12]

    Knop, Friedrich , Booktitle =. The

  13. [13]

    Plongements d'espaces homog\`enes , Url =

    Luna, Dominique and Vust, Thierry , Date-Added =. Plongements d'espaces homog\`enes , Url =. Comment. Math. Helv. , Number =. 1983 , Bdsk-Url-1 =. doi:10.1007/BF02564633 , Fjournal =

    work page doi:10.1007/bf02564633 1983

  14. [14]

    Principles of algebraic geometry , Url =

    Griffiths, Philipp and Harris, Joseph , Date-Added =. Principles of algebraic geometry , Url =. 1994 , Bdsk-Url-1 =. doi:10.1002/9781118032527 , Isbn =

    work page doi:10.1002/9781118032527 1994

  15. [15]

    Pukhlikov, Alexander and Khovanskii, Askold , Date-Added =. The. Algebra i Analiz , Number =

  16. [16]

    and Horja, R

    Borisov, Lev A. and Horja, R. Paul , TITLE =. Snowbird lectures on string geometry , SERIES =. 2006 , MRCLASS =. doi:10.1090/conm/401/07551 , URL =

    work page doi:10.1090/conm/401/07551 2006

  17. [17]

    The number of roots of a system of equations , Volume =

    Bernstein, David , Date-Added =. The number of roots of a system of equations , Volume =. Funkcional. Anal. i Prilo

  18. [18]

    Newton polyhedra , Volume =

    Bernstein, David and Kushnirenko, Anatoly and Khovanskii, Askold , Date-Added =. Newton polyhedra , Volume =. Uspehi Mat. Nauk , Number =

  19. [19]

    Poly\`edres de

    Kouchnirenko, Anatoly , Date-Added =. Poly\`edres de. Invent. Math. , Number =. 1976 , Bdsk-Url-1 =. doi:10.1007/BF01389769 , Fjournal =

    work page doi:10.1007/bf01389769 1976

  20. [20]

    Uspekhi Mat

    Panina, Gaiane and Streinu, Ileana , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 2015 , NUMBER =

    2015

  21. [21]

    Annals of mathematics , pages=

    Some theorems on actions of algebraic groups , author=. Annals of mathematics , pages=. 1973 , publisher=

    1973

  22. [22]

    The Ring of Conditions for Horospherical Homogeneous Spaces , Year =

    Hofscheier, Johannes , Booktitle =. The Ring of Conditions for Horospherical Homogeneous Spaces , Year =

  23. [23]

    Ziegler, G. M. , Doi =. Lectures on polytopes , Url =. 1995 , Bdsk-Url-1 =

    1995

  24. [24]

    Sur la cohomologie des espaces fibr\'

    Borel, Armand , Doi =. Sur la cohomologie des espaces fibr\'. Ann. of Math. (2) , Pages =. 1953 , Bdsk-Url-1 =

    1953

  25. [25]

    , Isbn =

    Bott, Raoul and Tu, Loring W. , Isbn =. Differential forms in algebraic topology , Volume =

  26. [26]

    Fonctions d'une infinit

    Gateaux, Ren. Fonctions d'une infinit. Bulletin de la Soci

  27. [27]

    Finitely additive measures of virtual polyhedra , Volume =

    Pukhlikov, Alexander and Khovanskii, Askold , Fjournal =. Finitely additive measures of virtual polyhedra , Volume =. Algebra i Analiz , Number =

  28. [28]

    Pasquier, Boris , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 2008 , NUMBER =

    2008

  29. [29]

    arXiv preprint arXiv:2101.03643 , year=

    Primary Decomposition with Differential Operators , author=. arXiv preprint arXiv:2101.03643 , year=

    arXiv

  30. [30]

    arXiv preprint arXiv:2104.10146 , year=

    Linear PDE with Constant Coefficients , author=. arXiv preprint arXiv:2104.10146 , year=

    arXiv

  31. [31]

    Elias, Juan and Rossi, Maria Evelina , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2017.04.025 , URL =

    work page doi:10.1016/j.aim.2017.04.025 2017

  32. [32]

    Schulze, Mathias and Tozzo, Laura , TITLE =. J. Algebra , FJOURNAL =. 2019 , PAGES =

    2019

  33. [33]

    and Rossi, M

    Elias, J. and Rossi, M. E. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2012 , NUMBER =

    2012

  34. [34]

    2011 , publisher=

    Toric varieties , author=. 2011 , publisher=

    2011

  35. [35]

    Gr. On the. ACM Commun. Comput. Algebra , FJOURNAL =. 2010 , NUMBER =

    2010

  36. [36]

    Lecture Note for KIAS special lectures, Seoul, South Korea , year=

    Introduction to spherical varieties and description of special classes of spherical varieties , author=. Lecture Note for KIAS special lectures, Seoul, South Korea , year=

  37. [37]

    Complete symmetric varieties

    De Concini, Corrado and Procesi, Claudio , Booktitle =. Complete symmetric varieties

  38. [38]

    The transversality of a general translate , Volume =

    Kleiman, Steven , Fjournal =. The transversality of a general translate , Volume =. Compositio Math. , Pages =

  39. [39]

    Cohomology of toric bundles , Url =

    Sankaran, Parameshvaran and Uma, Vikraman , Doi =. Cohomology of toric bundles , Url =. Comment. Math. Helv. , Number =. 2003 , Bdsk-Url-1 =

    2003

  40. [40]

    Piecewise polynomial functions, convex polytopes and enumerative geometry , Volume =

    Brion, Michel , Booktitle =. Piecewise polynomial functions, convex polytopes and enumerative geometry , Volume =

  41. [41]

    Homogeneous spaces and equivariant embeddings , Url =

    Timashev, Dmitri , Doi =. Homogeneous spaces and equivariant embeddings , Url =. 2011 , Bdsk-Url-1 =

    2011

  42. [42]

    Moment polytopes, semigroup of representations and

    Kaveh, Kiumars and Khovanskii, Askold , Doi =. Moment polytopes, semigroup of representations and. J. Fixed Point Theory Appl. , Number =. 2010 , Bdsk-Url-1 =

    2010

  43. [43]

    Note on cohomology rings of spherical varieties and volume polynomial , Volume =

    Kaveh, Kiumars , Fjournal =. Note on cohomology rings of spherical varieties and volume polynomial , Volume =. J. Lie Theory , Number =

  44. [44]

    Kaveh, Kiumars and Villella, Elise , TITLE =. J. Comb. Algebra , FJOURNAL =. 2022 , NUMBER =. doi:10.4171/jca/56 , URL =

    work page doi:10.4171/jca/56 2022

  45. [45]

    Demazure, Michel , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1974 , PAGES =

    1974

  46. [46]

    Demazure, Michel , TITLE =. Bull. Sci. Math. (2) , FJOURNAL =. 1974 , NUMBER =

    1974

  47. [47]

    and Brion, M

    Alexeev, V. and Brion, M. , Date-Modified =. Toric degenerations of spherical varieties , Volume =. Selecta Mathematica , Number =

  48. [48]

    Cones, crystals, and patterns , Volume =

    Littelmann, Peter , Date-Modified =. Cones, crystals, and patterns , Volume =. Transformation groups , Number =

  49. [49]

    2016 , PAGES =

    Pun, Anna Ying , TITLE =. 2016 , PAGES =

    2016

  50. [50]

    Tensor product multiplicities, canonical bases and totally positive varieties , Url =

    Berenstein, Arkady and Zelevinsky, Andrei , Date-Modified =. Tensor product multiplicities, canonical bases and totally positive varieties , Url =. Invent. Math. , Mrclass =. 2001 , Bdsk-Url-1 =. doi:10.1007/s002220000102 , Fjournal =

    work page doi:10.1007/s002220000102 2001

  51. [51]

    Inventiones mathematicae , volume=

    Projective modules over polynomial rings , author=. Inventiones mathematicae , volume=. 1976 , publisher=

    1976

  52. [52]

    Arnold Math

    Hofscheier, Johannes and Khovanskii, Askold and Monin, Leonid , TITLE =. Arnold Math. J. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s40598-023-00233-6 , URL =

    work page doi:10.1007/s40598-023-00233-6 2024

  53. [53]

    Uspekhi Matematicheskikh Nauk , volume=

    Newton polyhedra and tropical geometry , author=. Uspekhi Matematicheskikh Nauk , volume=. 2021 , publisher=

    2021

  54. [54]

    arXiv preprint arXiv:2106.15562 , year=

    Gorenstein algebras and toric bundles , author=. arXiv preprint arXiv:2106.15562 , year=

    arXiv

  55. [55]

    Linear Algebra and its Applications , volume=

    VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A6 , author=. Linear Algebra and its Applications , volume=. 2018 , publisher=

    2018

  56. [56]

    Journal of Commutative Algebra , volume=

    A criterion for isomorphism of Artinian Gorenstein algebras , author=. Journal of Commutative Algebra , volume=. 2016 , publisher=

    2016

  57. [57]

    Arnold Mathematical Journal , volume=

    Volume polynomials and duality algebras of multi-fans , author=. Arnold Mathematical Journal , volume=. 2016 , publisher=

    2016

  58. [58]

    Schubert calculus and

    Kirichenko, Valentina and Smirnov, Evgeny and Timorin, Vladlen , journal=. Schubert calculus and. 2012 , publisher=

    2012

  59. [59]

    Convex polytopes,

    Davis, Michael and Januszkiewicz, Tadeusz , journal=. Convex polytopes,. 1991 , publisher=

    1991

  60. [60]

    An analogue of the

    Timorin, Vladlen , journal=. An analogue of the. 1999 , publisher=

    1999

  61. [61]

    Tohoku Math

    Sankaran, Parameswaran , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 2008 , NUMBER =. doi:10.2748/tmj/1232376162 , URL =

    work page doi:10.2748/tmj/1232376162 2008

  62. [62]

    Finite-dimensional representations of the group of unimodular matrices , JOURNAL =

    Gel. Finite-dimensional representations of the group of unimodular matrices , JOURNAL =. 1950 , PAGES =

    1950

  63. [63]

    Topics in cohomological studies of algebraic varieties , SERIES =

    Brion, Michel , TITLE =. Topics in cohomological studies of algebraic varieties , SERIES =. 2005 , MRCLASS =. doi:10.1007/3-7643-7342-3_2 , URL =

    work page doi:10.1007/3-7643-7342-3_2 2005

  64. [64]

    2005 , PAGES =

    Brion, Michel and Kumar, Shrawan , TITLE =. 2005 , PAGES =

    2005

  65. [65]

    2015 , PAGES =

    Beck, Matthias and Robins, Sinai , TITLE =. 2015 , PAGES =. doi:10.1007/978-1-4939-2969-6 , URL =

    work page doi:10.1007/978-1-4939-2969-6 2015

  66. [66]

    Bott, Raoul , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1957 , PAGES =. doi:10.2307/1969996 , URL =

    work page doi:10.2307/1969996 1957

  67. [67]

    Makhlin, Igor , title =. Funct. Anal. Appl. , issn =. 2016 , language =. doi:10.1007/s10688-016-0135-2 , keywords =

    work page doi:10.1007/s10688-016-0135-2 2016

  68. [68]

    Brion, Michel , title =. Ann. Sci. 1988 , language =. doi:10.24033/asens.1572 , keywords =

    work page doi:10.24033/asens.1572 1988

  69. [69]

    1979 , PAGES =

    Schubert, Hermann , TITLE =. 1979 , PAGES =

    1979

  70. [70]

    Kleiman, Steven and Laksov, Dan , TITLE =. Amer. Math. Monthly , FJOURNAL =. 1972 , PAGES =

    1972

  71. [71]

    1998 , PAGES =

    Manivel, Laurent , TITLE =. 1998 , PAGES =

    1998

  72. [72]

    Bernstein, Joseph N. and Gel. Schubert cells, and the cohomology of the spaces. Uspehi Mat. Nauk , FJOURNAL =. 1973 , NUMBER =

    1973

  73. [73]

    Symmetry and flag manifolds , BOOKTITLE =

    Lascoux, Alain and Sch. Symmetry and flag manifolds , BOOKTITLE =. 1983 , MRCLASS =. doi:10.1007/BFb0063238 , URL =

    work page doi:10.1007/bfb0063238 1983

  74. [74]

    Polyn\^omes de

    Lascoux, Alain and Sch. Polyn\^omes de. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1982 , NUMBER =. doi:10.1090/conm/088/1000001 , URL =

    work page doi:10.1090/conm/088/1000001 1982

  75. [75]

    , TITLE =

    Fomin, Sergey and Kirillov, Anatol N. , TITLE =. Discrete Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1016/0012-365X(95)00132-G , URL =

    work page doi:10.1016/0012-365x(95)00132-g 1996

  76. [76]

    , TITLE =

    Fomin, Sergey and Kirillov, Anatol N. , TITLE =. J. Algebraic Combin. , FJOURNAL =. 1997 , NUMBER =. doi:10.1023/A:1008694825493 , URL =

    work page doi:10.1023/a:1008694825493 1997

  77. [77]

    1997 , PAGES =

    Fulton, William , TITLE =. 1997 , PAGES =

    1997

  78. [78]

    , TITLE =

    Stanley, Richard P. , TITLE =. 2012 , PAGES =

    2012

  79. [79]

    , TITLE =

    Stanley, Richard P. , TITLE =. 1999 , PAGES =. doi:10.1017/CBO9780511609589 , URL =

    work page doi:10.1017/cbo9780511609589 1999

  80. [80]

    , TITLE =

    Wachs, Michelle L. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1985 , NUMBER =. doi:10.1016/0097-3165(85)90091-3 , URL =

    work page doi:10.1016/0097-3165(85)90091-3 1985

Showing first 80 references.