Schubert calculus and intersection theory of flag manifolds
- Autores: Duan H.1,2,3, Zhao X.4
-
Afiliações:
- Yau Mathematical Sciences Center, Tsinghua University
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences
- School of Mathematical Sciences, Dalian University of Technology
- Capital Normal University
- Edição: Volume 77, Nº 4 (2022)
- Páginas: 173-196
- Seção: Articles
- URL: https://journals.rcsi.science/0042-1316/article/view/133707
- DOI: https://doi.org/10.4213/rm10059
- ID: 133707
Citar
Resumo
Palavras-chave
Sobre autores
Haibao Duan
Yau Mathematical Sciences Center, Tsinghua University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, Dalian University of Technology
Email: dhb@math.ac.cn
Xuezhi Zhao
Capital Normal University
Email: zhaoxve@mail.cnu.edu.cn
Bibliografia
- М. Ф. Атья, Ф. Хирцебрух, “Векторные расслоения и однородные пространства”, Математика, 6:2 (1962), 3–39
- H. F. Baker, Principles of geometry, v. 6, Cambridge Univ. Press, Cambridge, 1933, ix+308 pp.
- P. F. Baum, “On the cohomology of homogeneous spaces”, Topology, 7 (1968), 15–38
- И. Н. Бернштейн, И. М. Гельфанд, С. И. Гельфанд, “Клетки Шуберта и когомологии пространств $G/P$”, УМН, 28:3(171) (1973), 3–26
- S. Billey, M. Haiman, “Schubert polynomials for the classical groups”, J. Amer. Math. Soc., 8:2 (1995), 443–482
- A. Borel, “Sur la cohomologie des espaces fibres principaux et des espaces homogènes de groupes de Lie compacts”, Ann. of Math. (2), 57 (1953), 115–207
- A. Borel, “Kählerian coset spaces of semisimple Lie groups”, Proc. Nat. Acad. Sci. U.S.A., 40:12 (1954), 1147–1151
- A. Borel, F. Hirzebruch, “Characteristic classes and homogeneous spaces. I”, Amer. J. Math., 80:2 (1958), 458–538
- Н. Бурбаки, Группы и алгебры Ли, гл. 1–3, Элементы математики, Мир, М., 1976, 496 с.
- C. B. Boyer, A history of mathematics, John Wiley & Sons, Inc., New York–London–Sydney, 1968, xv+717 pp.
- A. S. Buch, “Mutations of puzzles and equivariant cohomology of two-step flag varieties”, Ann. of Math. (2), 182:1 (2015), 173–220
- A. S. Buch, A. Kresch, K. Purbhoo, H. Tamvakis, “The puzzle conjecture for the cohomology of two-step flag manifolds”, J. Algebraic Combin., 44:4 (2016), 973–1007
- M. Chasles, “Construction des coniques qui satisfont à cinq conditions”, C. R. Acad. Sci. Paris, 58 (1864), 297–308
- C. Chevalley, “Sur les decompositions cellulaires des espaces $G/B$”, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math, 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, 1–23
- J. L. Coolidge, A history of geometrical methods, Oxford Univ. Press, New York, 1940, xviii+451 pp.
- I. Coskun, “A Littlewood–Richardson rule for two-step flag varieties”, Invent. Math., 176:2 (2009), 325–395
- W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR – A computer algebra system for polynomial computations
- Y. Dold-Samplonius, “Interview with Bartel Leendert van der Waerden”, Notices Amer. Math. Soc., 44:3 (1997), 313–320
- Haibao Duan, “The degree of a Schubert variety”, Adv. Math., 180:1 (2003), 112–133
- Haibao Duan, “Multiplicative rule of Schubert classes”, Invent. Math., 159:2 (2005), 407–436
- Haibao Duan, “Multiplicative rule in the Grothendieck cohomology of a flag variety”, J. Reine Angew. Math., 2006:600 (2006), 157–176
- Haibao Duan, “On the Borel transgression in the fibration $Gto G/T$”, Homology Homotopy Appl., 20:1 (2018), 79–86
- Haibao Duan, Banghe Li, Topology of blow-ups and enumerative geometry, 2016 (v1 – 2009), 30 pp.
- Haibao Duan, Xuezhi Zhao, Appendix to “The Chow rings of generalized Grassmannians”, 2007 (v1 – 2005), 27 pp.
- Haibao Duan, Xuezhi Zhao, “Algorithm for multiplying Schubert classes”, Internat. J. Algebra Comput., 16:6 (2006), 1197–1210
- Haibao Duan, Xuezhi Zhao, “A unified formula for Steenrod operations in flag manifolds”, Compos. Math., 143:1 (2007), 257–270
- Haibao Duan, Xuezhi Zhao, Schubert calculus and cohomology of Lie groups. Part I. $1$-connected Lie groups, 2015 (v1 – 2007), 32 pp.
- Haibao Duan, Xuezhi Zhao, “Erratum: Multiplicative rule of Schubert classes”, Invent. Math., 177:3 (2009), 683–684
- Haibao Duan, Xuezhi Zhao, “The Chow rings of generalized Grassmannians”, Found. Math. Comput., 10:3 (2010), 245–274
- Haibao Duan, Xuezhi Zhao, “Schubert calculus and the Hopf algebra structures of exceptional Lie groups”, Forum Math., 26:1 (2014), 113–139
- Haibao Duan, Xuezhi Zhao, “Schubert presentation of the cohomology ring of flag manifolds $G/T$”, LMS J. Comput. Math., 18:1 (2015), 489–506
- Haibao Duan, Xuezhi Zhao, “On Schubert's problem of characteristics”, Schubert calculus and its applications in combinatorics and representation theory, Springer Proc. Math. Stat., 332, Springer, Singapore, 2020, 43–71
- C. Ehresmann, “Sur la topologie de certains espaces homogènes”, Ann. of Math. (2), 35:2 (1934), 396–443
- D. Eisenbud, J. Harris, 3264 and all that. A second course in algebraic geometry, Cambridge Univ. Press, Cambridge, 2016, xiv+616 pp.
- S. Fomin, S. Gelfand, A. Postnikov, “Quantum Schubert polynomials”, J. Amer. Math. Soc., 10:3 (1997), 565–596
- S. Fomin, A. N. Kirillov, “Combinatorial $B_n$-analogues of Schubert polynomials”, Trans. Amer. Math. Soc., 348:9 (1996), 3591–3620
- W. Fulton, Young tableaux, With applications to representation theory and geometry, London Math. Soc. Stud. Texts, 35, Cambridge Univ. Press, Cambridge, 1997, x+260 pp.
- W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, 2nd ed., Springer, Berlin, 1998, xiv+470 pp.
- W. Fulton, P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Math., 1689, Springer-Verlag, Berlin, 1998, xii+148 pp.
- G. Z. Giambelli, “Risoluzione del probema degli spazi secanti”, Mem. Accad. Sci. Torino (2), 52 (1903), 171–211
- D. R. Grayson, A. Seceleanu, M. E. Stillman, Computations in intersection rings of flag bundles, 2022 (v1 – 2012), 37 pp.
- D. Grayson, A. Stillman, Macaulay2: a software system for research in algebraic geometry
- D. Hilbert, “Mathematical problems”, Bull. Amer. Math. Soc., 8:10 (1902), 437–479
- D. Husemoller, J. C. Moore, J. Stasheff, “Differential homological algebra and homogeneous spaces”, J. Pure Appl. Algebra, 5:2 (1974), 113–185
- S. Katz, Enumerative geometry and string theory, Stud. Math. Libr., 32, Amer. Math. Soc., Providence, RI, 2006, xiv+206 pp.
- A. N. Kirillov, H. Naruse, “Construction of double Grothendieck polynomials of classical types using IdCoxeter algebras”, Tokyo J. Math., 39:3 (2017), 695–728
- S. L. Kleiman, “Problem 15: rigorous foundation of Schubert's enumerative calculus”, Mathematical developments arising from Hilbert problems (Northern Illinois Univ., De Kalb, IL, 1974), Proc. Sympos. Pure Math., 28, Amer. Math. Soc., Providence, RI, 1976, 445–482
- S. L. Kleiman, “Book review: W. Fulton ‘Intersection theory’ // W. Fulton ‘Introduction to intersection theory in algebraic geometry’”, Bull. Amer. Math. Soc. (N. S.), 12:1 (1985), 137–143
- S. L. Kleiman, “Intersection theory and enumerative geometry: a decade in review”, With the collaboration of A. Thorup, Algebraic geometry, Bowdoin, 1985, Part 2 (Brunswick, ME, 1985), Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987, 321–370
- L. Lascoux, M. P. Schützenberger, “Polynômes de Schubert”, C. R. Acad. Sci. Paris Ser. I Math., 294:13 (1982), 447–450
- S. Lefschetz, “Intersections and transformations of complexes and manifolds”, Trans. Amer. Math. Soc., 28:1 (1926), 1–49
- D. E. Littlewood, A. R. Richardson, “Group characters and algebra”, Philos. Trans. Roy. Soc. London Ser. A, 233 (1934), 99–141
- Ю. И. Манин, “К пятнадцатой проблеме Гильберта”, Проблемы Гильберта, Наука, М., 1969, 175–181
- R. Marlin, “Anneaux de Chow des groupes algebriques $operatorname{SU}(n)$, $operatorname{Sp}(n)$, $operatorname{SO}(n)$, $operatorname{Spin}(n)$, $mathrm G_{2}$, $mathrm F_{4}$; torsion”, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 119–122
- Дж. Милнор, Дж. Сташеф, Характеристические классы, Мир, М., 1979, 371 с.
- S. I. Nikolenko, N. S. Semenov, Chow ring structure made simple, 2006, 17 pp.
- J. Scherk, Algebra. A computational introduction, Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton, FL, 2000, x+319 pp.
- H. Schubert, “Zur Theorie der Charakteristiken”, J. Reine Angew. Math., 1870:71 (1870), 366–386
- H. Schubert, “Anzahl-Bestimmungen für lineare Räume. Beliebiger Dimension”, Acta Math., 8 (1886), 97–118
- H. Schubert, “Lösung des Characteristiken-Problems für lineare Räume beliebiger Dimension”, Mitt. Math. Ges. Hamburg, 1 (1886), 134–155
- H. Schubert, Kalkül der abzählenden Geometrie, Reprint of the 1879 original, Springer-Verlag, Berlin–New York, 1979, 349 pp.
- F. Severi, “Sul principio della conservazione del numero”, Rend. Circ. Mat. Palermo, 33 (1912), 313–327
- F. Severi, “Sui fondamenti della geometria numerativa e sulla teoria delle caratteristiche”, Atti Ist. Veneto Sci. Lett. Art., 75 (1916), 1121–1162
- E. Smirnov, A. Tutubalina, Pipe dreams for Schubert polynomials of the classical groups, 2020, 36 pp.
- F. Sottile, Schubert calculus, Springer Encyclopedia of Mathematics, 2012
- H. Tamvakis, “Giambelli and degeneracy locus formulas for classical $G/P$ spaces”, Mosc. Math. J., 16:1 (2016), 125–177
- H. Toda, “On the cohomology ring of some homogeneous spaces”, J. Math. Kyoto Univ., 15 (1975), 185–199
- B. L. van der Waerden, “Topologische Begründung des Kalküls der abzählenden Geometrie”, Math. Ann., 102:1 (1930), 337–362
- B. L. van der Waerden, “The foundation of algebraic geometry from Severi to Andre Weil”, Arch. History Exact Sci., 7:3 (1971), 171–180
- A. Weil, Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publ., XXIV, Rev. ed., Amer. Math. Soc., Providence, RI, 1962, xx+363 pp.
- J. Wolf, “The cohomology of homogeneous spaces”, Amer. J. Math., 99:2 (1977), 312–340
Arquivos suplementares
