Chern class formulas for $G_2$ Schubert loci
HTML articles powered by AMS MathViewer
- by Dave Anderson PDF
- Trans. Amer. Math. Soc. 363 (2011), 6615-6646 Request permission
Abstract:
We define degeneracy loci for vector bundles with structure group $G_2$ and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by BernsteinâGelfandâGelfand and Demazure. This has been extended to the setting of general algebraic geometry by GiambelliâThomâPorteous, KempfâLaksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham.
In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham.
References
- Ilka Agricola, Old and new on the exceptional group $G_2$, Notices Amer. Math. Soc. 55 (2008), no. 8, 922â929. MR 2441524
- D. Anderson, Degeneracy loci and $G_2$ flags, Ph.D. thesis, University of Michigan, 2009, http://www.math.washington.edu/$\sim$dandersn/papers/thesis.pdf
- D. Anderson, âDegeneracy of triality-symmetric morphisms,â arXiv:0901.1347.
- Michael Aschbacher, Chevalley groups of type $G_2$ as the group of a trilinear form, J. Algebra 109 (1987), no. 1, 193â259. MR 898346, DOI 10.1016/0021-8693(87)90173-6
- John C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145â205. MR 1886087, DOI 10.1090/S0273-0979-01-00934-X
- I. N. BernĆĄteÄn, I. M. GelâČfand, and S. I. GelâČfand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3â26 (Russian). MR 0429933
- Sara Billey and Mark Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), no. 2, 443â482. MR 1290232, DOI 10.1090/S0894-0347-1995-1290232-1
- R. Bott and H. Samelson, The cohomology ring of $G/T$, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 490â493. MR 71773, DOI 10.1073/pnas.41.7.490
- Robert L. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987), no. 3, 525â576. MR 916718, DOI 10.2307/1971360
- H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561â578. MR 19111
- Michel Demazure, DĂ©singularisation des variĂ©tĂ©s de Schubert gĂ©nĂ©ralisĂ©es, Ann. Sci. Ăcole Norm. Sup. (4) 7 (1974), 53â88 (French). MR 354697
- Dan Edidin and William Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995), no. 2, 277â299. MR 1333501, DOI 10.1215/S0012-7094-95-07812-0
- LĂĄszlĂł M. FehĂ©r and RichĂĄrd RimĂĄnyi, Schur and Schubert polynomials as Thom polynomialsâcohomology of moduli spaces, Cent. Eur. J. Math. 1 (2003), no. 4, 418â434. MR 2040647, DOI 10.2478/BF02475176
- Sergey Fomin and Anatol N. Kirillov, Combinatorial $B_n$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591â3620. MR 1340174, DOI 10.1090/S0002-9947-96-01558-9
- William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381â420. MR 1154177, DOI 10.1215/S0012-7094-92-06516-1
- William Fulton, Schubert varieties in flag bundles for the classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 241â262. MR 1360506
- William Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom. 43 (1996), no. 2, 276â290. MR 1424427
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik 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 in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 1639468, DOI 10.1007/BFb0096380
- R. Skip Garibaldi, Structurable algebras and groups of type $E_6$ and $E_7$, J. Algebra 236 (2001), no. 2, 651â691. MR 1813495, DOI 10.1006/jabr.2000.8514
- William Graham, The class of the diagonal in flag bundles, J. Differential Geom. 45 (1997), no. 3, 471â487. MR 1472885
- William Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), no. 3, 599â614. MR 1853356, DOI 10.1215/S0012-7094-01-10935-6
- Stephen Griffeth and Arun Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 (2004), no. 8, 1263â1283. MR 2095481, DOI 10.1016/j.ejc.2003.10.012
- Joe Harris and Loring W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71â84. MR 721453, DOI 10.1016/0040-9383(84)90026-0
- F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR 1045637
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- Atanas Iliev and Laurent Manivel, The Chow ring of the Cayley plane, Compos. Math. 141 (2005), no. 1, 146â160. MR 2099773, DOI 10.1112/S0010437X04000788
- T. JĂłzefiak, A. Lascoux, and P. Pragacz, Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 3, 662â673 (Russian). MR 623355
- Allen Knutson and Ezra Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245â1318. MR 2180402, DOI 10.4007/annals.2005.161.1245
- A. Kresch and H. Tamvakis, Double Schubert polynomials and degeneracy loci for the classical groups, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1681â1727 (English, with English and French summaries). MR 1952528
- J. M. Landsberg and L. Manivel, The sextonions and $E_{7\frac 12}$, Adv. Math. 201 (2006), no. 1, 143â179. MR 2204753, DOI 10.1016/j.aim.2005.02.001
- Alain Lascoux and Piotr Pragacz, Operator calculus for $\widetilde Q$-polynomials and Schubert polynomials, Adv. Math. 140 (1998), no. 1, 1â43. MR 1656481, DOI 10.1006/aima.1998.1757
- I. G. Macdonald, Notes on Schubert Polynomials, Publ. LACIM 6, Univ. de Québec à Montréal, Montréal, 1991.
- Adam ParusiĆski and Piotr Pragacz, Chern-Schwartz-MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc. 8 (1995), no. 4, 793â817. MR 1311826, DOI 10.1090/S0894-0347-1995-1311826-0
- Holger P. Petersson, Composition algebras over algebraic curves of genus zero, Trans. Amer. Math. Soc. 337 (1993), no. 1, 473â493. MR 1108613, DOI 10.1090/S0002-9947-1993-1108613-X
- S. PumplĂŒn, Albert algebras over curves of genus zero and one, J. Algebra 320 (2008), no. 12, 4178â4214. MR 2464101, DOI 10.1016/j.jalgebra.2008.09.011
- Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974, DOI 10.1007/978-3-662-12622-6
- Burt Totaro, The Chow ring of a classifying space, Algebraic $K$-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 249â281. MR 1743244, DOI 10.1090/pspum/067/1743244
Additional Information
- Dave Anderson
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Address at time of publication: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 734392
- Email: dandersn@umich.edu, dandersn@math.washington.edu
- Received by editor(s): August 31, 2009
- Received by editor(s) in revised form: February 2, 2010
- Published electronically: July 19, 2011
- Additional Notes: This work was partially supported by NSF Grants DMS-0502170 and DMS-0902967.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6615-6646
- MSC (2010): Primary 14N15; Secondary 14M15, 20G41, 05E05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05317-1
- MathSciNet review: 2833570