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SL2-Regular Subvarieties of Complete Quadrics

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Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Part of the book series: Fields Institute Communications ((FIC,volume 71))

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Abstract

We determine SL n -stable, SL2-regular subvarieties of the variety of complete quadrics. We use the methods of Akyıldız and Carrell given in Proc Natl Acad Sci USA 86(11):3934–3937, 1989 to give a factorization of Poincaré polynomials of these regular subvarieties.

We dedicate our work with admiration to Professors Mohan Putcha and Lex Renner on the occasion of their 60th birthday.

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References

  1. Akyıldız, E., Carrell, J.B.: Cohomology of projective varieties with regular SL2 actions. Manuscr. Math. 58(4), 473–486 (1987)

    Article  MATH  Google Scholar 

  2. Akyıldız, E., Carrell, J.B.: A generalization of the Kostant-Macdonald identity. Proc. Natl. Acad. Sci. U. S. A. 86(11), 3934–3937 (1989)

    Article  MATH  Google Scholar 

  3. Akyıldız, E., Carrell, J.B., Lieberman, D.I., Sommese, A.J.: On the graded rings associated to holomorphic vector fields with exactly one zero. In: Singularities, Part 1, Arcata, 1981. Volume 40 of Proceedings of Symposia in Pure Mathematics, pp. 55–56. American Mathematical Society (1983)

    Google Scholar 

  4. Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98(2), 480–497 (1973)

    Article  MATH  Google Scholar 

  5. Bifet, E., De Concini, C., Procesi, C.: Cohomology of regular embeddings. Adv. Math. 82(1), 1–34 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourbaki, N.: Lie Groups and Lie Algebras, Chapters 4–6. Elements of Mathematics. Springer, Berlin (2002). (Translated from the 1968 French original by Andrew Pressley)

    Book  MATH  Google Scholar 

  7. Brion, M., Joshua, R.: Equivariant Chow ring and Chern classes of wonderful symmetric varieites of minimal rank. Transform. Groups 13(3–4), 471–493 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carter, R.W.: Finite Groups of Lie Type. Wiley Classics Library. Wiley, Chichester (1993). Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication

    Google Scholar 

  9. De Concini, C., Procesi, C.: Complete symmetric varieties. In: Invariant Theory, Montecatini, 1982. Volume 996 of Lecture Notes in Mathematics, pp. 1–44. Springer, Berlin (1983)

    Google Scholar 

  10. De Concini, C., Springer, T.A.: Betti numbers of complete symmetric varieties. In: Geometry Today, Rome, 1984. Volume 60 of Progress of Mathematics, pp. 87–107. Birkhäuser, Boston (1985)

    Google Scholar 

  11. De Concini, C., Goresky, M., MacPherson R., Procesi, C.: On the geometry of quadrics and their degenerations. Comment. Math. Helv. 63(3), 337–413 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kleiman, S.L., Laksov, D.: Schubert calculus. Am. Math. Mon. 79, 1061–1082 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  14. Laksov, D.: Completed quadrics and linear maps. In: Algebraic Geometry, Bowdoin, Brunswick, 1985. Volume 46 of Proceedings of Symposia in Pure Mathematics, pp. 371–387. American Mathematical Society, Providence (1987)

    Google Scholar 

  15. Littelmann, P., Procesi, C.: Equivariant cohomology of wonderful compactifications. In: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Paris, 1989. Volume 92 of Progress of Mathematics, pp. 219–262. Birkhäuser, Boston (1990)

    Google Scholar 

  16. Macdonald, I.G.: The Poincaré series of a Coxeter group. Math. Ann. 199, 161–174 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pezzini, G.: Lectures on spherical and wonderful varieites. In: Actions Hamiltoniennes: Invariants et Classification. No. 1 of Les cours du CIRM, vol. 1, pp. 33–53 (2010) http://www.cedram.org

  18. Renner, L.: An explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. Can. Math. Bull. 46(1), 140–148 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schubert, H.: Kalkül der abzählenden Geometrie. Springer (1979). Reprint of the 1879 original with an introduction by Steven L. Kleiman. Berlin, Heidelberg, NewYork

    Google Scholar 

  20. Semple, J.G.: On complete quadrics. J. Lond. Math. Soc. 23, 258–267 (1948)

    Article  MathSciNet  Google Scholar 

  21. Shimomura, N.: A theorem on the fixed point set of a unipotent transformation on the flag manifold. J. Math. Soc. Jpn. 32(1), 55–64 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  22. Spaltenstein, N.: The fixed point set of a unipotent transformation on the flag manifold. Nederl. Akad. Wetensch. Proc. Ser. A 38(5), 452–456 (1976)

    Article  MathSciNet  Google Scholar 

  23. Strickland, E.: Schubert-type cells for complete quadrics. Adv. Math. 62(3), 238–248 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Strickland, E.: Equivariant cohomology of the wonderful group compactification. J. Algebra 306(2), 610–621 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tyrrell, J.A.: Complete quadrics and collineations in S n . Mathematika 3, 69–79 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  26. Vainsencher, I.: Schubert calculus for complete quadrics. In: Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981). Progress of Mathematics, vol. 24, pp. 199–235. Birkhäuser, Boston (1982)

    Google Scholar 

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Acknowledgements

We thank Ersan Akyıldız and Lex Renner for many fruitful conversations on the group embedding problems. The first author is partially supported by a Louisiana Board of Regents Research and Development grant.

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Authors and Affiliations

  1. Tulane University, New Orleans, LA, USA

    Mahir Bilen Can & Michael Joyce

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  1. Mahir Bilen Can
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Correspondence to Mahir Bilen Can .

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Editors and Affiliations

  1. Department of Mathematics, Tulane University, New Orleans, Louisiana, USA

    Mahir Can

  2. Department of Mathematical Sciences, University of South Carolina, Aiken, South Carolina, USA

    Zhenheng Li

  3. Department of Mathematics, City College of New York, New York, New York, USA

    Benjamin Steinberg

  4. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Qiang Wang

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Can, M.B., Joyce, M. (2014). SL2-Regular Subvarieties of Complete Quadrics. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0938-4_12

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