Rational points and Coxeter group actions on the cohomology of toric varieties

Gustav I. Lehrer[1]

  • [1] University of Sydney School of Mathematics and Statistics N.S.W. 2006 (Australia)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 671-688
  • ISSN: 0373-0956

Abstract

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We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

How to cite

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Lehrer, Gustav I.. "Rational points and Coxeter group actions on the cohomology of toric varieties." Annales de l’institut Fourier 58.2 (2008): 671-688. <http://eudml.org/doc/10328>.

@article{Lehrer2008,
abstract = {We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.},
affiliation = {University of Sydney School of Mathematics and Statistics N.S.W. 2006 (Australia)},
author = {Lehrer, Gustav I.},
journal = {Annales de l’institut Fourier},
keywords = {Toric varieties; cohomology; Hodge theory; rational points; toric varieties},
language = {eng},
number = {2},
pages = {671-688},
publisher = {Association des Annales de l’institut Fourier},
title = {Rational points and Coxeter group actions on the cohomology of toric varieties},
url = {http://eudml.org/doc/10328},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Lehrer, Gustav I.
TI - Rational points and Coxeter group actions on the cohomology of toric varieties
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 671
EP - 688
AB - We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.
LA - eng
KW - Toric varieties; cohomology; Hodge theory; rational points; toric varieties
UR - http://eudml.org/doc/10328
ER -

References

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  2. Roger W. Carter, Finite groups of Lie type, (1985), John Wiley & Sons Inc., New York Zbl0567.20023MR794307
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  8. William Fulton, Introduction to toric varieties, 131 (1993), Princeton University Press, Princeton, NJ Zbl0813.14039MR1234037
  9. Mark Kisin, Gus I. Lehrer, Equivariant Poincaré polynomials and counting points over finite fields, J. Algebra 247 (2002), 435-451 Zbl1039.14005MR1877860
  10. Mark Kisin, Gus I. Lehrer, Eigenvalues of Frobenius and Hodge numbers, Pure Appl. Math. Q. 2 (2006), 497-518 Zbl1105.14026MR2251478
  11. Gus I. Lehrer, The l -adic cohomology of hyperplane complements, Bull. London Math. Soc. 24 (1992), 76-82 Zbl0770.14013MR1139062
  12. Gus I. Lehrer, Rational points and cohomology of discriminant varieties, Adv. Math. 186 (2004), 229-250 Zbl1077.14025MR2065513
  13. Chris Macmeikan, The Poincaré polynomial of an mp arrangement, Proc. Amer. Math. Soc. 132 (2004), 1575-1580 (electronic) Zbl1079.14026MR2051116
  14. C. Procesi, The toric variety associated to Weyl chambers, Mots (1990), 153-161, Hermès, Paris Zbl1177.14090MR1252661
  15. John R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. 106 (1994), 244-301 Zbl0838.20050MR1279220

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