Rational points and Coxeter group actions on the cohomology of toric varieties
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 671-688.

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.

On donne une formule simple pour l’action d’un groupe de Coxeter fini crystallographique sur la cohomologie de la variété torique complexe associée. La méthode utilise la structure de Hodge sur la cohomologie pour relier le nombre des points rationnels sur un corps fini à cette action. On utilise la formule pour quelques applications, telles que la détermination de la multiplicité graduée de la représentation par réflexions dans la cohomologie.

DOI: 10.5802/aif.2364
Classification: 14M25, 14F40, 14G05, 20G40, 14L30
Keywords: Toric varieties, cohomology, Hodge theory, rational points
Mot clés : variétés toriques, cohomologie, théorie de Hodge, points rationnels
Lehrer, Gustav I. 1

1 University of Sydney School of Mathematics and Statistics N.S.W. 2006 (Australia)
@article{AIF_2008__58_2_671_0,
     author = {Lehrer, Gustav I.},
     title = {Rational points and {Coxeter} group actions on the cohomology of toric varieties},
     journal = {Annales de l'Institut Fourier},
     pages = {671--688},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     doi = {10.5802/aif.2364},
     zbl = {1148.14026},
     mrnumber = {2410386},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2364/}
}
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
SP  - 671
EP  - 688
VL  - 58
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2364/
DO  - 10.5802/aif.2364
LA  - en
ID  - AIF_2008__58_2_671_0
ER  - 
%0 Journal Article
%A Lehrer, Gustav I.
%T Rational points and Coxeter group actions on the cohomology of toric varieties
%J Annales de l'Institut Fourier
%D 2008
%P 671-688
%V 58
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2364/
%R 10.5802/aif.2364
%G en
%F AIF_2008__58_2_671_0
Lehrer, Gustav I. Rational points and Coxeter group actions on the cohomology of toric varieties. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 671-688. doi : 10.5802/aif.2364. http://archive.numdam.org/articles/10.5802/aif.2364/

[1] Bourbaki, N. Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 | MR | Zbl

[2] Carter, Roger W. Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1985 (Conjugacy classes and complex characters, A Wiley-Interscience Publication) | MR | Zbl

[3] Danilov, V. I. The geometry of toric varieties, Uspekhi Mat. Nauk, Volume 33 (1978), pp. 85-134 English translation:Russian Math. Surveys 33 (1978), 97–154 | MR | Zbl

[4] De Concini, C.; Goresky, M.; MacPherson, R.; Procesi, C. On the geometry of quadrics and their degenerations, Comment. Math. Helv., Volume 63 (1988) no. 3, pp. 337-413 | DOI | MR | Zbl

[5] De Concini, C.; Procesi, C. Cohomology of compactifications of algebraic groups, Duke Math. J., Volume 53 (1986) no. 3, pp. 585-594 | DOI | MR | Zbl

[6] Dimca, A.; Lehrer, G. I. Purity and equivariant weight polynomials, Algebraic groups and Lie groups (Austral. Math. Soc. Lect. Ser.), Volume 9, Cambridge Univ. Press, Cambridge, 1997, pp. 161-181 | MR | Zbl

[7] Dolgachev, Igor; Lunts, Valery A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra, Volume 168 (1994) no. 3, pp. 741-772 | DOI | MR | Zbl

[8] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993 (The William H. Roever Lectures in Geometry) | MR | Zbl

[9] Kisin, Mark; Lehrer, Gus I. Equivariant Poincaré polynomials and counting points over finite fields, J. Algebra, Volume 247 (2002) no. 2, pp. 435-451 | DOI | MR | Zbl

[10] Kisin, Mark; Lehrer, Gus I. Eigenvalues of Frobenius and Hodge numbers, Pure Appl. Math. Q., Volume 2 (2006) no. 2, pp. 497-518 | MR | Zbl

[11] Lehrer, Gus I. The l-adic cohomology of hyperplane complements, Bull. London Math. Soc., Volume 24 (1992) no. 1, pp. 76-82 | DOI | MR | Zbl

[12] Lehrer, Gus I. Rational points and cohomology of discriminant varieties, Adv. Math., Volume 186 (2004) no. 1, pp. 229-250 | DOI | MR | Zbl

[13] Macmeikan, Chris The Poincaré polynomial of an mp arrangement, Proc. Amer. Math. Soc., Volume 132 (2004) no. 6, p. 1575-1580 (electronic) | DOI | MR | Zbl

[14] Procesi, C. The toric variety associated to Weyl chambers, Mots (Lang. Raison. Calc.), Hermès, Paris, 1990, pp. 153-161 | MR | Zbl

[15] Stembridge, John R. Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math., Volume 106 (1994) no. 2, pp. 244-301 | DOI | MR | Zbl

Cited by Sources: