Rational points and Coxeter group actions on the cohomology of toric varieties
[Points rationnels et action d’un groupe de Coxeter sur la cohomologie des variétés toriques]
Lehrer, Gustav I.
Annales de l'Institut Fourier, Tome 58 (2008), p. 671-688 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

Publié le : 2008-01-01
DOI : https://doi.org/10.5802/aif.2364
Classification:  14M25,  14F40,  14G05,  20G40,  14L30
Mots clés: variétés toriques, cohomologie, théorie de Hodge, points rationnels 
@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},
     volume = {58},
     year = {2008},
     pages = {671-688},
     doi = {10.5802/aif.2364},
     zbl = {1148.14026},
     mrnumber = {2410386},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 58 (2008) pp. 671-688. doi : 10.5802/aif.2364. http://gdmltest.u-ga.fr/item/AIF_2008__58_2_671_0/

[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, Hermann, Paris, Actualités Scientifiques et Industrielles, No. 1337 (1968) | MR 240238 | Zbl 0186.33001

[2] Carter, Roger W. Finite groups of Lie type, John Wiley & Sons Inc., New York, Pure and Applied Mathematics (New York) (1985) (Conjugacy classes and complex characters, A Wiley-Interscience Publication) | MR 794307 | Zbl 0567.20023

[3] Danilov, V. I. The geometry of toric varieties, Uspekhi Mat. Nauk, Tome 33 (1978), pp. 85-134 (English translation:Russian Math. Surveys 33 (1978), 97–154) | MR 495499 | Zbl 0425.14013

[4] De Concini, C.; Goresky, M.; Macpherson, R.; Procesi, C. On the geometry of quadrics and their degenerations, Comment. Math. Helv., Tome 63 (1988) no. 3, pp. 337-413 | Article | MR 960767 | Zbl 0693.14023

[5] De Concini, C.; Procesi, C. Cohomology of compactifications of algebraic groups, Duke Math. J., Tome 53 (1986) no. 3, pp. 585-594 | Article | MR 860662 | Zbl 0614.14013

[6] Dimca, A.; Lehrer, G. I. Purity and equivariant weight polynomials, Algebraic groups and Lie groups, Cambridge Univ. Press, Cambridge (Austral. Math. Soc. Lect. Ser.) Tome 9 (1997), pp. 161-181 | MR 1635679 | Zbl 0905.57022

[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, Tome 168 (1994) no. 3, pp. 741-772 | Article | MR 1293622 | Zbl 0813.14040

[8] Fulton, William Introduction to toric varieties, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 131 (1993) (The William H. Roever Lectures in Geometry) | MR 1234037 | Zbl 0813.14039

[9] Kisin, Mark; Lehrer, Gus I. Equivariant Poincaré polynomials and counting points over finite fields, J. Algebra, Tome 247 (2002) no. 2, pp. 435-451 | Article | MR 1877860 | Zbl 1039.14005

[10] Kisin, Mark; Lehrer, Gus I. Eigenvalues of Frobenius and Hodge numbers, Pure Appl. Math. Q., Tome 2 (2006) no. 2, pp. 497-518 | MR 2251478 | Zbl 1105.14026

[11] Lehrer, Gus I. The l-adic cohomology of hyperplane complements, Bull. London Math. Soc., Tome 24 (1992) no. 1, pp. 76-82 | Article | MR 1139062 | Zbl 0770.14013

[12] Lehrer, Gus I. Rational points and cohomology of discriminant varieties, Adv. Math., Tome 186 (2004) no. 1, pp. 229-250 | Article | MR 2065513 | Zbl 1077.14025

[13] Macmeikan, Chris The Poincaré polynomial of an mp arrangement, Proc. Amer. Math. Soc., Tome 132 (2004) no. 6, p. 1575-1580 (electronic) | Article | MR 2051116 | Zbl 1079.14026

[14] Procesi, C. The toric variety associated to Weyl chambers, Mots, Hermès, Paris (Lang. Raison. Calc.) (1990), pp. 153-161 | MR 1252661 | Zbl 1177.14090

[15] Stembridge, John R. Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math., Tome 106 (1994) no. 2, pp. 244-301 | Article | MR 1279220 | Zbl 0838.20050