Invariants, torsion indices and oriented cohomology of complete flags
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, p. 405-448

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s ${K}_{0}$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁\left(G/B\right)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the torsion index of $G$ is inverted, $c$ is surjective and its kernel is generated by elements invariant under the Weyl group of $G$. As an application, we provide an algorithm to compute the ring structure of $𝗁\left(G/B\right)$ and to describe the classes of desingularized Schubert varieties and their products.

Soit $G$ un groupe algébrique linéaire semi-simple déployé sur un corps et soit $T$ un tore maximal déployé de $G$. Étant donnée une cohomologie orientée $𝗁$ (anneau de Chow, ${K}_{0}$ de Grothendieck, $K$-théorie connective, etc.) et sa loi de groupe formel $F$, nous construisons un anneau appelé anneau de groupe formel, associé à $F$ et aux caractères de $T$, puis un homomorphisme caractéristique $c$ de cet anneau de groupe formel vers l’anneau $𝗁\left(G/B\right)$$G/B$ est la variété des sous-groupes de Borel de $G$. Le résultat principal de cet article montre que, lorsque l’indice de torsion du groupe $G$ est inversé, $c$ est surjectif et son noyau est engendré par des éléments invariants sous l’action du groupe de Weyl de $G$. En guise d’application, nous fournissons un algorithme qui permet de calculer la structure d’anneau de $𝗁\left(G/B\right)$ et d’y calculer les classes de variétés de Schubert désingularisées et leur produits.

DOI : https://doi.org/10.24033/asens.2192
Classification:  14F43,  14L05,  14M15,  19L41,  20G10
Keywords: linear algebraic group, oriented cohomology, formal group law
@article{ASENS_2013_4_46_3_405_0,
author = {Calm\es, Baptiste and Petrov, Viktor and Zainoulline, Kirill},
title = {Invariants, torsion indices and oriented cohomology of complete flags},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 46},
number = {3},
year = {2013},
pages = {405-448},
doi = {10.24033/asens.2192},
language = {en},
url = {http://www.numdam.org/item/ASENS_2013_4_46_3_405_0}
}

Calmès, Baptiste; Petrov, Viktor; Zainoulline, Kirill. Invariants, torsion indices and oriented cohomology of complete flags. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, pp. 405-448. doi : 10.24033/asens.2192. http://www.numdam.org/item/ASENS_2013_4_46_3_405_0/`

[1] R. Bott & H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. | MR 105694

[2] N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, 1970; réédition Springer, 2007.

[3] N. Bourbaki, Éléments de mathématique. Algèbre commutative. Chapitres 1 à 4, Masson, 1985; réédition Springer, 2006. | MR 782296

[4] P. Bressler & S. Evens, The Schubert calculus, braid relations, and generalized cohomology, Trans. Amer. Math. Soc. 317 (1990), 799-811. | MR 968883

[5] P. Bressler & S. Evens, Schubert calculus in complex cobordism, Trans. Amer. Math. Soc. 331 (1992), 799-813. | MR 1044959

[6] B. Calmès & V. Petrov, Cohomology of Borel varieties, a Macaulay 2 package, http://www.math.uni-bielefeld.de/~bcalmes/M2packages/cohbovar.html, 2009.

[7] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301. | MR 342522

[8] M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. 7 (1974), 53-88. | MR 354697

[9] W. Fulton, Intersection theory, second éd., Ergebn. Math. Grenzg. 2, Springer, 1998. | MR 1644323

[10] D. R. Grayson & M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.

[11] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, 1977. | MR 463157

[12] M. J. Hopkins, N. J. Kuhn & D. C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553-594. | MR 1758754

[13] J. Hornbostel & V. Kiritchenko, Schubert calculus for algebraic cobordism, J. reine angew. Math. 656 (2011), 59-85. | MR 2818856

[14] M. Levine & F. Morel, Algebraic cobordism, Springer Monographs in Math., Springer, 2007. | MR 2286826

[15] A. Merkurjev, Algebraic oriented cohomology theories, in Algebraic number theory and algebraic geometry, Contemp. Math. 300, Amer. Math. Soc., 2002, 171-193. | MR 1936372

[16] I. Panin, Oriented cohomology theories of algebraic varieties, $K$-Theory 30 (2003), 265-314. | MR 2064242

[17] I. Panin, Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov), Homology, Homotopy Appl. 11 (2009), 349-405. | MR 2529164

[18] A. Preygel, Algebraic cobordism of varieties with G-bundles, preprint arXiv:1007.0224.

[19] T. A. Springer, Schubert varieties and generalizations, in Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad. Publ., 1998, 413-440. | MR 1653040

[20] B. Totaro, The torsion index of ${E}_{8}$ and other groups, Duke Math. J. 129 (2005), 219-248. | MR 2165542

[21] B. Totaro, The torsion index of the spin groups, Duke Math. J. 129 (2005), 249-290. | MR 2165543

[22] A. Vishik, Symmetric operations in algebraic cobordism, Adv. Math. 213 (2007), 489-552. | MR 2332601

[23] A. Vishik & N. Yagita, Algebraic cobordisms of a Pfister quadric, J. Lond. Math. Soc. 76 (2007), 586-604. | MR 2377113

[24] M. Willems, Cohomologie équivariante des tours de Bott et calcul de Schubert équivariant, J. Inst. Math. Jussieu 5 (2006), 125-159. | MR 2195948