Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès; Viktor Petrov; Kirill Zainoulline

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 405-448
  • ISSN: 0012-9593

Abstract

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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 ( G / B ) 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 ( G / B ) and to describe the classes of desingularized Schubert varieties and their products.

How to cite

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Calmès, Baptiste, Petrov, Viktor, and Zainoulline, Kirill. "Invariants, torsion indices and oriented cohomology of complete flags." Annales scientifiques de l'École Normale Supérieure 46.3 (2013): 405-448. <http://eudml.org/doc/272231>.

@article{Calmès2013,
abstract = {Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $\mathsf \{h\}$ 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 $\mathsf \{h\}(G/B)$ 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 $\mathsf \{h\}(G/B)$ and to describe the classes of desingularized Schubert varieties and their products.},
author = {Calmès, Baptiste, Petrov, Viktor, Zainoulline, Kirill},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {linear algebraic group; oriented cohomology; formal group law},
language = {eng},
number = {3},
pages = {405-448},
publisher = {Société mathématique de France},
title = {Invariants, torsion indices and oriented cohomology of complete flags},
url = {http://eudml.org/doc/272231},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Calmès, Baptiste
AU - Petrov, Viktor
AU - Zainoulline, Kirill
TI - Invariants, torsion indices and oriented cohomology of complete flags
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 3
SP - 405
EP - 448
AB - Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $\mathsf {h}$ 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 $\mathsf {h}(G/B)$ 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 $\mathsf {h}(G/B)$ and to describe the classes of desingularized Schubert varieties and their products.
LA - eng
KW - linear algebraic group; oriented cohomology; formal group law
UR - http://eudml.org/doc/272231
ER -

References

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