Baptiste Calmes, Victor Petrov, Kirill Zainoulline: Invariants, torsion indices and oriented cohomology of complete flags

Submission: 2010, Apr 7

In the present notes we generalize the classical work of Demazure [Invariants sym\'etriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let $G$ be a split semisemiple linear algebraic group over a field and let $T$ be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of $T$ with the cohomology of the variety of Borel subgroups of $G$. The main result of the paper says that the kernel of this map is generated by $W$-invariant elements, where $W$ is the Weyl group of $G$. As one of the applications we provide an algorithm (realized as a Macaulau2 package) which can be used to compute the ring structure of an oriented cohomology (algebraic cobordism, Morava $K$-theories, connective $K$-theory, Chow groups, $K_0$, etc.) of a complete flag variety.

2000 Mathematics Subject Classification: 20G10; 14F43; 14L30

Keywords and Phrases: torsion index, linear algebraic group, oriented cohomology, algebraic cobordism, projective homogeneous variety

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