Invariants, torsion indices and oriented cohomology of complete flags

TL;DR

This paper extends classical invariant theory to arbitrary oriented cohomology theories of flag varieties, providing a generalized characteristic map and an algorithm for computing their cohomology rings.

Contribution

It generalizes Demazure's work to formal group laws, constructing a characteristic map and an algorithm for cohomology ring computations of flag varieties.

Findings

Kernel of the characteristic map is generated by W-invariant elements.

Provides a Macaulay2 package for explicit computations.

Applicable to various cohomology theories like algebraic cobordism and K-theories.

Abstract

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.