Invariants, torsion indices and oriented cohomology of complete flags (Q2854291)

Let \(H\) be an algebraic cohomology theory endowed with Chern cases \(c_i\) such that for any two line bundles \({\mathcal L}_1\) and \({\mathcal L}_2\) over a variety \(X\) we have NEWLINE\[NEWLINEc_1({\mathcal L}_1\otimes{\mathcal L}_2)= c_1({\mathcal L}_1+ c_2({\mathcal L}_2).NEWLINE\]NEWLINE The basic example of such a theory is the Chow group of algebraic cycles modulo rational equivalence. This situation has been generalized by several authors to the case of an arbitrary oriented cohomology theory \({\mathbf h}\), i.e. such that \(c_1(({\mathcal L}_1\otimes{\mathcal L}_2)= F(c_1({\mathcal L}_1))+ F(c_2({\mathcal L}_2))\), where \(F\) is the formal group law associated to \({\mathbf h}\). Examples of such theories include algebraic \(K\)-theory, étale cohomology \(H^*_{\mathrm{et}}(-,\mu_n)\) for varieties over a field of characteristic prime to \(n\), connective \(K\)-theory and the algebraic cobordism \(\Omega\), which is the universal oriented cohomology theory.NEWLINENEWLINE In this paper, the authors consider the variety \(G/B\) of Borel subgroups with respect to a split maximal torus \(T\). Here \(G\) is a split semisimple linear algebraic group \(G\) over a field \(k\) and \(T\) a split maximal torus inside \(G\) contained in a Borel subgroup \(B\). \textit{M. Demazure} [Invent. Math. 21, 287--301 (1973; Zbl 0269.22010)] considered the cohomology ring \(H(G/B,\mathbb{Z})\) and provided an algorithm to compute it in terms of generators and relations, using the characteristic map \({\mathbf c}: S^*(M)\to H(G/B.\mathbb{Z})\), where \(S^*(M)\) is the symmetric algebra of the character group \(M\) of \(T\). Then purpose of this paper is to generalize Demazure's results to the case of an arbitrary oriented cohomology theory. They introduce a new combinatorial object, the formal group ring \(R[M]_F\), where \(R={\mathbf h}\) (point) is the coefficient ring. This formal group ring may be viewed as a substitute of the cohomology ring of the classyfying space \({\mathbf h}(BT)\) of \(T\). The characteristic map c turns into the map NEWLINE\[NEWLINE{\mathbf c}: R[M]_F\to{\mathcal H}(M)_F,\tag{1}NEWLINE\]NEWLINE where \({\mathcal H}(M)_F\) is a combinatorial substitute for the cohomology ring \({\mathbf h}(G/B;\mathbb{Z})\). Then the Weyl group \(W\) acts naturally on \(R[M]_F\). The main result is the followingNEWLINENEWLINE Theorem 1. If the torsion index of \(G\) is invariable in \(R\) and \(R\) has no 2-torsion, then the characteristic map in (1) is surjective and its kernel is generated by \(W\)-invariant elements in the augmentation ideal.NEWLINENEWLINE As an immediate application of their results the authors provide an efficient algorithm for computing the cohomology ring \({\mathcal H}(M)_F={\mathbf h}(G/B;\mathbb{Z})\). In particular, in the last section of the paper, the authors list the multiplication tables for rings \(\Omega^Z*(G/B)\), where \(G\) has rank 2.