Basic polynomial invariants, fundamental representations and the Chern class map (Q418687)

Let \(\Phi\) be a crystallographic root system of rank \(n\) with weight lattice \(\Lambda\) and Weyl group \(W\). Then \(W\) acts on \(\Lambda,\) hence on the integral group ring \(\mathbb{Z}\left[ \Lambda\right] \) and the symmetric algebra \(S^{\ast}\left( \Lambda\right).\) By a result of Chevalley, the ring of invariants \(\mathbb{Z}\left[ \Lambda\right] ^{W}\) is a polynomial ring over \(\mathbb{Z}\) in classes of fundamental representations \(\rho_{1} ,\dots,\rho_{n};\) also, \(S^{\ast}\left( \Lambda\right) \otimes\mathbb{Q}\) is a polynomial ring over \(\mathbb{Q}\) in basic polynomial invariants \(q_{1},\dots,q_{n}.\) There are augmentation maps \(\mathbb{Z}\left[ \Lambda\right] \rightarrow\mathbb{Z}\) and \(S\left( \Lambda\right) \rightarrow\mathbb{Z}\): the powers of the augmentation ideals \(I_{m}\) and \(I_{a}\) respectively allow for filtrations of both rings, and these \(I\)-adic filtrations respect the action of \(W\) on these rings. From the collection of ring isomorphisms \(\phi_{i}:\mathbb{Z}\left[ \Lambda\right] /I_{m} ^{i+1}\rightarrow S^{\ast}\left(\Lambda\right) /I_{a}^{i+1}\) the authors are able to find connections between the \(\rho_{i}\)'s and the \(q_{i}\)'s.NEWLINENEWLINESpecifically, let \(I_{m}^{W}\) (resp. \(I_{a}^{W}\)) be the ideal generated by \(W\)-invariant elements from \(I_{m}\) (resp. \(I_{a}\)). We say the action of \(W\) of \(\Lambda\) has finite exponent in degree \(i\) if there is a positive integer \(N_{i}\) such that \(N_{i}\left( I_{a}^{W}\right) ^{\left( i\right) } \subset\phi^{\left( i\right) }\left( I_{m}^{W}\right) ,\) where \(\phi^{\left( i\right) }\) is the composition \(\mathbb{Z}\left[ \Lambda\right] \rightarrow\mathbb{Z}\left[ \Lambda\right] /I_{m} ^{i+1}\rightarrow S^{\ast}\left( \Lambda\right) /I_{a}^{i+1}\rightarrow S^{i}\left( \Lambda\right).\) In such a case we denote the greatest common divisor of all such \(N_{i}\) by \(\tau_{i}.\) As \(\mathbb{Z}\left[ \Lambda\right] ^{W}\) is the representation ring of a Lie algebra \(\mathfrak{g},\) it is shown that \(\tau_{2}\) is the Dynkin index of \(\mathfrak{g},\) so \(\tau_{2}=1,2,6,12,\) or \(60\) depending on the type of \(\mathfrak{g}.\) Also, \(\tau_{3}\) and \(\tau_{4}\) both divide \(\tau_{2}\). Finally, if \(\Lambda\) is the group of characters of a maximal split torus of a split simple simply connected group over a field \(k,\) and \(X\) is the variety of Borel subgroups of \(G\) with Grothendieck filtration \(\left\{ \gamma ^{i}\left( X\right) \right\} ,\) then \(\tau_{i}\left( i-1\right) !\) annihilates the torsion of \(\gamma^{i}\left( X\right) /\gamma^{i+1}\left( X\right) \) for \(i=2\) or \(3;\) this result also holds for \(i=4\) unless \(G\) is of type \(B_{n}\) \(\left( n\geq3\right) \) or \(D_{n}\) \(\left( n\geq5\right) \) -- in these cases the torsion is annihilated by \(24\).