Cauchy decompositions and invariants
Let \(k\) be an algebraically closed field. Given a pair of classical algebraic groups \((G_1,G_2)\) contained in \(\text{GL}_n\) and \(\text{GL}_m\) over \(k\) respectively, let \(X(G_1,G_2)\) be the affine variety of \(n\times m\)-matrices \(M\) such that \(M^tJ_{G_1}M=0\) and \(MJ_{G_2}M^t=0\), where \(J_{G_1}\) (resp. \(J_{G_2}\)) is the defining matrix for \(G_1\) (resp. \(G_2\)) in \(\text{GL}_n\) (resp. \(\text{GL}_m\)). \(X(G_1,G_2)\) is naturally a \(G_1\times G_2\) variety and its coordinate algebra \(A(G_1,G_2)\) is a graded algebra on which \(G_1\times G_2\) acts homogeneously. The question is whether \(A(G_1,G_2)\) has a good filtration as a \(G_1\times G_2\)-module (i.e., each homogeneous component has a good filtration), what are the factors. This is a generalization of Cauchy decomposition and the standard basis theorem for general linear groups to other classical groups. One of the main results is that, under certain restrictions on the appearance of a group of type \(D_n\) and the characteristic, \(A(G_1,G_2)\) has a good filtration with subquotients isomorphic to \(\nabla_{G_1}(\lambda^*)\otimes\nabla_{G_2}(\lambda)\), where the weights \(\lambda\) are described as partitions with number of parts at most \(\min(\text{rk}(G_1),\text{rk}(G_2))\) and \(\lambda^*=-w_0(\lambda)\) for the longest element \(w_0\) of the Weyl group element for \(G_1\). The second main result is an application of the first main result to the case \(G_2=\text{GL}_n\) and \(G_1\) arbitrary, to give a set of generators of the algebra \(A(G_1,G_2)^H\) of \(H\)-invariants with \(H\) being a subgroup of \(\text{GL}_n\) of the forms \(\text{SL}_n\), \(\text{Sp}_n\) (for \(n\) even), and \(\text{SO}_n\).
scientific article; zbMATH DE number 1559187