Action on flag varieties: 2-dimensional case
Let \(A\) be a finite dimensional commutative semisimple algebra over a field \(k\), and let \(V\) be a finitely generated \(A\)-module. We examine the action of the general linear group \(GL_ A(V)\) on the set of flags of \(k\)-subspaces of \(V\). Also, let \((V,B)\) be a finitely generated `symplectic module' over \(A\). We also investigate the action of the symplectic group \(Sp_ A(V,B)\) on the set of flags of \(B'\)-isotropic \(k\)-subspaces of \(V\), where \(B'=\varphi \circ B\) is the \(k\)-symplectic form induced by a nonzero \(k\)-linear map \(\varphi:A\to k\). Studied earlier were the cases of \(GL_ A(V)\) acting on the set of \(k\)-subspaces of \(V\) and of \(Sp_ A(V,B)\) acting on the set of \(B'\)-isotropic \(k\)- subspaces of \(V\). So this paper is a natural extension of the previous works. In both cases, the orbits are completely classified in terms of certain integer invariants provided that \(\dim_ k A=2\), from which one can determine the precise structure of orbits, compute the exact number of orbits and give typical representatives for each orbit, etc.
action of the general linear group on flags
scientific article; zbMATH DE number 166878