ACTION OF GENERAL LINEAR-GROUPS ON GRASSMANNIANS

Let A be a finite dimensional commutative semisimple algebra over a field k and let V be a finitely generated A-module. In previous work the author examined the action of the general linear group GL(A)(V) on the Grassmannians of k-subspaces of V. The present paper examines the structure of the orbits in greater detail, in particular by working out the structure of the stabilizers in each of the cases when dim(k) A less-than-or-equal-to 3. From an algebraic point of view the most interesting situation occurs for A a cubic extension field of k.