Action on Grassmannians associated with commutative semisimple algebras

Abstract:

Let $A$ be a finite-dimensional commutative semisimple algebra over a field $k$ and let $V$ be a finitely generated faithful $A$-module. We study the action of the general linear group ${\text {GL}}_A(V)$ on the set of all $k$-subspaces of $V$ and show that, if the field $k$ is infinite, there are infinitely many orbits as soon as $A$ has dimension at least four. If $A$ has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of ${\text {GL}}_A(V)$ on the poset of $k$-subspaces of $V$.