Action on Grassmannians associated with a field extension

Abstract:

We examine the action of the general linear group ${\text {GL}}_L(V)$ on the set of all $K$-subspaces of $V$, where $L/K$ is a finite field extension and $V$ is a finite-dimensional vector space over $L$. The orbits are completely classified in the case of quadratic and cubic extensions; for infinite fields, the number of orbits is shown to be infinite if the degree of the extension is at least four. As an application we obtain $q$-analogues of tranformation and evaluation formulas for hypergeometric functions due to Gessel and Stanton.