On Borel orbits of quadratic forms in characteristic 2

Abstract

We consider the spherical variety of quadratic forms over a quadratically closed field of characteristic 2, and determine its orbits for the action of the Borel subgroup of upper triangular matrices. We exhibit a connection between these orbits and the Catalan triangle numbers. In addition, we describe explicitly a natural Weyl group action on the set of Borel orbit double covers