On Borel orbits of quadratic forms in characteristic 2

TL;DR

This paper studies the structure of Borel group orbits on quadratic forms over fields of characteristic 2, revealing connections to Catalan numbers and describing Weyl group actions.

Contribution

It characterizes Borel orbits on quadratic forms in characteristic 2 and links them to Catalan triangle numbers, providing explicit descriptions of Weyl group actions.

Findings

Borel orbits are classified and connected to Catalan triangle numbers.

Explicit description of Weyl group actions on orbit double covers.

The structure of quadratic forms in characteristic 2 is elucidated.

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