Affine polar spaces derived from symplectic spaces, their geometry and representations: alternating semiforms

TL;DR

This paper explores the structure and symmetries of affine polar spaces derived from symplectic spaces, introducing a generalized form called semiforms and analyzing their geometric properties and automorphisms.

Contribution

It introduces the concept of an \afsempol\, a generalized geometry from semiforms, and characterizes the automorphism groups of these spaces.

Findings

Properties of affine polar spaces from symplectic spaces are established.

Automorphism groups of these geometries are characterized.

Generalization from alternating forms to semiforms is demonstrated.

Abstract

Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an \afsempol\ arises as a result of generalization of the notion of an alternating form to a semiform. Some properties of these two geometries are given and their automorphism groups are characterized.