Notes on Plucker's relations in Geometric Algebra

TL;DR

This paper reinterprets Plucker's relations within Clifford's geometric algebra, providing a unified and simplified characterization that avoids complex traditional formalisms used in projective and algebraic geometry.

Contribution

It offers a novel geometric algebra-based characterization of Plucker relations, streamlining their understanding and application across related mathematical fields.

Findings

Plucker relations can be fully characterized using the geometric product.

The approach simplifies the understanding of Plucker relations in projective geometry.

The method unifies various mathematical formalisms into a single geometric algebra framework.

Abstract

Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Plucker relations in Clifford's geometric algebra. We discover that the Plucker Relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature.