The thickness of Schubert cells as incidence structures

TL;DR

This paper investigates the structure of Schubert cells in finite Chevalley groups, linking finite geometry and representation theory, and characterizes those with properties similar to ovoids, potentially aiding cross-disciplinary understanding.

Contribution

It provides a characterization of Schubert cells with ovoid-like properties in finite Chevalley groups, bridging finite geometry and representation theory.

Findings

Characterization of Schubert cells with ovoid-like properties

Bridge between finite geometry and representation theory

Potential new insights into ovoid structures

Abstract

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.