Weak Order on Complete Quadrics

TL;DR

This paper investigates the weak order structure on the variety of complete quadrics using monoid actions, explicitly determines maximal chains, and relates findings to cohomology classes in flag varieties.

Contribution

It introduces a novel approach using Richardson-Springer monoid actions to analyze the weak order on complete quadrics and explicitly finds maximal chains.

Findings

Maximal chains in the weak order are explicitly determined.

The study describes cohomology classes in the complete flag variety.

The approach links monoid actions to geometric and cohomological properties.

Abstract

Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.