Labelling Schubert intersections in the Grassmanian

TL;DR

This paper establishes a unified, elementary method for naturally labelling intersection points of Schubert varieties in the Grassmannian using combinatorial objects, confirming consistency across previous approaches.

Contribution

It demonstrates that different existing methods for labelling Schubert intersections with combinatorial objects are equivalent and provides an elementary description of this labelling.

Findings

Different methods produce the same labelling.

Elementary description of the labelling method.

Confirmation of consistency in real osculating flags.

Abstract

Points in the intersection of Schubert varieties are counted by various combinatorial objects, for example standard tableaux. This paper consider the problem of producing a natural labelling of intersection points by these combinatorial objects. When the Schubert varieties are being taken with respect to flags osculating at real points, several different methods have appeared implicitly in the literature (specifically in work of Mukhin-Tarasov-Varchenko, Speyer and Marcus). In this paper we show these various methods produce the same labelling and we describe it in an elementary way.