Diagonal Orbits in a Type A Double Flag Variety of Complexity One

TL;DR

This paper classifies the structure of diagonal $SL(n)$-orbit closures in certain flag varieties when the action has complexity one, revealing a finite set of posets and bounding the number of orbits.

Contribution

It explicitly determines 28 posets for complexity one cases and shows the maximum number of orbits is at most 10, advancing understanding of orbit structures in flag varieties.

Findings

28 explicitly determined posets for complexity one

Maximum of 10 orbits in these posets for any n

Contrast with complexity 0 case where posets can have arbitrary height

Abstract

We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some cases, the resulting posets attain arbitrary heights.