Bumpless Pipe Dream Fragments -- Equivariant Geometry of Clans

TL;DR

This paper introduces a new geometric framework linking bumpless pipe dreams, clan polynomials, and equivariant geometry of certain orbits, resolving an open problem in Schubert calculus.

Contribution

It defines clan polynomials from bumpless pipe dream fragments and connects them to the equivariant geometry of ($GL_p\times GL_q$)-orbit closures, solving a previously open problem.

Findings

Clan polynomials are the coefficients in equivariant Schubert expansions.

Clan polynomials arise naturally in the geometry of ($GL_p\times GL_q$)-orbits.

The work resolves an open problem by Wyser and Yong.

Abstract

In this paper, we establish a new geometric setting for bumpless pipe dreams and double Schubert polynomials. Building on the notion of bumpless pipe dream fragments, we define clan polynomials as their weight generating functions. It turns out that clan polynomials arise naturally in the equivariant geometry of ($GL_p\times GL_q$)-orbits over the flag variety $Fl_{p+q}$ parametrized by $(p,q)$-clans. Furthermore, we show that the coefficients in the equivariant Schubert expansion of the fundamental classes of ($GL_p\times GL_q$)-orbit closures are exactly clan polynomials, which resolves an open problem posed by Wyser and Yong.