Bumpless Pipe Dream Fragments -- Equivariant Geometry of Clans

Abstract: In this paper, we establish a new geometric setting for bumpless pipe dreams and double Schubert polynomials. We introduce the notion of bumpless pipe dream fragments, and 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.