Bumpless Pipe Dream Fragments -- Equivariant Geometry of Clans

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 ($G{L}_p\times G{L}_q$)-orbits over the flag variety $F{l}_{p+q}$ parametrized by $(p,q)$-clans. Furthermore, we show that the coefficients in the equivariant Schubert expansion of the fundamental classes of ($G{L}_p\times G{L}_q$)-orbit closures are exactly clan polynomials, which resolves an open problem posed by Wyser and Yong.