The Grothendieck Group of the Variety of Spanning Line Configurations

We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski–Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a localization statement in $K$-theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton–Lascoux degeneracy loci formula, we prove that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski–Rhoades varieties with Grothendieck polynomials associated to words $w\in [k{]}^n$. Motivated by this identification, we develop models of classical and bumpless pipe dreams for words. We show that Schubert and Grothendieck polynomials of words are monomial-weight generating functions for these pipe dreams, extending the classical story from permutations to words and ordered set partitions.