The Grothendieck Group of the Variety of Spanning Line Configurations

TL;DR

This paper establishes a geometric and algebraic connection between the Grothendieck group of spanning line configuration varieties and generalized coinvariant algebras, extending classical results to new combinatorial models.

Contribution

It proves an isomorphism between the Grothendieck group of these varieties and the generalized coinvariant algebra, and develops pipe dream models for words.

Findings

$K_0(X_{n,k})$ is isomorphic to $R_{n,k}$

Grothendieck polynomials are generating functions for pipe dreams

Extended classical permutation results to words and set partitions

Abstract

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.