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The Grothendieck Group of the Variety of Spanning Line Configurations

Michael Ruofan Zeng

TL;DR

The paper proves that the Grothendieck group of the spanning line configuration variety $X_{n,k}$ is canonically isomorphic to the generalized coinvariant algebra $R_{n,k}$, extending the classical $K_0(Fl_n)\cong R_n$ correspondence. It identifies the structure sheaves of Pawlowski–Rhoades strata $X_w$ with Grothendieck polynomials $\mathfrak{G}_w$ of words, and develops word-based classical and bumpless pipe dreams to realize Schubert and Grothendieck bases for $R_{n,k}$. The work integrates cellular localization in $K$-theory with Fulton–Lascoux degeneracy-loci formulas to derive the main isomorphism and to provide combinatorial models extending the permutation case to words. This yields a new bridge between geometry, $K$-theory, and pipe dream combinatorics for generalized coinvariant algebras and their bases.

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.

The Grothendieck Group of the Variety of Spanning Line Configurations

TL;DR

The paper proves that the Grothendieck group of the spanning line configuration variety is canonically isomorphic to the generalized coinvariant algebra , extending the classical correspondence. It identifies the structure sheaves of Pawlowski–Rhoades strata with Grothendieck polynomials of words, and develops word-based classical and bumpless pipe dreams to realize Schubert and Grothendieck bases for . The work integrates cellular localization in -theory with Fulton–Lascoux degeneracy-loci formulas to derive the main isomorphism and to provide combinatorial models extending the permutation case to words. This yields a new bridge between geometry, -theory, and pipe dream combinatorics for generalized coinvariant algebras and their bases.

Abstract

We study the Grothendieck group of the variety of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra . Our first result is a localization statement in -theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton--Lascoux degeneracy loci formula, we prove that is canonically isomorphic to , extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski--Rhoades varieties with Grothendieck polynomials associated to words . 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.
Paper Structure (21 sections, 28 theorems, 106 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 21 sections, 28 theorems, 106 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Algebro-geometric Prerequisites
  4. The Chow ring
  5. Chern classes in the Grothendieck group $K_0$
  6. Chern classes in the Chow ring
  7. Chern classes in cohomology
  8. Cellular varieties
  9. Schubert Polynomials and Grothendieck Polynomials
  10. Degeneracy Loci Formulae
  11. Classical and Bumpless pipe dreams
  12. The Variety of Spanning Line Configurations
  13. The Grothendieck Group of Spanning Line Configurations
  14. Key lemmas for cellular varieties
  15. The Grothendieck polynomial of a word
  16. ...and 6 more sections

Key Result

theorem 1

Suppose $X$ is a smooth variety over $\mathbb{F}$ that admits a cellular decomposition. Let $Z \subseteq X$ be a union of cell closures. Let $I(Z)$ be the ideal in $K_0(X)$ generated by the fundamental classes of cell closures $[\cO_{\overline{C}}]$ for all cells $C \subseteq Z$. The inclusion $Z

Figures (2)

  • Figure 1: Visualization of chute moves (left) and $K$-theoretic chute moves (right).
  • Figure 2: Visualization of droop moves (left) and $K$-theoretic droop moves (right).

Theorems & Definitions (53)

  • definition 1
  • theorem 1
  • theorem 2
  • definition 2
  • theorem 4
  • definition 3
  • theorem 5: Localization sequence, weibel2013k Application 6.4.2
  • theorem 6: Homotopy Invariance, weibel2013k Fundamental Theorem 6.5.1
  • theorem 7
  • proof
  • ...and 43 more