Grothendieck Group of Spanning Line Configurations and Generalized Coinvariant Algebras
Published:Dec 28, 2025 04:15
•1 min read
•ArXiv
Analysis
This paper explores the Grothendieck group of a specific variety ($X_{n,k}$) related to spanning line configurations, connecting it to the generalized coinvariant algebra ($R_{n,k}$). The key contribution is establishing an isomorphism between the K-theory of the variety and the algebra, extending classical results. Furthermore, the paper develops models of pipe dreams for words, linking Schubert and Grothendieck polynomials to these models, generalizing existing results from permutations to words. This work is significant for bridging algebraic geometry and combinatorics, providing new tools for studying these mathematical objects.
Key Takeaways
- •Establishes an isomorphism between the K-theory of the variety of spanning line configurations and the generalized coinvariant algebra.
- •Develops models of pipe dreams for words, extending the classical theory from permutations.
- •Connects Schubert and Grothendieck polynomials of words to monomial-weight generating functions for these pipe dreams.
Reference
“The paper proves that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety.”