Grothendieck polynomials and quiver formulas

TL;DR

This paper provides a combinatorial formula for coefficients in K-theoretic quiver formulas, revealing their alternating signs, and applies this to derive new expansions for Grothendieck polynomials.

Contribution

It introduces an explicit combinatorial formula for coefficients in K-theoretic quiver formulas, advancing understanding of Grothendieck polynomials.

Findings

Coefficients have alternating signs.

New expansions for Grothendieck polynomials.

Explicit combinatorial formula derived.

Abstract

Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs. Our result is applied to obtain new expansions for the Grothendieck polynomials of Lascoux and Sch\"utzenberger.